"Mathematics was born from art."
Introduction
“Do not deal in passing. It is the privilege of fools. A wise man delights in fellowship and is grateful in rest.”
In this text, there is to be found the field of mathematics explained to its full and complete fundamental measure. The different terms which are to be housed herein will be the constituent components of the general rudimentations to ensue, and as the whole of the coming prose comes to form, the individual concepts which reside as integral to the field of mathematics will eventually grant a full and complete frame for all of that which is to be held in the generalist breadth of a comprehensive fielded analytic of mathematical knowledge and understanding. As the different chapter sections are written, the overarching terms that compose their body content will be defined, explained, and then variantly provisioned in the form of examples, so as to make clear what is to be articulated from their individuate terming and formulaic construction.
As a point of note for the whole of how it is that this text finds direct and implicate applicability within the fundamental philosophy of mathematics, the configuring principles of its underlying design are ones that stringently articulate the understanding and comprehension revolving around base arithmetic and its complimentary rudimentations. In this intercombinative marriage of foundational mathematics and science, there is found the integral dimensionality to the school and discipline of thought termed throughout this library's narrative as a fundamental philosophy, and it serves heavily in the light of how it is that general, complex, dynamic, and wealthful narratives of knowledgeable basis find placing within the folds of a powerful framework which is instrumentational to the aforewritten design and composite matrix of ensuing writ.
Numbers
Written out universally in mathematics as a collection of ten symbols, numbers are the figures which represent values of different measure in the field. There are many different ways that numbers can look, but for the intents and purposes of this doctoral library, there is a set sequence that lists as,
and 
.Seen elsewhere, they may look like 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, but the values hold to the exact same measure within the different configurations. Interestingly, the second set of curved numbers are termed as zero, one, two, three, four, five, six, seven, eight, and nine. The above list can be named as such the same, but the universally interpreted meaning from their value comes from the "amount" they are purposed with representing. Before correlating those amounts with their termed counterpart, the measure described above comes from the number of junctions present upon the figure. The first symbol in the set is designed as it is because it is to be equated to having no value. This is so because there are no junctions on the symbol. At no point along the symbol's axes or designs of measure is there a connection that results in an angle or "cornered area". As the list progresses in its design, the amount of angles increases by one for each nextive configuration in the sequence. Counting from the first symbol, each symbol afterward contains an angular configuration that increases by one each time. This is the case until the list arrives at the final value - nine.
As was written earlier, the ten symbols listed are the only numbers needed to compose any numeric configuration that would be seen in any field, anywhere, in any compositional dynamic. When compared to the list that follows, the numbers match by value - and value alone. They do not match by draft or design. Considering their terming, though, the two lists maintain in the way that they can each be named by the numbers written out in the word format seen after. From zero to nine, each number holds in the exact same value, but their design remains as different wherein the "linear" based figures are not drawn the same as the suppositionally "curved" ones. Numbers liken unto those described above are the seeded basis for mathematics, and with their writ, there is the capacity to engage mathematics from a universal position of communication and expression. Ten numbers grant all configurations, and the resultancy of that linear spectrum of reasoning flowers further with the whole of all other artful constructions present within the field being depictions of the philosophical fundamentals articulated from the interpretations had in the ten numbers described and listed previously. Zero, one, two, three, four, five, six, seven, eight, and nine are the terms which root this rudimentary principle, and from their construction, the fundamental philosophy of mathematics builds further the termed arrays written prior. This is, again, done with the rationale described earlier in this section, and it is also emphasized by the symbols which are to serve as this text’s universal fieldings.
and...are the first listing, and as a cross comparison, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 serve afterward - in writ - as analogous measures of value to be termed all in all as numbers.
Positive Numbers
Positive numbers are numbers that operate with a unique emphasis upon their meaning and value. When written out, they are tasked with indicating a number that does not take away value in any way, unless it is otherwise indicated. Positive numbers list in different valued configurations, but as a central point of interpretation a "(+)" symbol will be seen next to the number in order to indicate that it is a positive value. A positive number "9" will be written as "+ 9" or "(+) 9". When positioning that figure in the throes of a numerologically based mapping, the numbers, along a linear spectrum, would read as positive in one direction and then, as the countering dynamic, negative in the other direction. From a balanced position, positive numbers always operate along the lines of increase, and when taken a step further, regardless of their formulaic positioning, their signed value places their measured calculation in the universal configuration for addition. This is so for the potential equations rooted in additive and subtractive basis being incorporant of positive values along the axes described prior. In an addition equation, positive values would read as "(+) 8 + (+) 7 =..." wherein the answer is (+) 15. The furthering lesson in the above is found in the rudimentary expression of positive numbers arching into a one-directional flow of dynamic. When seen in negative equations, the positive signage does not change, but the dynamics of interpretational calculation do. An example equation can be written as "(+) 9 - (+) 5 =...". Positive nine minus positive five equals positive four, or (+) 4. In that particular instance, though, the negative symbol was used, but the result was one that arrived from the use of two positive figures.
Linear metrics aid in the interpretation that is to be extracted from that which is written prior by way of positioning the balancing dynamics of a line graph in step with how one would count out values in ordered sequence. Beginning at "0", or zero, the positive numbers, or values, of the graph would be directed in an infinite measure in one direction, and in another direction, the correlating negative numbers, or values, would be infinitely counted elseward. The two different directions on a line graph, or more so "number" line, would place "0" directly in the middle, and the other valued assignments of positive and negative at odds with each other. How it is that the numbers hold within the positive range, though, remains as the signaled emphasis, due to the different values maintaining in the way of universal measure from standard literary terming to static visual configurate. Reiterating the theoretical infinity present upon the line graph, or numbered line, moves the fundamental philosophies of the mathematics written so far unto the isolatudinous folds of an initiate linear matrix that denotes unique signing within standing knowledges. Positive signs are meant to articulate what is an upvalue, and as the prose of this textual narrative progresses, the concept of emphasized meaning will move further into the compounded dimensions of artful compositry which will make sense of the field's principled purpose of calculating along interpretable articulations of measured and measurable capacity. From where positive numbers place within this frame of quantifying numerological integrity, the whole of their purpose as upvalues allows for the dynamics described among the linear based numerals drafted in the first chapter to be analogously layered degrees which holistically articulate the importance of universal communication for the different elements that define, and make whole, mathematics and all of the fundamental philosophies found therein.
Negative Numbers
In the same way that positive numbers are explicit "upvalues", negative numbers are purposed with articulating "down" value quantants. They are the numbers which are to reside as opposite to positive values when calculating the different elements to an equation. Signed with the same symbol seen in subtraction, the values, when seen written in different formats, markedly signal the "taking away" from whatever established balance is present. Equations, which are the termed and defined phrases, denoted as formulas within fielded mathematics, follow a relatively linear modality of pathology when considering their suppositional calculation. When written out, their formulaic construction can be read to the effect of coding from literal interpretation. As an example, "(-) 5 + (-) 2 = (-) 7" is a route that is followed by different axes of spatial interpretation which grant the inevitable sum when followed correctly. Dissecting the equation written out above comes to start with knowing that one is going to move in a downward direction. The "(-)" at the start begins the conduction with the direction of down. The following question of "how far down?" arrives at the number next to the negative that is valued at "5". Remembering the linearly built angular numeric from earlier sees the measure of downshift counted at the same calculated matrix of junctions within the arrayed construction of the five numeric. The instructions in the equation then go on to say, "Move down", with the "(-)", or negative, symbol and then compound further with "Move down [(-)] five [5]", considering the additional five figure incorporated. From base rudimentations, the whole of the equation’s literal prose articulates to "Move down five and then add two more down movements to arrive at negative seven". An interwoven symbol bound matrix would place the aforewritten literations in the brace of "Move down [(-)] five [(5)] and then add [(+)] two more down movements [(-2)] to arrive [(=)] at negative seven [(-7)]". If the symbols from the brackets seen between the different wordings were to be lined up with the brackets taken away, the literally translated equation would read as, "(-)(5)(+)(-2)(=)(-7)". In order to make sense of it, the reorganization agreements of communicative linguistics seen throughout the above written equation's delineation would write back to something of the original equation reading to the effect of (-) 5 + (-) 2 = (-) 7, where the language that is present at the bedrock of mathematics' fundamental philosophy reveals itself to be a naturally spoken tongue. It remains, also, as integral to understand that negative numbers move in step as parted facilitants to that language by way of their symbolic design being a piece to the architectural puzzle of fielded quantification. By this notion, negative numbers are no different than their positive counterparts, save for the fundamental necessity of the negative symbol being the sole variant between the structural form of positive numbers and negative numbers. Situated knowledgeably in the above, then, there is a comprehensive extraction that builds further the mechanics of mathematics' fundamental philosophy (listed hereinsofar as the literate translatory principle of obeying different symbols' meanings in order to integrate and sustain the universality of their prose). In the case of the negative symbol, it is a narrative that indicates "down" or "less". That is what a negative symbol is. It is a down symbol. That down movement is one that can occur along linear or spatial interpretation, but it is a 'down' symbol regardless.
As the narrative of this text evolves and develops along these dynamics, the whole of what will come to be a fluently written manner of speech will embody the entirety of where the fundamentals of mathematics' philosophy will place onward. Fundamentalism is key. Philosophy is key, and all of that which will be defined in mathematics will follow in step along the same modes of interpretation in kind.
Addition (+)
Mathematics is capable of saying many things and with the different ways in which the infinite number of expressions find articulation, one can develop further enumerable compounded meanings purposed with making clear the myriad of messages and measures gifting of the above. Addition is one of the paths present in that which is written prior that is an analogously framed method of communication in accordance with not only the intervariant elements of mathematics' integrity matrix but also all of the universality embedded in its framework of modular design. Addition is defined as combining the different sums present in an equation, and in the same way that positive numbers signal upvalues and negative numbers signal downvalues, addition is the term used to indicate the action of putting values, or numbers, together. Certain equations require numbers, and those numbers, when framed within the context of additive mechanics, find their role in the purpose of summative fusion. A compounded dimension of delineation emphasizes the aforewritten phrasing of summative fusion by way of making clear and quantifying the linear components of gauge for the embedded systemitry drafted into listed numerals for this text. Read in listing as,
and.+=", or "4 + 4 = 8", where the four angles present in the rectangular figures count out to the eight junctions visually depicted on the symbol written after the equals sign. Writing out the analytical process of interpretation spans along the axes of interpretation as, "Four [from the first symbol written] plus [which is the sign for addition] four [returning to the mirrored count of four junctions described above] equals eight [arriving at the total value to be had from the equation written]”. A statement of initial value is given with the instruction to increase or compile its value by principles of summative complex, and following, an identical value is assigned marriage in that complex to arrive at the final count of all symbols included - eight. If one were to consider the "mathematical" pathology further, the explicit line of thinking pans as: A declaration of four; the instruction of increase; the degreed count of four provisioned for the aforewritten instruction of increase; and the final result of eight as the summative conclusion from all that was written. Fundamental understanding is what makes sense of the above and as the philosophies of its structure build further from linguistic knowledges, among other mediums of expression, the whole of what is to be based out of the ensuing mathematical litany of delineation will propagably find rightful variate by the fielded designs of the aforewritten science as well as the rudimentations of its larger, developmental throes of complexity.Subtraction (-)
Communicating addition's counterpart brings one to the concept of subtraction. It is the action, in mathematics, of taking away an amount from another amount. The most base of equations will usually read with a smaller amount being taken away from a larger amount, and in this arrangement, the result will always be a positive number, because the larger number, after having had the smaller number 'subtracted' from its value, will still reside with value afterward because the smaller amount does not completely deplete the larger one. Other instances, beyond this exchange, position negative numbers among the suppositional exchange described above wherein the resulting total can arrive at a positive one. This is also the case for positive values that have more subtracted from their measured amount than what they represent in terms of standard, interpreted value. A continuate dynamic can be had in the above with the difference of negative values being subtracted from positive values wherein the negative amount being subtracted would be, in essence, added by processes of mathematical principle. As a listing, all of the aforewritten examples can be shown as,
5 - 3 = 2, positive minus positive
3 - 5 = - 2, positive minus positive
5 - (-) 3 = 8, positive minus negative
3 - (-) 5 = 8, positive minus negative
- 5 - (-) 3 = - 2, negative minus negative
- 3 - (-) 5 = 2, negative minus negative
- 5 - 3 = - 8, negative minus positive
- 3 - 5 = - 8, negative minus positive
Each one varies by the two-sided dimensionality of the numbers being ordered in the linearity of either big to small or small to big. What follows after is the signing of either of the figures as negative or positive. Considering, then, the operation of subtraction, the results of the different equations follow along with what is analytically extracted from the communicative instructions seen in the 'route-based' coding had in the different formulas listed prior. There arrives in that delineation, then, the possibility of eight different base formulas with four different total possible solutions for that formulaic set. When all of the values place as equal to each other, the results change by slight dynamics. The below listing is an example,
4 - 4 = 0, positive minus positive
4 - (-) 4 = 8, positive minus negative
- 4 - 4 = - 8, negative minus positive
- 4 - (-) 4 = 0, negative minus negative
Positioning each element in the throes of a base literary translation frames a narrative where the directions point in the way of declaring a value with an assigned positive or negative configuration. That configuration then combines that number with another value of identical dichotomous form which results as a calculated sum. Declared value. Upward or downward movements. Declared value of dictated degreed movements. Result. The communicative dynamics of mathematics' literal components are wroughtly manifest in the defined and described narrative of subtraction outlined in the different aforelisted examples, and as a prominent build from the bases of fundamentality squarely placed throughout the whole of what would be the ongoing framework for mathematics, the intrinsic philosophies that succinctly articulate how it is that subtraction compounds that conversation come to operate as tandem correlates for all of the general conceptualizations relevant. A thorough reiterate branches into how it is that the spectrum of linguistic capacity allows for complex numerological and mathematical meticulations to properly develop wherein subtraction, and all of the preceding chapters, are where that complexity can be seen in accumulation, and the accompanying explanations, the same.
Multiplication
As an obviate of what is found among the inevitable complexities of mathematics, multiplication arises from the principled concept of compounded addition. Where more than one instance of operant addition is needed in an equation, multiplication houses its capacities there. As a mathematical equation, it is conducted by adding together several different amounts within a set, valued enumeration. For example, six multiplied by six equals thirty-six. The set value can be either prior written six, and the amounts to be added would sequence in the countering sixes. 'Six sixes' equals out to thirty-six by count of six (6), twelve (12), eighteen (18), twenty-four (24), thirty (30), and thirty-six (36). As each six fills one of the six different positions for value indicated by the elements of the equation, the sum arrives at the thirty-six written above. Six multiplied by six equals thirty-six. One of the developmental instruments that resides as an integral concept in the aforewritten process of multiplication is one that utilizes matrices as a propagating dynamic in the sumequate line of calculating analytics. If one were to write out the matrix of six multiplied by six, the array would visually amount to the number six placed at six different positions. Before they are combined, or 'summed up', they are rowed in a unilinear form as a sequence of six different values. If one were to add an additional six factor, the array would build from a line of sixes to a square composite of six figures. The grid would look like the visual below where each matrix embodies an arrayed representation of **, or 6 * 6 * 6.
Or
6 6 6 6 6 6
6 6 6 6 6 6
6 6 6 6 6 6
6 6 6 6 6 6
6 6 6 6 6 6
6 6 6 6 6 6
The answer, when counted out, is , or 216. A reiterate of the above houses the matrices seen in the tabled array as depictions of a fully articulated multiplication problem. Six multiplied by six multiplied by six equals out to two hundred and sixteen. As the factored variants increase, the number of elements dynamically increase in kind, until all of the equation charts to its appropriate value. Markedly unique in this delineation is the fact that no matter what the arrangement of factored multiples, as long as the assigned values are obeyed, the sum total will always amount to the exact same figure. Two multiplied by six is the same as six multiplied by two, and so on and so forth. The above literary example breaks down and communicates this principled capacity with open interpretation had for the other multiples that can be added to increase the sum and operant function of the mathematical conditions present. Increasing the figures listed in the equation may change the representative matrix, but it does not change the ending sum. Linguistic communicacy based in clarity seeds the writing of a multiplication equation as, "With the value x compose y counts of the x value inwhere the sum total amounts to z." Filling in the placeholding variables gives a working example for the prior outline. "With the value 
(9) compose (4) counts of the (9) value in where the sum total amounts to (36). Four different nine values, when combined, equals thirty-six. Emphasis on the representative symbolism of the numbers described earlier in this text makes sense of this dynamic, with the individual angles, or junctions, present within the forms of the numbers' structures being the depicted value that is to be interpreted. Nine junctions on the "nine" symbol counted to the array of four measures grants the sum of thirty-six. Following, then, with the fundamental philosophy that is buildantly embedded in this dynamic, one arrives at the wrought intrinsincies of furthered developmental conceptualization conducive to the aforewritten designs of mathematics' bedrock philosophical fundamentalism.
Division
Division is the process of breaking down numbers into smaller quantified constituents. When an operation including division is to be done, a dividend, the number to be broken down, is paired with a divisor, the factor by which the dividing is to be done. A division equation written out as (9) / (3) equals out to (3). The nine is the dividend and the divisor is three. The quotient is the ending sum which in this case - identical to the divisor - is three as well. Using linear matrices as an instrument of depiction, the equation written of above would be composed as, [ | | | ] [ | | | ] [ | | | ]. One tally mark for each junction on the nine numeric counts out to its purposed value, and grouping the marks into sets of three situates the implicate (9) dividend correctly among the triumvirate sets of three count tally marks. One tally mark for each junction on the nine numeral grants the value to be divided, and three even groups of tally marks serves as the represented divisor factor. With all of the tally marks grouped, the resulting collection of sets sums to be the quotient. When there is an amount left over, due to the divisor not evenly breaking apart all value in the dividend into equal sets and portions, it is called the 'remainder'. If the divisor from the earlier equation were to be changed to two, the problem would visually shift its grouping to four sets of two with the number one as the remainder. This change in formula would look like this, | [ | | ] [ | | ] [ | | ] [ | | ]. The nine tally marks would be sorted from 'three groups of three' to 'four groups of two' with 'one tally' as the remainder. The problems which maintain their solutions with remainders are written as,
, , , , , , , , , and .] amounts to a figure which can be framed along the capacitative dimensions of simply counting and breaking apart the whole of what is junctionally housed in any dividend-divisor relationship. Terming also the divisible of different dividends grants room for the interexchange of the aforewritten capacitative dimensions by which raw divisibility is measured. One way that this can be done is beginning with the count of angles present upon a figure and only sectioning in accordance with what can be evenly distributed - be it by odd or even measure. A (9) with nine angles, or junctions, has only the divisibles of (1), (3), and (9). That means that only those (3) numbers can be used to break down the number (9) without there being any remainder. A number can always be its own divisible - and the same holds for the number (1), as it is the only universal divisible - but all other variants are particular to not only the odd and even value of the dividend but also its general value. As a final rule, the order in division must be maintained in respect to the integrity of the calculated quotient. The sequence in which the numbers in a division equation are listed cannot be changed from their form, elsewise the quotient changes. The only exception is when the dividend and divisor are the same value.Long Division
As an operation, division is seen at its rudimentary basis among numbered counts that do not span past simple configurate dynamics. Addition, subtraction, and multiplication are the same in this light by all angles save for their operant functionality, but long division, the term of this section, is phrased as such because of the larger amounts that are conducted throughout its processes. In a long division problem, the calculating methods which produce the ending quotient stem past the isolatudinous step seen in basic division and go on to compose the representative complexity that is ascribed the title "long division" in mathematics. From wrought, fundamental articulation, if more than one level of divisory action is needed, the operation is to be considered long division. A division problem looks like this:
/ or 5/5
A long division problem looks like this:
,/ or 5,255/5
The difference between the two is the amount of dividing that is necessary for the problem to arrive at its quotient. For the first problem, the solution measures out to 1 and is conducted to its end with one step of division. For the second problem the processes of operation are conducted as follows,
In the above problem, the number of divisors that lead to the final solution stem from several sequences of division. Because there is more than one step, the problem is a long division problem. The literary verbiage also for the C and D bullets above emphasizes this defining element by way of embodying the concepts of developmental complexity naturally intrinsic in mathematics' design. The quotient in a long division problem is composed of constituent divisibles. Divisibles, again, are the factors which reside as viable for dividing solutions. A number is always divisible by itself, and as company, the number (1) maintains within division as the universal divisible. For the quotient written in the long division problem, several different divisibles compose the solution - all of which are results from the problem's divisor. The only change from this order of sequencing is when the suppositional "divisible matrix" leaves a remainder from the process of operations. The answer, then, termed again as the quotient, is one that utilizes factors from both fractioned and remaining values. Without all of the elements above, the different anatomical components of a division problem would not fit into a plausible, propagable framework of design. Returning to the numbers which compose the dimensions of the doctoral numerology of this text, the individual linearities that grant a clear and effective serial denotation for the universal values which are defined in a ten count narrative are ones that arrive at a representative inevitability that artistically expounds upon the purpose of mathematics as a fielded science. Each point, then, in the language developed from conceptualized terming so far, is one that moves the aforewritten agenda of understanding mathematics as a language forward into a raw built base of fundamentalist prose gifting of simple direction from developmental philosophical complexity. Symbolism is the root of the above, and branching further into the instructional analytics of methodological processes is where the wrought derivation of calculable principle resides in terms of how it is that the fundamental philosophies of mathematics are to be understood, interpreted, and articulated hereinsofar. Long division is a resonant embodiment of the above and is also a perpetuating mechanism of the integral functions to be termed and explained onward in the ongoing prose of discussion.
Fractions
Fractions are defined as the parted amounts of numeric figures that are understood within corresponding symbolism. A fraction is composed of two portions. There is an upper portion and a lower one. The upper portion is termed as the numerator, and the lower portion is termed as the denominator. When drawn out, a standard fraction design will position the numerator over a line that sits above the denominator, so that when viewed, the two components reside within their respective positions but are separated by a line. Fundamental extraction from the standard fraction structure described prior places the operation of division squarely within its purpose. A fraction is a division problem, but it does not always need to be solved. As an example 1/2 is a fraction that is equal to .5, but all it does to write it as such is equivalent to translating its representative form. 1/2 can also be written as 2/4 or 3/6, but the calculated mathematical value remains the same.
The value for fractions varies between partial amounts, whole numbers, and an intercombinative relation of the two. Written earlier was the example of 1/2, but there is also the possibility of a 2/2 which is equivalent to 1. Continuing on, 3/2 is a fraction that contains one whole number and also one half. As a translated figure, 3/2 can be written as 1.5 or 1 1/2. The one is the serial equivalent of 2/2 that can be taken from the 3/2, and the 1/2 value left over is the accompanying fraction.
Considering further the above mathematical concepts, the principles surrounding the value of fractions centers its base pillars of comprehension on the measured values ascribed to the numerator and denominator. When the numerator is larger than the denominator, the fraction represents a whole figure which can also be accompanied by an additional partial amount. Shifting the value of 3/2 to 6/2 changes the fraction's measured amount from 1 1/2 to 3. The increase of the numerator increases the value of the fraction. The denominator is the opposite. By increasing its value the fraction reduces in quantified measure. 1/2 is a larger amount than 1/4 even though 4 is a larger amount than 2. The amount by which the 1 is being divided is greater in the second fraction, so the resulting value of the fraction is loss. This rule is universally applicable to all fractions because of their principled construction being built on the operation of division.
Within this principled dimensionality, there exists, also, the concept of reciprocates. A reciprocate is the reverse version of a fraction. The reciprocate of 1/8 is 8/1, or just 8, inwhere the fraction's resulting reciprocate is a whole number because the numerator is written as the universal divisible, 1. When the denominator is able to be evenly distributed by the count of the numerator, regardless of the situant values present, the resulting reciprocate will always be a whole number. 2/4, when reciprocated, is equal to 2. 3/9 is equal to 3. 10/100 is equal to 10 and so on. All of the prior listed fractions equal out to a whole number when they are reciprocated, and this is, again, because the denominator is able to be completely broken down into a whole figure when switched with the numerator.
The ending principle in this section seats zero as the central point of discussion wherein the figure maintains as a universal solvent regardless of where it is placed within a fractional combinative. When zero is placed in the numerator, the value of the fraction is zero. When zero is placed in the denominator, the fraction equates to what can be termed as invalid. When zero is present in both the numerator and denominator, the value of the fraction is '1': [0/x = 0]; [x/0 = invalid]; and [0/0 = 1].
All of these principles are integral to the dynamics of fractions and found also among their wrought delineative brace is the developmental continuate of mathematics' fundamental philosophy wherein the narrative of this text progresses further to build upon a figure seen in the above written scribe - decimals.
Decimals
Just like fractions, decimals are purposed with articulating part of a whole or general figure. When written out they define part of the figure with numerals that span past a point mark. For example, in the same way that 1/2 is half of a whole number, .5 operates by liken dimensions. However, in the case of the decimal, the 5 present after the point mark is representative of a 5 count which translates to 1/2. This is done by way of the place where the 5 is seated being dubbed the 'tenths' place. A count of 5 tenths writes out to 5/10 which is 1/2 by simplified translation. From the tenths place one zero is added with each extension from the point mark - building development from the previous place. The tenths place is first and afterward comes the hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, billions, et cetera. Each mark, after the point, is one that follows the above path, until either a finity is established, a repeating figure results, or an infinity resides as the solution. No matter how large the number, though, the numerals of value follow the exact same code of linearity wherein the interpretation remains uniform and standard across all matters of implicate analysis. When incorporating the numbers that form before the point mark, their suppositional finity arises from the knowledge that only whole numbers can be placed in that location, and their infinity is maintained by the infinitesimality of the decimals spanning what is, by theory, an uncountable measure of places in the rightward direction. It is also that a whole number which is represented by infinity can be placed at the leftward position of the decimal wherein the entirety of the figure would be enraveled by a whole infinity. The infinitesimality of the decimals would no longer be applicable. The different mathematical operations involving decimals amount to principled relations built on the concepts outlined above. Addition, subtraction, multiplication and division are the four basic operations which are conducted in mathematics, and with decimals, the rules structurally maintain their brace and frame, but their functional course of operations is unique to their respective forms. When adding or subtracting decimals from one another, the point mark must align regardless of whether or not the numbers present in the figures don't. An example can be had in the following,
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Multiplication calls for the decimal to be moved leftward in the count of places that exist rightward for all figures involved in the equation. An example can be had in the following,
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For division the numbers operate by the same metric, but with dividing mechanics as opposed to multiplication. As an example, consider the following equations,
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The four does not need the decimals from the equation because it is a whole number. However, unlike multiplication the number of places used past the decimal can be any count as opposed to the limit set by the amount present in the relevant equation. As a finality, also, any sequence of zeroes seen past the decimal principally register as a set when there is an ending figure of substantive value. One thousand zeroes followed by a one after the decimal are examples of placeholders, and within fundamental philosophical instruction, the mathematics surrounding such resides as integral to all of its operant design.
Weight
The concept of weight comes from a myriad of theoretical positions, but the central theme in its prose is based in machining the principles surrounding the amount that something contributes to an environment in terms of its 'resistive' presence. A block of wood that maintains a certain weight would need to be overcome with a certain amount of force before it moves. Any naturally formed barriers present in unique ecosystems would morphologically present their structural compositry as such because the weight of their build would be a factor that has had an effect on environmental activity. Different animals assume different positions in the variant landscapes of the world, due to their selective biological dispositions granting way for their respective pathologies of life and the weight they amass in accordance with their own behaviors. Weight is an element that has an effect on the different forms which are measured in its science, and when considering the universality of the concept, all of nature falls in step with the phenomena associated with its fundamental necessity and subsequent capacity.
Weight, itself, is measured by different amounts, but the variantly termed unit based fieldings are meant to articulate the count of resistance a form uniformly gives within a set gravitational setting. Weight, by the aforewritten precepts of measure, is calculated through the relationship of mass and gravity. Mass is the amount of a form present in a certain space, and gravity is the force that acts upon that form to give it a degree of resistance in that space. Weight is equal to mass multiplied by gravity, and the dynamics of that relationship are compounded by the ongoing narrative of measure naturally incorporant in the factors which are seeded as constituent in the above. Weight changes when mass changes. Weight changes when gravity changes, and weight also changes when both gravity and mass change. For weight to change in relation to its mass would mean that the amount of the form being acted upon is either greater or lesser in comparison to its previous state. When the mass decreases, the weight decreases. When the mass increases, the weight increases. The relationship between the two is a positive correlation, so when one changes in relation to the other, the measured effect is one that sees an analogous shift in the corresponding mechanics and calculable dynamics. The exact same principled understanding rings true for the variable of gravity written in the prior articulated relationship. As gravity increases, weight increases. As gravity decreases, weight decreases, and going a step further sees the concept of weight pan out in unique principled dynamics. If one were to increase the mass and gravity present as variables in an equation of weight calculation, the measured amount would increase within the span of multiplicative arrays. Ratios of variant measure would maintain the proportional dynamics, but as an all-inclusive point of standardized fielding, the weight of a form would be multiplicatively impacted by either the lowering or highering of its quantifying values. As a final principle of delicate reciprocative dimensionality, the factor of gravity, by relatively analogous calculable measure, is influenced and even more so determined by the amount of form present in the measured and relevant gravitational field. The more mass there is localized to a region, the greater the gravity of that environment and mass, the same. This principle is a compounding notion to the multiplicative dynamics discussed above, and as a fundamental element to the whole of mathematics' base serialisms, it develops the growing concepts of its complexity and narrative in kind.
Mass
Mass is defined as the amount of space that a subject occupies. It is measured within general arrangements that range in standard intervals, but the purpose of the metric is to be able to determine how much of a particular substance is present in an area, or environment, and also the amount of that substance within that environment. Starting along the narrative of perceivable space, mass can be said to begin with grams - a standard unit of measurement. From there, different prefixes change to adjust the amount of mass being articulated across different arrangements of measure. From nanograms to kilograms, the base, or ‘root’, term is meant to set a form to the standardizations which make sense of the physical world by means of ordering and organizing scientific phenomena in the parted way of serially translating the different grades of occupied matter by scheme and recognizable signature. Mass is integral in mathematics, because it aids in fully reinforcing the principles seen in the natural conduction of any and all physical environments. As a mathematical concept, it is one which is not only necessary to measure and contextualize the metric of weight, it is also a base fielding that moves the whole of spatial interpretation by gauge of how it is that one would be able to fully compose different forms in proper accordance with not only their morphologies but physiologies the same. Different organic and inorganic chemical components are founded in the above wherein what would otherwise be without full and proper assignment is able to be stringently and strictly composed to a full gauge of identity and measure. The intermarriage of the different variables that then see mass carry out its role in step with other seeded fieldings integral to scientific study is one that bases its implications in alignment with not only all of that which resides as a natural derivation from existing phenomena (i.e. weight, volume, surface area, et cetera) but also a staple in rudimentary sciences that evolve and develop further how it is that physical, and by extension ‘non-physical’, environments can be studied, interpreted, and understood.
Arcs and Infinity Waves
Within mathematics, there is the fielded concept of arcs and waves which span between finite and infinite measure. The two do not differ by any real variance save for the structures which they are respectively termed to articulate. Arcs are rounded shapes which form part, and not all, of an enclosed linear curvature. Following along the curve of a circle would pose an example by way of sectioning the circle into thirds and only drawing out one-third of the circle. The arc would then be a linear curvature of one-third circular measure with a perimeter that would analogously be distanced at a one-third circle ratio. An arc is not a wave. A wave is an infinitesimal pattern which is composed of arcs. The arcs found in wave architecture are built around unique degrees of measure, and when considering their connection, the linearity which composes the continual patterns of flux is made sense of through shape and length-based design. Arcs, then, can be written as wave constituents, and within that composite matrix, different geometric orientations can be termed under base morphological constructions. From an initial position, those rudimentary morphologies span two different categorizations - circular and elliptical. There are, among those two, six different classifications of curvature which term under acute, obtuse, and equidistant. Circular arcs are pieces of a circular perimeter that fall under the above classifications by sorted means of a triumvirate sectioning. Acute circular arcs fall within the 1 degree to 179 degree range. Obtuse circular arcs fall within the 181 degree to 359 degree range, and equidistant circular arcs stand at a level 180 degrees. These different measures would visually articulate to a collection of arcs that would list in respective sequence as an arc which is less than half of a circle, an arc which is more than one half of a circle, and an arc that is a perfect half circle. Those different visualizations would represent an acute circular arc, an obtuse circular arc, and an equidistant arc. For the elliptical classifications, there would be a round which, in three different depictions, would measure short of a whole elliptical by way of a less than median round, a perfect median round, and a more than median round. As interpretable drafts, the six gauge sequence would look like the following,
Moving from left to right, the individual drawings above grant examples of what the individual wave constituents (arcs) would look like in their compositional patterning. Other forms are plausible the same, in that the symmetry would not be as prevalent in other abstract wave sequencings. Consider the following visuals as an example,
The equidistant classifications roughly hold the same, but the other four illustrations embody the dynamic variance present in the structural library artfully housed within the prose of the capacitative potentialities depicted prior. Between the two different visuals, symmetry, asymmetry, circularism, ellipticalism, acute angulature, equidistant angulature, and obtuse angulature, a fundamental philosophical buildancy of the mathematical conceptualizations described hereinsofar is found, as well as those which are to make sense of the full narrative to be found present throughout the ongoing text and its propagable conclusivity. Liken dimensions within the ensuing prose of accrual are designed to reinforce these rudimentary notions based in fielding script.
Points
Seen in part as integral dimensionalities present in the angular junctions of the numbered system defined in this text, points are denoted as measurable positions of spatial context which articulate a specific node of existence. A point meticulates the unique geometric rudimentations of a region that reside as its constituent matrix. It is, by theory, the "bit" or "piece" which operates as the finity of periodical inspection for any and all of time and space. When viewed or measured, by whatever interpretation that may be, points size their structure to the degree that appropriately grades the environment. Suppositionally, static matrices would be defined by static points. Dynamic matrices would be defined by dynamic points, and as the whole of what would otherwise be seen as solely theoretical mathematical plausibility propagates by the analogous dimensions outlined above, the entirety of what would stand as pointilist architecture develops further with the compounded complexity naturally inherent in the calculable throes of quantifiable fielding resulting in a subsequentiality of serial design.
Points are a part of the aforewritten conceptual configuration purely for the purpose of performing the role described. As each element of their frame is enunciated to the capacitative brace of fitting appropriate to the field of relevant housing, the conduction of pointilist form by constituent prose arrives at the grace of perpetual definition. No matter how large or small the field, the "points" which compose its intercombinative arrangement are responsible for making sense of the order found present in the relevant frame of scape. Moving onto quantancy, points take on the characteristic of measured value. A point that is followed by others to compose a series is one that constructs a line. The line that is made from those points is typically what builds an environment, and the space that houses all of those lines is one that gives greater context and scope to the purpose of points in geometric configuration. Points are seen in graphs, charts, models, designs, matrices, maps, angles, planes, grids, and even shapes. Situated at the different corners and axes of their interpretable form and function, points build the most minute of architectural givings from their capacity as general fixtures in composite structuralism. The distance of a relevant field's stretch and the size of its space are the two integral facets in the aforewritten gauge of geospatial construction, and as each component of the areatic narrative finds placing, the modular conformity seen typical in standardized terminology aids in granting a substantive interpretation of what is integral to mathematics as a fundamental conceptualization and philosophical proponent of universal linguistics. In terms of the language found naturally in mathematics, points grow from formulas that define not only spaces and regions but also specific collections of integrity that can only be defined by points and pointilist structure. Two examples that hold in fluent mathematical prose arrive at coordinates as an initial scenario and physics based photoscopics as a second frame of model. A point defined by coordinates can be composed by one, two, three, four, or more variables, but the purpose of reference is fulfilled by a variable, or coordinate, operating in reference to a larger field. For physics-based light mechanics, coordinates maintain as relevant, but other factors hold as integral as well. Consider, then, the following visuals as broad rudimentary articulations of what is written of throughout this section as well as those to come.
The three directional ray arrangement seen following shows a point at the center, and it is the standard depiction of points that are plotted within a grid. The point plotted in the depiction is termed as the 'origin' - which is written as (0,0). Considering the physics based example to the right of the three directional ray arrangement, the four main components to the equation outline what is occurring within the visual. The sphere, itself, is to be considered the point of interest, and the velocity of its movement is tracked with the measure of twenty-five centimeters per second. The volume is an approximate 33.51 centimeters cubed, and with the sphere's theoretical movement along the axis, the point it represents is one that is emphasized by its composite linear matrix being made apparent with interwoven linearity.
Lines
Lines are geometric constructions that articulate straight formations which function along the axis of different shapes, or 'structures'. Posing them by their theoretical formulation, if one were to draw a series of points into a matrix with one row and infinite columns or one column and infinite rows, then the depiction would be one of a line, or lines. Lines compose the most fundamental of mathematical metrics because they are integral structuralisms positioned within the shapes and functions of geometry. All of that which is seen in form, regardless of its design, is, in part, an element of linearity, and they are composed as such by the dimensions of their build. Within all of that which then goes on to build from the aforewritten fundamental, there is found the propagating purpose of seeding lines at the base of all calculations performed - outside of numerological serialism. Along with numbers, lines grow to orchestrate the constituent componentry of mathematics' myriadated fieldings that span into the aforewritten subject of geometry as well as calculus, trigonometry, physics and especially heavenly studies such as astronomy. Knowing not only the shape of different forms but also their direction of travel and the accompanying composite dimensionality is an element of design that is integral to all of any fielded theories, and from its grade of integrity, their is the holist prose of philosophically housing fundamentalism in accordance with the range of uses lines have. Shapes are composed of lines. Graphs are composed of lines. Grids are composed of lines, and near all structures drafted in mathematics have a common element with lines. From this commonality, fundamentalism resonates by all philosophies granting of its associative incorporance.
X-Axis
The x-axis, within any spatial context, is the plane of orientation that operates along a horizontal alignment. Archetypal depictions see this understanding as one which places horizontal orientations and horizontal axes within the framed dimensionality termed as the aforewritten x-axis. The different means by which it is interpreted span arrayed variations but all of them build from one standardized common element. As the different dimensions incorporated in spatial understanding and/or context grow in number, so too do the correlating complexities of the x-axis. Horizontal planarity can shift in position, which may not alter anything but perspective, but this is still enough to foster a sufficient differentiation granting of developmental axial mechanics. The machined variables built around interpreting the x-axis alone span no further than the above, though they maintain propagable applicability, and with all components of an intercombinative matrix present, the structuralisms that naturally make sense of the aforewritten spatial environments go on to be ones that construct the integral portions of any fielded dynamic. All of that which is visually interpreted from a base level of depiction moves in step with that which is written prior, and the fundamental evolutions, from a suppositional two-dimensional architecturalism, are those imparting of support regarding the soundly framed theories articulated thus far. All of that concerning horizontal orientation holds to the delineative bedrock of the above principled constructions wherein reiterate buildancy, beginning with only one dimension and beyond, places the x-axis as a continuate in step with how those terms find collective fielding.
Y-Axis
The y-axis is the vertically aligned plane present in any visual portrayal. Seen within variant degrees of longitudinality, it articulates the suppositional 'height' or 'depth' of any form of measure present along the relevant alignments of its dimensionality. The y-axis is usually seen as perpendicular to the x-axis wherein the four different angles created by their intersection all stand at 90 degrees. From the position of a perpendicular orientation, if the whole of a two-dimensional structure were to be rotated 90 degrees, then the respective x- and y- axes would switch in role wherein the reiterate of perpendicularity would be emphasized in its measurements by the above motion. The y-axis forms within one dimension as an isolatudinous plane, and also within two dimensions as an associative axis. The incorporation of a third dimension builds to contextualize interpretable space, and from that environment, further developmental progressions construct complex geometric arrangements that grow from the rudimentary principles of linear fundamentalism. Vertical axes seat their base structural narrative in the propagable knowledges associated with the whole of this spatial interpretation, and the axial arrangements that stem beyond the intrinsic integrity of the above may build further from the prose outlined prior. The full range of analysis is one that poises longitudinality at the founding level of meticulation granting of how one makes sense of orientation and axial dynamics. Machining within the prose of the y-axis' standardized terming, vertical sectioning is the finality for its form, and from its structure, there is resonant the necessary integrity of geometric contextualization.
Z-Axis
The z-axis is the axis of depth seen in three-dimensional orientations. It is one that extends forward and backward by theoretically infinitesimal arrays, and is responsible for giving form to two-dimensional depictions. Spatial articulations are rooted in the above because by theory, there is no 'real' one-dimensional plane, or two dimensional plane, even, in existence due to the measure of depth having to be present. Drawing on a canvas pad or designing a digital matrix are both wroughtly housed in a two-dimensional environment, but regardless of the two axes that would intrinsically be situated in the above two examples, the digital microns and fibrously parted nanometers that give the screen and pad form, respectively, are proof that the interpretations, embedded at the very least in physical science, are ones dependent on depth as a constituent. The two examples only propagate within the dimensions of tangible contextualization, though, so when going further and incorporating the intangible environments where spatial interaction still manifests, depth holds there, the same, to some degree of integrity, as it is still resonantly foundational in physical settings. Depth is rightly termed as integral for this reason, and the standardizations of its incorporant dynamics are built upon the above with natural developmental mechanics being built on the general narrative of geometric structuralism. The machined commonality based in any rudimentation regarding structure is framed within all formulations inclusive of the above, and in the terming of their fundamental necessity, the different rates of their dimensionality move by correlatory grade.
Coordinates
Coordinates are the combinative constituents which articulate the different spatial positions seen in planar dynamics. For each numbered or labeled fixture present within a fielded arrangement, the corresponding coordinate combination composes itself within the traces of a correlating lineature. One plane, or dimension, demands one coordinate. Two planes, or dimensions, demands two coordinates. Three planes, or dimensions, demands three coordinates, and so on and so forth. From dimensional contextualization, coordinates serve as the instrumentation of spatial interpretation which reinforce the integrity of grid structure by serially sequencing all of the components that are housed within its form. An example set holds in the following listing,
One-dimensional coordinates, (5)
Two-dimensional coordinates, (5,4)
Three-dimensional coordinates, (5,4,2)
The parentheses surrounding the numbered sequences are purposed with isolating the coordinates placed within their grouping, and the sequence within which the numbers are listed correspond to the axes upon which the different values are found. Starting from the left and moving right, the values term as the x-, y-, and z-axis. The one-dimensional coordinate houses only one value, so it can be ascribed any of the axes written above, but the other two are typically seen in standardized terming as (X,Y) and (X,Y,Z). Novelly, it may sometimes hap that the one- and two-dimensional coordinates are indicative of planes that don't correspond with the standardized axial listing, but the values still hold to the same light. The '(5)' could be along the z-axis, and the '(5,4)' could be along the y- and z-axes, respectively. Coordinates are integral for this reason, and the role they perform within planar analytics operates in kind as resonantly fundamental.
Angles
Angles are the geometric constructions seen in mathematics that articulate cornered spatialities. Generally, they measure within the circumference of 360 degrees, but given the dynamics of calculus, and the corresponding disciplines in mathematics, an angle can measure outside of what is typically seen in a circular patterning of orientation. As a standard depiction, it can be visualized as two straight lines meeting to form a junction that is enclosed, or meted at a point, and completely open to the degree of angulature that corresponds with the joining lines. As a worded example, if one were to see an angle that is one-quarter the measure of a circle, the degreed measure of that angle would be 90. The angle would be 90 degrees. There are also different measurement classifications for angles that fall within three categories. Those categories are acute angles, right angles, and obtuse angles. Acute angles are angles that measure between 1 and 89 degrees. Right angles are angles that measure at 90 degrees, and obtuse angles are angles that measure between 91 and 179 degrees. The different spatial orientations where angles find their relevance are situated within the three different classifications above, and moving from their categorization, there is a means by which to see where it is that they place within geometric configurations which compose larger mathematical fieldings. Grids, planes and other spatial constructions all demand the use of angles to make sense of their forms, and as the terms used to interpret their variant shapes compound to build the base fundamentals of different spatial articulations, angles naturally gravitate into the implicate picture as structurally integral dimensions to the enumerated arrangements that are standardized for terming.
Planes
Planes are dimensional constructions that compose spatially articulable scapes of existence. They can be contextualized to the different axes seen in standardized terming, but their central purpose in being termed is to signify a place of spatial interpriality. The x-axis, as an example, would be built as an accompanying construct to its respective plane by way of serving as its linear measure. The plane, itself, would be an infinitesimally expanding space that would, by theory, reach as far as the observing body would measure, and even further, wherein its structural dimensionality would be theoretically defined by its perpetually reaching spatial expanse. The architectural composition of this suppositional x-axis plane would, then, be one that is defined as an ever-expanding, ever-reaching, ever-’growing’ space which is only able to be defined by constraints that operate within finite contexts. The concept holds true for all other axes that may be termed as well as those which may not be standardized but still hold to the geometric organization that houses their standardized dimensionality. One of the most interesting facts about this particular mathematical concept is that they can exist on any level of measure. Planes can be large or small, but as elements of geometry, calculus, physics, and other fieldings seen in the science of mathematics, planes reside as integral because of their composite architecturalism within the infinitesimal matrices described above. Theoretically, there are two different ways to define any fundamental philosophical concept throughout the whole of existence. They either articulate space or time. Planes do space.
Dimensions
Dimensions are uniquely tied to planes, because of their intrinsic compositry being built on the spatial expanses that are, in part, defined by planes. As a difference though, planes find relegation to the restricting constraint of a suppositionally infinitesimal stretch of space that expands to different ratios and measures, and dimensions, though they can be modeled in that light, maintain to the capacity of mirroring more abstract constructions. Dimensions are interestingly reflected between the aforewritten definition and the theoretically complex structuralism of a mathematically unique ‘plane’ of existence. Within wrought mathematical interpretation, they are usually purposed with classically articulating the planes that define the spatial constructions oriented around the x-, y-, and z- axes, but they can also be an intercombinative array of irregularly shaped formations that contextually compose dynamic frameworks. Rudimentary interpretations seed this concept with the intercombinative matrices that can result from different angular composites which articulate respectively unique forms, but more complex notions seed further with novel metrics that span an infinitesimal number of possibilities.
Dimensions, interestingly enough, are also only capable of defining space. In no way can they define time. Because time is a distortion of space, the space that is distorted in order for time to be something which can be measured must first be contextualized within some dimensional dynamic. Time, then, can take place within a dimension, but it is the actual movement of that dimension that articulates, or ‘gifts’, time. Spatial understanding then falls upon dimensional interpretation as an integral fundamental for mathematics and the philosophies found present in its prose.
Two-Dimensionality
Two-dimensionality is the concept of two different spatial environments interacting at a discernable level of intercombinative relation. Within a standardized mathematical setting, two dimensions will usually be seen as an x- and y- axis. With the x-axis being visually represented by a horizontal orientation and the y-axis being represented with a vertical orientation, the two would connect at a point of perpendicular arrangement. These two axes would be the aforewritten two-dimensional construction indicative of a planarity of two-dimensionality. Along with the above arrangement, there are also two-dimensional configurations that seat the intercombinative connection of a suppositionally x- and y- axis outside of a standardized fielding. The usual four fold 90 degree quartering can be had in acute-obtuse sectionings, the same, wherein the origin (or the point of initial measure for the entire dimensional scape) is surrounded by variant degrees of angulature that still measure out to a complete 360 degrees. Dimensionality is powerful for this reason, because as a constituent of spatial description, it houses the environment where furthering componentry is able to not only be positioned but articulated in full and complete form. Two-dimensionality is an example of that articulation, because its forms are relegated to no other restraint than there being two different spatial environments. One dimension houses one form of uniquely identifiable space, and another dimension houses a second kind of uniquely identifiable space. Two dimensions are built on this dynamic, and all of the space that they articulate falls within the above as an interpretation granting of the aforetermed spatial structuralism.
Three-Dimensionality
Three dimensionality is seen within standardized geometric depictions as an intercombinative matrix that is made further sense of with the x-, y-, and z- axes listings. These three terms articulate the latitudinal, longitudinal, and depth wise orientation, respectively, wherein their forms aid in describing the unique spatial environment that is a three-dimensional field. Three-dimensionality is the ascribed descriptor of all fields that maintain the aforewritten construction, and as a geometrically unique structural dynamic, its governing architecture is compounded by the capacity to exist outside of ‘standard depictions’. Two dimensions are usually seen in perpendicular arrangements, so with three dimensions, the additive axis of depth is one that merely builds further the equitably based structure. Each individual axis within this structure is then associated with a plane that spans within whatever the dimensions of the relevant expanse may be. Three dimensions, though, articulates three expanses, and each one aligns with the plane wherein the dimension is housed and the axis is ‘lined’. If one were to view a three dimensional structure, it would be no feat outside of considering the physical world in which we, people, live. The two-dimensions that are seen in initial construction would be liken unto opening one’s eyes in the morning, and the depth would become apparent as the light of the surrounding environment would warm the occipital anatomy. As a movie moving to motion that is catalyzed by natural constituents, the intrinsically built three-dimensions of the implicate spatial context would be made sense of within one’s own position of relevance. Existence is, theoretically, the suppositionally ‘first’ three-dimensional environment that one may truly comprehend within a setting that is immediately and perpetually relevant, and because of this reason, three-dimensionality holds along its axial depictions to the same gradated effect that may be enunciated at different periods of its interpretation and articulation - however it is that, that is experienced within its narrative of prose.
Two Dimensional Coordinates
As a natural element of two-dimensional spatial environments, positioning seeds the first measures of how it is that one would engage the particular constituents of a standard two plane arrangement. The positions that are found within two-dimensional fields conduct their role under the operative term of ‘coordinates’. Coordinates are the literal point at which a specific region exists, wherein their mathematical articulation comes to reside as an integral component in making sense of different spatial environments within the dynamics of interpretation and analysis. Two-dimensional coordinates are made up of numerals that are grouped in twos - with one numeral being the x-axis measure, and the other being the y-axis measure. Written out as an example, they would read as (0,0), (3,4), (-3,4), (3,-4), and (-3,-4). The first coordinate is an example of what an ‘origin’ coordinate would look like for a two-dimensional grid. The two different positions are written within an (x,y) format, insothat the first coordinate - before the comma - indicates where the first position is to be measured along the x-axis, and the second coordinate indicates the y-axis measure. The coordinates that are then listed after vary in accordance to their negative and positive signage - with that signage being the indication of whether or not the axis is to be measured on its positive or negative side. A positive x-axis measure exists on the right hand side of its origin, and a negative x-axis measure exists on the left hand side of its origin. A positive y-axis measure exists above its origin, and a negative y-axis measure exists below its origin. Two-dimensional coordinates are wroughtly framed within the above aforewritten parameters as fundamental elements to be interpreted exactly as one reads, and with their terming, the mathematics to be propagated herein continues as a narrative of rudimentary design and prose.
Three Dimensional Coordinates
Three-dimensional coordinate arrangements are written in the exact same format as two-dimensional coordinates, but there is a third coordinate that is added to its structure that articulates the z-axis - the axis of depth. A three-dimensional coordinate then sees its form listed in the order of the latitudinal, longitudinal, and depthwise axes. By lettering, it would read as (x,y,z). With numbered examples, the different coordinates could be written as (0,0,0), (2,3,4), (-2,-3,-4), or (2,3,-4). The four different examples articulate four different concepts within ‘coordinate science’ that help in holistically contextualizing what is found naturally present in the different interpretations that arise from its working. The first coordinate - (0,0,0) - articulates the origin of a three-dimensional coordinate system. It is the centering focus of the entire three-dimensional complex about which the whole of the coordinate matrix makes sense. From the origin focal point, the entirety of the coordinate matrix operates along the same metrics seen in two-dimensional coordination. A positive figure places a latitudinal coordinate in rightward measures, a longitudinal coordinate in upward measures, and a depth wise coordinate in forward measures. A negative figure reverses the above dynamics, in that the x-axis based coordinates would be seen along its leftward portions, the y-axis based coordinates would be seen along its downward portions, and the z-axis based coordinates would be seen along its rearward portions. Each axis, within this delineation, still articulates the standard three-planed environment held within typical mathematical fielding, and from their writing, the three-dimensional coordinate system facilitates how it is that one would interact and make sense of a three-dimensional system.
Grids
Grids are structures seen in mathematics that compose arrays which are often counted within standardized contexts, so as to give form to unique spatial environments. Grids can be composed of any number of dimensions, but away from what would be counted as a naturally intrinsic element, there is the capacity of their variant forms to be architecturally comprised of a multitudinous plethora of shapes. Usually, grids will be made up of quadrilateral constructions, namely squares, that will, again, be intercombinatively formed as an array - by whatever dimensional measure that array may be. However, grids can also be made up of circles, or diamonds, and even much more complex geometric structures like pentagons, heptagons, and nonagons. Each of these shapes articulate a fielding that is not conducive to a wroughtly ‘square’ or ‘even’ distribution - which is where grids are often situated - but they are able to orchestrate the dynamics naturally intrinsic to grid structure all the same. Grids, at their fundamental philosophical base, are arrays, but they are arrays that are purposed with articulating a spatial environment. That environment is one that requires some element of measure, elsewise it would not be able to be understood in the light that it may demand wherein the pretenses of an array arise as an integral underpinning to the more inclusive objective purposed with being of greater description. Grids are then succinctly placed within mathematical designs and fieldings as measurable arrays which aid in articulating spatial environments in whatever form they may exist.
Two Dimensional Grids
Two-dimensional grids are two-dimensional arrays that field two axes of interpretation. Within standard depictions, the two axes will be the x-axis and the y-axis, but there are other ways in which a two-dimensional grid can be oriented. However, within base articulation, a two-dimensional grid is a complex that arranges its form about two junctioned axes. The perpendicular origin would be one that houses not only the two-dimensional coordinate (0,0) but also the intersection variation which gives birth to the larger structural form which is the grid, itself. Reconstructive processes would then seat the aforewritten delineation within the contextualized dynamic of a two-axial symmetry giving of a discernable and interpretable fielding of design. Two different planes, within a general basis, compose that design, and from those two suppositionally infinitesimal rootings of coordination, there arises the corresponding metric of measure that is squarely associated within typical grid metrics. A grid, again, at its fundamental philosophical basis is a measured array, and that array is one that composes the dimensions to the larger geometric fielding that is defined by the grid’s architecture. Engineering the different components to its geometric form falls upon the metric with which it is measured in order to make sense of its prose wherein the whole of the construct comes to be one that defines a wroughtly applicable spatial interpretation that otherwise would not be possible if the intercombinative elements of structural dimensionality were not found termed within its narrative. The linearity that then goes along with this descriptive compositry aids further, as the rudimentary scape for what is fittingly forded in the spatial environment housed aimly within the above infound dichotomous grid-dimension framework emphasizes its architectural givings by compounding prose.
Three Dimensional Grids
Three-dimensional grids are measured arrays which are formed from the x-, y-, and z- axes. The perpendicularity that typically constitutes the junction of the x- and y- axes is not necessarily mirrored by the z-axis, but the concept of symmetry reigns wroughtly within the dynamics of a three-dimensional mathematical construction. With the above understanding surrounding two-dimensional grids constituting how it is that they are built as such with the embedded componentry of measure at their base, a select change incorporant of depth presents the only difference between a two-dimensional environment and a three-dimensional environment. The two of them both define a spatial environment, and the environment is one that is denoted by either two planar dimensions or three. Considering a strict three-dimensional association, though, there is the capacity to consistently attribute a firm standardization of symmetry unto its prose, but the previously discussed concept of a dynamic spatial environment remains statically applicable in the light of their being an asymmetrical arrangement to the arrayed spatial environment that is measured within the context of a measured three-dimensional scape. A grid is measured by discernable metrics, and a three-dimensional environment presents the expanse wherein those metrics are to be placed by whatever dimensionality that is found as appropriate for its reach. Symmetry and asymmetry are uniquely applicable in this light, because of their respective geometric configurations presenting frameworks that are classified along different spectrums. The whole of what is to be understood, or mayhaps, gauged from this is that the different mathematical arrangements often placed within standardized fieldings of recognizable array are ones that hold to variant degrees wherein the infinitesimality wroughtly associated with mathematics finds furthered arraignment.
Shapes
Shapes are seen markedly relevant in geometry and are defined as enclosed spaces that find further articulation in contextualized linear environments. Shapes are built from different forms, but at their bared core, there are a select few that initiate the first givings into the natural, developmental complexity associated with their structural composition. They list as triangles, squares, rectangles, rhombi, parallelograms, pentagons, hexagons, heptagons, octagons, nonagons, and decagons. Triangles have three sides. Squares, rectangles, rhombi, and parallelograms have four sides. Pentagons have five sides. Hexagons have six sides. Heptagons have seven sides. Octagons have eight sides. Nonagons have nine sides, and decagons have ten sides. There are other shapes that span into more dynamic portrayals and arrangements, but the general grouping of most shapes comes in the form of the above listing within the variance existing where they are either combined or parted to compose other shapes and forms. Shapes are unique within mathematics, specifically geometry, for this reason. They are termed according to their structure, and however it is that, that structure frames its composition is how it finds standardization within both mathematics and geometry. The arrayed matrix of shapes that then spans across a countless spectra of compounded and complex arrangements is one that houses all of the potentialities for general shape depiction constituent of all visuals seen in mathematics.
The principled dynamic that propagates shapes by their innately continued analytical summatance goes on to seed all of form defined within mathematics as a pretensive denotation tasked with defining space in whatever form it may exist. Shapes define space, and as a markedly interesting note, shapes can be used to define the distorting of space, too, wherein shapes can define time as well.
Regular Shapes
Regular shapes are shapes that are defined within the metrics of a symmetrical and denotable context. Regular shapes list to the termed constructions outlined above in the 'shapes' section, and for each of those termed structures, there is the ascribed classification of their symmetrical forms residing as 'regular' shapes. Triangles, squares, rectangles, rhombi, parallelograms, pentagons, hexagons, heptagons, octagons, nonagons, and decagons all have regular forms, and they all fall along the orientation of being enclosed, symmetrical structures that articulate a definable form within space. Those defined spaces serve as the spatial environments in which regular shapes serve also configurations partly granting of the above described capacity to contextualize temporality within certain degreed dimensions of geometric understanding. A shape defines space, and time is a distortion of space, so a changing shape defines time to variant degrees of measure. Regular shapes house this concept along the dimensions of different dimensional combinatives that aid as defining constitutions for all of what can be defined as ‘space’ in geometry. Mathematics is the general fielding for this concept, and as a rudimentary basis for all of what could be found present and posed within its form, there is the continued constituency of perpetually propagating metrics which emphasize how it is that shapes articulate the serial fieldings of mathematics by the same token of its preliminary fundamentalisms. Shapes, in certain lights, can be equated to numbers, and the numbers that define those shapes accentuate their respective forms within the light of standardized fieldings and measures.
Irregular Shapes
Irregular shapes are enclosed spaces that operate as forms within geometric science which conduct their spatial processes along the lines of both symmetrical and asymmetrical alignment. Irregular shapes differ from regular shapes in that they are structurally composed as forms which do not maintain configurations which are of even distribution. Regular shapes, regardless of the side count that frames not only their shape but also their terming, house both symmetrical construction and equal spatial distribution about their form. Irregular shapes can be symmetrical, but their spatial distribution is not equal. They are unevenly spaced, though, again, they can still be symmetrical. The relationship between different kinds of shapes is built on this dynamic, because all shapes fall within either of the aforewritten classifications. Irregular shapes are markedly important by this notion, because they compose the majority of shapes. There are more irregular shapes than regular shapes, because there are more abstract configurations than regular configurations. Both asymmetrical and symmetrical configurations fall within this category, and as a strikingly profound fact, both classifications count greater than regular configuration. This is because the spatial distribution which qualifies a shape as a regular shape numbers fewer than the number of configurations which count among both the asymmetrical and symmetrical configurations of irregular shapes. Irregular shapes operate as functional elements of geometric integrity for this reason, and as their intrinsic roles develop further, the entire field of geometry seeds its base rudimentary complexities in irregularity and abstract expressionism. Figuring and classifying the different categories that are associated with this dynamic fall in line with the compounded metrics of geometry, and with the whole of what would be found present in this delineation being an essential element of spatial interpretation, the natural algorithms that follow along build to be the subsequent bases of mathematical fundamentalism traced to not only geometric sciences but also the base philosophical principles of mathematics.
Two Dimensional Shapes
Two-dimensional shapes are shapes that exist within two planes of interpretation. Those two planes are length and width. A shape seen within these two dimensions would appear as a figure placed upon a plate, but the plate would have no depth. Within serial grid architecture, the two planes that would be relevant within the situation described above would be labeled as the x- and y- axes, and as the interpreting mechanisms for the two standard planes that compose two-dimensional shapes, each axis serves in the way of articulating not only the planar dimensionality but also the relevant measures and spatial environments, the same. Two dimensions are composed of two environments, and when the spaces within those environments are housed within enclosed framing, they articulate two-dimensional shapes.
Two-dimensionality is also wroughtly based in the same dynamics of time and space described earlier in the first ‘shapes’ section. The intercombinative asymmetrical and symmetrical matrices that compose two-dimensional shapes move into the arena of tempospatial fielding by measure of there being a consistent theme of comprehensively distorting the space in which the shapes are defined, so as to make a clear reference and depiction of time. Time is a distortion of space, and for two-dimensional shapes, the different arrangements of time that are housed within their framework are ones that inevitably come to articulate the variant complexes which make complete sense of different spatial environments. As the space which is defined by the shapes moves, or distorts, and shifts about the enclosed and housing spaces, time operates the suppositional dimensional axes of facilitation to be a mechanism of conduction, the same.
Two Dimensional Regular Shapes
Two-dimensional regular shapes are shapes that are symmetrically configured and also evenly spaced within their enclosed areas. The two axes that compose the two dimensions which house two-dimensional constructions within their standard measure are the x- and y- axes, and with their fielding, there is a consistent means by which to engage the different planar organizations that arise out of two-dimensional environments. Regular shapes, within this light, operate along the base premise of being the lesser counted shape classification - which, regardless, is a general truth - and for the standardized shape arrangements that amount to the categorizations described above, there is an even further means by which to extract what could be defined as ‘unique’ from traits intrinsic to shape interpretation and identification. Triangles, squares, rectangles, rhombi, parallelograms, pentagons, hexagons, heptagons, octagons, nonagons, and decagons are the shapes that fall within that fielding, and the accompanying traits that root alongside their respective forms emphasize further the incorporant dimensionality that is naturally necessary for making sense of their structures. Those traits amount to symmetrical configurations and an equal spatial distribution of the different areas associated with the different side counts wherein regular shapes are encompassed effectively between the two aforewritten general descriptors. The inclusion of a dimensional dynamic is one that sees the interpretation of the spatial environment wherein the prior described shapes would be found in the throe of a wroughtly squared framing that articulates the metrics associated with two-dimensional fielding. Two-dimensional shapes are built on these architectural parameters, and for the overarching philosophies of geometry, the fundamentals they represent evolve further the developmental mechanics of both rudimentary and complex mathematics.
Two Dimensional Irregular Shapes
Irregular shapes are enclosed spaces that are either symmetrical or asymmetrical which do not frame their respective enclosures in evenly distributed environments. Those environments are defined within and along different metrics, but for two-dimensional configurations, they reside within two axial arrangements. Standardized fieldings see those arrangements in the light of the x- and y- axis, but there are other configurations which can house a two-dimensional matrix of differing orientation. The typical angulature measures about a perpendicular origin, but when the angles surrounding the origin are positioned at different intersections, they can be placed at variant points depicting of enumerable configurations. A standard metric places the origin (0,0) at a 90 degree measure about each of the four angles that compose the intersection. There are four quadrants in total that title under the ascribed terms of quadrant I, quadrant II, quadrant III, and quadrant IV, and each one is commonly seen within a 90 degree configuration. Irregular shapes, given their composition, are seeded within this two fold dynamic of standardized fielding and irregular dimensionality by way of how it is that they place within their spatial contextualizations. Two-dimensional irregularity calls from an axial arrangement that, at the very least, is able to translate to planar environments wherein the differential is granting of an interpretable space capable of articulating enclosed geometric architecture. The capacity to do so is how one fields the denoted propagacy of two-dimensional contextualization, and from its inclusivitous dynamics, there emanates the intrinsic narrative of a dynamic spatial environment made sense of by not only symmetric interpretation and its accompanying depictions but also abstract expressions and their associative articulations.
Three Dimensional Shapes
Three-dimensional shapes are defined within standard articulation along the lines of three-axes. Those axes are termed as the x-, y-, and z- axes, and when framed within their relevant planar arrays, any and all enclosed spaces that reside within their visual borders of interpretation arise as configurations of three-dimensional construction. Shapes within this kind of three-dimensional arrangement are built on the housing principles of intercombinative matrices which contain the three incorporant dimensions of depth, length, and width, and the individual components that define their respective structures all house these three axial structures as integral constituents to their respective geometric forms. The forms which result from the different assortment of arrays then stand as the continuant propagacy of what a three-dimensional shape constitutionally imparts upon the field of geometry and mathematics as a whole.
Three-dimensional shapes are squarely based in multifaceted linear triumvirates like those described above, and when incorporating that which would be integral for the whole of what one may find embedded in the frame of the dual dichotomy of space and time relations - addressed partly in previous sections - the concept of developing further the role that three-dimensional configurations play in larger fielded principles arrives at a point of rudimentary scope and clarity. Two-dimensions articulate a two-dimensional tempospatiality, and three-dimensions articulate an analogous three planar correlate. The distorting principles that then operate along the lines of shifting environments, so as to give ‘form’ to time, inevitably serve in the way of building, at the very least, a definitive structural compositry gifting of a universally interpretable modularity.
Three Dimensional Regular Shapes
Regular shapes are defined as shapes that are organized symmetrically about an even distribution of their enclosed environment. Three-dimensional regular shapes then compound this definition with the added dynamic of their respective forms being geometrically situated within the fielded axial dimensions of three different planar arrangements that are seen within standard depictions as the x-, y-, and z- axes. The first two axial environments are usually seen within a perpendicular context, and the next axis, the z-axis, is one that incorporates the dimension of depth into its fielded design. Three-dimensional regularity, then, succinctly houses the aforewritten delineation within the uniquely situant construction of three-dimensional structures which are symmetrical and maintain their different spatial environments with equal and even spatial distribution. Circles form to spheres. Triangles form to tetrahedrons. Squares form to cubes. Rectangles, or ‘quadrilaterals’, form to prisms, and so on and so forth. The key mechanism which defines the geometric sequencing which is granting of the above, again, falls along the axes of a wroughtly two fold dynamic which is seeded in symmetry and even spatial distribution. Regularity is built on this functional dynamic, and when compounded further by the capacity to construct three-dimensional environments outside of their typical ‘standardized’ fielding, the propagable narrative of unique three-dimensional spatial construction composes a uniquely focal component to the larger mathematical arrays which result from variant geometric articulations.
Mathematics grooms the above from a two fold geometric linearity of fundamentalist rudimentation and corresponding philosophical principle, and when building up from where standardization and complexity meet regularity and uniquely afforded spatial articulation, the whole of what is housed arrives at the interpretable environment of not only integrity based geometric understanding but also the natural components to comprehending space as it is rooted and seen within its infinitesimal narrative.
Three Dimensional Irregular Shapes
Three dimensional irregular shapes are housed within definable spatial environments that naturally supplement the theoretically infinitesimal variancy associated with irregular composition and form. Within standard articulation, three-dimensional shapes are built to perpendicular form, with the third axis, or the z-axis, being the forming axis of depth, wherein the full three-dimensionality is found. The manner in which irregular shapes are geometrically fielded, though, wroughtly compounds the modular compositry which aids in how the complexity of spatial environments are articulated. Irregular shapes are defined by their dichotomous symmetry and asymmetry, as well as the uneven distribution of space within their enclosed structures, and when coupled with the different combinative arrays which are formed from the structures which result as formulaic compositions in three-dimensional architecture, there is a natural dynamic of gradated developmental ‘chemistry’ that builds to the effect of a propagably progressive narrative of mathematical capacity. The whole of what results from the spatial fielding described above is that of a greater enumerated measure than what is seen for regular shape configurations, and given the growth of potentiality that moves in step with both two- and three- dimensional environments, irregularity poses a careful depiction of metric that otherwise would not be matched outside of the depictions and visuals seen in other geometric articulations. The fundamental idea to be conveyed in this delineation is that, when considering shape configurations which are seen outside of regular, or ‘ordinary’, form, their count, within mathematical throes, amounts to a philosophically resonant profundity by correspondingly calculant dynamics.
Ratios
Ratios are proportional measures that multiply or divide varying factors by specific degrees. When considering the different segments that compose specific dynamic forms, the means by which they develop, grow, expand, and contract in relation to one another is quantifiable by specific gradients of fractional count. The multiplicative and divisional dynamics are actionable extensions and reductions of proportional gauge that operate under finite pathways. Intervaled exchange would then house shifts in the aforewritten proportionally static dynamics, but the fundamental principle to be had as an inherent intrinsincy is one telling of prepositioned and predictable measure. Machined ratios are giving of artificial fractionals, and naturally formed ratios are the result of measures which are pulled from the spatial regions crafted from developmental processes. As they naturally come about, natural ratios form within the relevant structures that compose the organic processuations of aforewritten description. Ratios hold to their form because they are designed for maintaining a specific serial integrity. Each combinative is a unique sequencing that articulates a full coordinative depiction conducive of an interactive scape of design. This principle of design is one that incorporates different elements of intervaled arrangement into a framework gifting of proportion, scale, and sized measure. An example holds in the ratio 1:2. A rectangle with a length-width measure of 1 inch by 2 inches, utilizing the above written ratio, would shift the dimensions of its make from 1 by 2 to 2 by 4. This is because the ‘one’ numeral represents the original dimensions and the ‘two’ numeral represents the factor by which the rectangle is to be increased. 1 is the original, and 2 is the result. As the different numerals that compose the ratio are altered, the dimensions by which the unique measures pan out come to conform to the variant descriptors which compose the dimensionalities responsible for serially transfiguring all geometries found relevant in the fractional equations of multiplicative and divisional calculation.
Area
Area within mathematics is denoted as the amount of space that is present within an enclosed defined environment. One that is articulable at some level of calculable communicacy. Also, area only applies to two-dimensional environments, but it can be used to formulate scapes that would be found in a multiplicative assortment of arrangements. The purpose of its calculation is to know how much space is being occupied by the suppositional enclosure that is being not only calculated but formulaically defined. Circles have areas. Squares have areas. Rectangles have areas. Rhombi have areas. Parallelograms have areas. Quadrilaterals have areas. Pentagons have areas. Hexagons have areas. Heptagons have areas. Octagons have areas. Nonagons have areas. Decagons have areas, and all other shapes, both regular and irregular, have areas. Again, though, areas are defined within two-dimensional contexts, so each environment that would be found within a regular or irregular ascription would be one that is confined to the standard x- and y- axes. Per descriptions found in the prior sections, the standard orientation for the x- and y- axes is seen as a perpendicular arrangement, but there are an enumerable number of theoretical depictions for how it is that a two-dimensional environment can be oriented. Supposedly, the combinatives range to an infinitesimal sector, but it is still integral to be able to discern that there are two different dimensions. Area within this arrangement would then be the amount of space that is present within the variant folds and contortions which would result in defining the two-dimensional spaces present in the infinite arrangement of environments possible among the configurations granting of enclosed and definable spaces.
Perimeter
Perimeter is the amount of distance around a definable space. It is calculated with not only base linearity but also dynamic and irregular curvature. Perimeter, in this form, then, arrives at the associated complexity indicative of what one would see within a typical geometric environment. The whole of what it is meant to pose within mathematical fielding is markedly unique for this reason, because of the dynamic formulations which are often found as integral necessitants for the entirety of what would be posed in fundamental mathematical architecture. From circles, one discerns circumference, and from squares, one discerns an angular based lineature. Curvature and straight forward path alignment are the only two variations seen within this dynamic, but the two-dimensionality that fields perimeter is defined by an array which spans beyond the standardized associations seen for lines and especially curves. Formulaically, perimeter moves to the beat of a minute pathology which methodologically articulates how it is that two-dimensionality compounds in delineative developmental dynamic outside of where its incorporant infinitesimal variance grows to the tune of the environments where it is enclosed and defined. The profundity in this statement is seeded in how it is that lines are definable, but when they are curved, there is the need to incorporate other mathematical elements, so as to be able to discern what the metric and measure for the summative perimeter would be. Perimeter, and every other conceptual fielding in mathematics, is based on this fundamental understanding, wherein the serialisms which are associated with this aforewritten rudimentary base are philosophically fundamental for understanding mathematics at its core.
Volume
Volume is the amount of space present in an enclosed defined three-dimensional space. It is present only within three-dimensional spaces but can be found in spatial interpretations that branch into depictions granting of scapes and contexts seeding of its form outside of archetypal articulations. As a reiterate, the only space where volume can be found present is in a three-dimensional environment, but the accompanying spatial associations that are centered or oriented about its locale can vary in structure to a degree sufficient of making clear and apparent its naturally intrinsic dynamism of movement and configuration. The use of volume as a geometric instrumentation is based in what would be necessary for calculating the space visually articulated in three-dimensional environments. Different three-dimensional orientations find their metrics composed in different arrangements, and with the entirety of what is present in their variant structures arriving at the continuous potentiality for finity or infinity, volume serves in the way of being the formulaic modularity granting of not only standardized terming but mathematical componentry and modularity, the same. Form is integral to this delineative narrative, and as it grows to make sense of the metrics which are based in rooted measures, the intrinsically associated three-dimensional fieldings that give frame and context to the different structures built within the aforewritten spatial environment are ones which arrive at the philosophical fundamental brace of a core mathematical rudimentation. Geometry is built on this philosophy, given that the different shapes and calculating formulations pervading of its study are built upon the discipline of making sense of space within context strictly composed to make clear the infinitesimal variations that exist within every spatial system.
Surface Area
The surface area of a geometric configuration is the amount of space that measures over the full scope of its outer dimensions. If you were to take a cube and unfold it to see the six squares which compose its face, the area of each square, when added as a sum total, would be the surface of the cube. This concept is important as a principled dynamic within understanding the fielded arrangement of different structures. When composing different configurations oriented around specific geometric schemes, the coordinated measures utilized in their composition come to make whole the serial elements present in the aforewritten calculable scheme. As the degree to which specific surface areas are articulated increases, the characteristics which compose their form inevitably come to be gradated delineations granting of variant applicabilities and relevancies. Three-dimensionality is one of the centering descriptors within this delineation wherein the structures which are present within its dynamics reside as the only ones where surface area would be found relevant - area is found in two-dimensional configurations, and surface area is found in three-dimensional configurations. Form is what is to be found as the extractable analytic in the above, due to the whole of what would be posed along different metrics within mathematics needing some seed of initial interpretation. Elsewise, the whole of its individuate constituency would be steeped in a perpetual uniformity granting of no real comprehensive interpretation but instead a smorgasbord of overarching generalizations. Surface area resides as one of those utilitarian constituents, and incorporating the developmental propagation of compounding dynamics found in dimensional contextualization is one method for basing the whole of its role within mathematics as one of a fundamental structuralism rooted in rudimentary philosophies.
Time
Within the confines of finity, time serves as a mechanic that aids in making sense of how events, or spatial shifts, transition from one point to the next via linear construction. As an architectural component of existence, time organizes different interpretive experiences via the immunological thermodynamics which orchestrate their different unique dimensions of incorporant matrix into fieldings of perceptive disposition and design. Time is perceived in a one-track line because the moments that occur in succession are naturally experienced in accordance with how space moves in order to see the interpreting mechanism come about. Written out plainly, time is a distortion of space, so when space ‘distorts’, time occurs. From the different elements that then go on to define the variant constituents of time, the whole of their composite modularity bases an integral function within any and all areas where it would be found present in mathematical fieldings conducive to incorporating its dynamics. Two-dimensionality and three-dimensionality pose the first spatial environments of suppositional applicability where time would be found directly implicated, and when thinking of how it is that movement within those environments would be the activity necessary to see time actively machined, they come to pose as the rudimentary frameworks wherein time is holistically based. Time is built on preliminary spatial notions, so asserting that which is written prior resides as the suppositional holistic conclusion that transitions the conversation of ‘spacetime’ or ‘time and space’ into the appropriate dimensions of comprehensive delineation and interpretable narrative. Time reigns in this regard, and the mechanisms around its propagacy met by the same metrics.
Velocity
Velocity, also known as ‘speed’, is the amount of distance covered over a specific measurement of time. The equation is written out as d/t where distance is the variable d and time is the variable t. Velocity within mathematics holds to this equation across several different axes of interpretation - with the central axes being the two-dimensional axial coordinate plane and the three-dimensional axial coordinate plane. Velocity is a rudimentary term which places directly within the throes of how it is that movement is calculated within different physical and non-physical environments. Seen along axial schemes, the pattern of movement found relevant in different equations and formulaic backdrops will be situated to the tune of a progressional matrix wherein the time-distance intercombinative written as ‘velocity’ will be seen as the metric associated with progressional axial dynamics. Distance will vary by termings based in the amount to be measured, and time is of the same metric, but the two will always operate, when considering velocity, as two factors which articulate the tracked movement of an object, or subject, rather, across an interpretable spatial field. For the whole of what is found philosophically rooted in mathematics, the different fundamental structures that go on to compose the structural nuances accentuating of what velocity is integral for articulating serve by notion of how the entirety of what is base in its formulation branches by dimensions of naturally intrinsic complexity. Whether it is more distance or more time, the base formulaic metric evolves in mathematics by that developmental light.
Acceleration
Acceleration is the rate of change that occurs within the dynamics of speed as it is recorded in differentiating scales. Speed, or as written ‘velocity’, is the amount of distance an object covers over a specific measurement of time. Acceleration, within these throes, is the change in velocity that finds increase or decrease within specific intervals. A positive acceleration would be a shift from one rate of speed to a higher rate of speed. A negative acceleration is a shift from one rate of speed to a lower rate of speed. One is termed as acceleration, and the other is termed as deceleration - with acceleration being the increase and deceleration being the decrease. Acceleration is written as distance over time over time (or d/t*t; d/t2) where the initial distance over time portion is one that articulates the measured velocity, and the final time portion is one that measures the change in velocity over a second time metric. The key to interpreting the associated values is found in understanding the progressional dynamics framed within the written version of the formula. If the relevant distance can be gauged, then the subsequent change in time that is directly associated with that metric would base its following movement over the shift in temporality squarely indicated by the accompanying time interval. The metric of acceleration is found relevant along these delineative braces within what would be posed as an infinitesimal fielding of theoretical prose, and for reasons of implicate continuance, all of that which then goes on to be measured by its formulation bases each associative fundamentality in that brace by analogous dimension.
Slope
Along the slant of a line that carries within three-dimensionality, there resides a formula for its form. It reads as y=mx+b wherein ‘y’ is the y-coordinate found along the y-axis. By variant principles it is to be calculated with input from the x-coordinate wherefound as the ‘x’ variable in the equation. When ‘x’ is put in place, the ‘m’ variable, which reigns as the measurable distance between the relevant intervaled points of measure, is counted as the slope written of above. The slope is written as the measure of vertical distance from the previous coordinate over the measure of horizontal distance from the previous coordinate wherein a four unit height change with a five unit horizontal shift would be a slope of m = ⅘ . The ‘b’ variable is the y-intercept which is the constant found where the graphed line intersects the y-axis. ‘y’ is the resulting y-axis coordinate. ‘m’ is the shift in coordinate measure. ‘x’ is the x-axis coordinate, and ‘b’ is the point at which the graphed equation intersects the y-axis - known again as the ‘y-intercept’. Slope is important for measuring the different rates of change that occur over any measurable period where shift is implicated as an integral process of occurrence. Slope is one example within physical study of how it is that movement along linear metrics is tracked with static variance determined by only two-dimensional dynamics. Even if the formula were to be layered within a three-dimensional context, the only shift would be which dimension to incorporate along its formulable throes wherein the base fundamental dynamics remain the same. A slope formula within a three-dimensional environment, then, would still maintain to two-dimensional throes, though the housing axial space would still be constitutionally constructed with length, width, and depth.
Physics
Physics is a branch of mathematics that stems from fundamental scientific disciplines. Science, itself, is the study of the natural world and the phenomena which are found present within its setting, and physics, by connection, is the branch of mathematics that articulates the different quantifiable combinatives that serially articulate the above written natural world and the subsequent phenomena which reside therein. Articulating the occurrences that reside within the contextually fitted realm of tempospatiality is where physics finds its footing, as the unique equations typically localized to its structures are where the practice of integrating general mathematical disciplines can be said to find its first fielding. Physics incorporates all of the previous dynamics of slope, acceleration, velocity, time, surface area, volume, area, perimeter, ratios, shapes, three-dimensionality, two-dimensionality, grids, coordinates, planes, lines, points, waves, decimals, fractions, numbers, addition, subtraction, multiplication, division, and especially long division all within its form. As a field found within the branch of mathematics, all of the calculable measures heralded as vital to the above listing of concepts holds as the fundamental philosophical basis for any and all theoretical suppositionings seeded as integral for the relevant matrices of foundational numerological base, and the infinitesimal theoretical metrics that move to the rhythm of what evolves naturally in the above rites by way of consistently implementing what gradates with mathematics as the embedded scientific element granting of how serial numeration and its associative formulations articulate the world and the natural phenomena found therein. Science is one example of how physics conducts this narrative, and a fundamentalist perspective within mathematics builds by this theoretical orientation, the same.
Two Dimensional Physics
Upon a two axis plane, seen within standardized depictions as an x-axis and a y-axis, there are typically a perpendicular set of lines which extend into what is a theoretical infinity. The set of lines is composed of a lengthwise line and a widthwise line, and if flattened out to an equally infinite planar arrangement, those two lines would compose what is dubbed as a two-dimensional field. Two-dimensional physics is responsible for making sense of all that occurs upon the two planes described above, and by synonymous contextual transition, the planes written of prior are defined solely as dimensions. Within those dimensions, physics gradates the different events along a spectrum of uniquely interpretable perceptual givings. The x- and y-axis are the typical terms found as standard associations to those givings, but their actual purpose resides within the philosophical role of articulating width and length, respectively. Mathematics utilizes two-dimensional fieldings in arrangements by this dynamic, so that as the different spatial environments where two-dimensional physics builds to house unique structural dynamics of infinitesimally incorporant design, the entirety of what would be found relevant within the reiterate context of either a two plane or two axes environment translates the phenomena and metrics of that environment into groupings of comprehensive design. Two-dimensional physics requires all elements of its prose to be situated at some level within its rudimentary architecture, but the fundamentals that result grant way for other complex developmentals to form by propagable effects wherein delineative characteristics that stem from this conceptual narrative, and its associative prose, are responsible for all notions compounding of the constitutional bedrocks intrinsically squared accordingly.
Three Dimensional Physics
Three-dimensional physics continues on from the previous section as a term that defines the increasingly compounding depth and dimensionality found within the general mathematics of integral formulation and calculus. Three-dimensionality is wroughtly characterized by this defining architecture, so for the elements that construct the fielded dynamics directly implicated in physics, the spatial environment that is interpreted from the aforewritten narrative is one that aids in contextualizing how it is that three-dimensionality traces within the frame of physics. Rudimentary bases established as principled workings within physics hold the architectural prose of mathematical measure in the frame of gradated axial scope, so that as the implications develop further, the onset of congruous functionality makes clear what would otherwise be left in parted systemic reduction. The processual graphics found in the aforewritten description of cross-sections holds further when synthesizing the capacity for depthwise articulation found in two-dimensional imagery with the inherent purpose of three-dimensionality communicating real-world visualizations and occurrences. Across the broad spectrum of mathematical extractions which exist within this period of analytics, the whole of what is found within the three-dimensional complexity - hierarchically situated in both tangential and intangential sciences - is wrought with clear and effective functionality. If it is that movement, mathematics, dynamism, or any sort of geometric form or scape is needing to be made sense of within a three-dimensional context, three-dimensional physics is the instrumentation by which it is done. Where the specific constituents of depthwise dimensionality hold in bodily integrity, the serial account grows in kind as an intrinsically fortified mapping of shifts, changes, transitions, and alterations squarely placed among the other phasal bracings inbuilt as mechanisms for representing all of the inner and outer workings of mathematics.
Colors and Numbers
Colors and numbers are both naturally considered as forms of art, and when going further, their direct interrelation holds as an integral facet to the interpretive practices necessary for making sense of mathematical morphology. Front to back and back to front, numbers compose colors, and light - defined by colors - composes the photoscopic result of quantification. From zero, one, two, three, four, five, six, seven, eight, and nine comes the numerature seated in the above, and within the frame of their standardized numerological use, there is a means by which to make sense of different wavelengths. Those wavelengths are what define light, and numbers are utilized as their modular communicants. For the mathematics that is fundamentally rooted in existential experience, there is a means by which to serially articulate every dynamic of its different elements along unique axes of interpretation. Building the theoretical framework positioned around the intercombinative arrangement of colors and numbers initially grants blue, violet, red, orange, yellow, and green, by whatever way they may be viewed within physical articulation, and for the physical geometry that accompanies the metrics associated with their wave based composition, numbers are the sole means by which to place not only their measure but their coloration into form. Quantifying serves by gifting what would be lost in the translational movement from visual depictions to interpretable figures. The configurations that grant the modular intervals which evolve the facilitating environments granting of how it is that colors are made sense of do so by that which is localized to the suppositionally ‘linear’ orientations described prior. Along the lines that count numbers, when it is that they are accompanied by their associated wavelengths of proper sequence and form, the corresponding intervaled sequencing builds how it is that numbers and colors are understood by their connective communicacy.
Color Theory and Numbers
Color Theory is the artful theory of color combinatives that offers calculable environments wherein the colors can be predicted in their mixing. Red and blue are giving of violet. Blue and yellow are giving of green, and yellow and red are giving of orange. Each of these combinations are written under the two-fold banner of primary and secondary colors. The primary colors are red, blue, and yellow, and the secondary colors are violet, green, and orange. Between these six different color types, there are an infinite number of color combinations that are extracted from unique equations. The numbers used to predict and articulate the different equations are sequenced in such a way that their form operates under the liken dimensionalities of a basic mathematics equation. If red were to be 1 and blue were to be 2, then provisioning the number 3, so as to discern what would result should gift violet by extraction. Under this fundamental principle of delineation, color prediction and color theory place as concepts termed to make sense of how arithmetic pans into art from a wrought serial basis. The finite mathematical principles which then build this arrangement are rooted in how visible wavelengths span across a quantified spectrum of fielded light conductions. The actual wavelength measures are within a realm that counts particles at a minute level wherein the distance intervals would be analogous to the fibrous partings of stretched cotton bundles. Photons are the particles used to define light, so by natural principles of extraction, photons oscillating, or moving, at different frequencies produce variant color combinatives. Color theory utilizes these combinatives within formulaic constructions that articulate the suppositional ‘sea’ of color that imparts quantified hues, pitches, tones, shades, et cetera. From what is, in essence, complete darkness to full and radiant clarity, the color spectrum found in between is the calculable result of Color Theory.
Frames
When viewing something within a specific range of dimensionality or even more so across the breadth of an infinite scape, there is the implication of their being a sequenced perceptivity to these different interpretable and even more so articulable states. As the full breadth of a particular image, visual, visualization, picture, depiction, et cetera comes to actually be interpretable, the unique sequencing or constituents of its make are definable by frames. Frames are the controlled or, at the very least, sizeable panelings which house the above described artful dynamics within the folds of an architectural throe. As the panelings find increased serial depiction within unique spectral gradients, their constituent builds convey what is to be communicated from the gradated serial elements of the frames design. When thinking of a specific moment along a temporal measure, the state in which the moment is interpreted, understood, and articulated is one that gifts modular graftings imparting of what has been described within framed bases. Frames are like cross-sections or three-dimensional molds. They are giving of finity within scapes that may be finite or infinite, but their general purpose is one that holds to their conceptual base being a rudimentation of discernable imagery. At this conceptual base the idea propagates further, in that frames can hold significance among culminations that are intangible and even non-physical all together. A frame of view from the eye may be different from what is found within the intangible machinations oft articulated within spatial environments, but the overarching generalism holds within an environment inclusive of calculable modularity.
Two Dimensional Frames
Two-dimensional frames find an example in cross-sections that communicate visual imagery within two-axial scapes. If one were to consider a perpendicular axial arrangement labeled with the x- and y- axis terms, the orientation would hold within a gridded measure of frame granting of two-dimensional imagery. Frames, within two-dimensions, are integral for the same purposes one would associate their composition with cross-sectional analyses. They show sequenced mathematical events that hold in intervaled segments, and as the events, organized within processual progression, succeed one another in order, they each form part of a narrative that is rooted in active communication. The communicative events of their visual prose are compounded by furtherings that are gauged from what is found in the consistent pathological methodology associated with uniformity. As each element of the two-dimensional frame is given placing within the unique constituent mapping of the above described visual morphology, there is a regulated reiterate of constant mathematical prose. Geometry is the fundamental basis for this reiterate, and in its utilitarian design, frames are integral to connective specification. Individuations that span into generalisms, as well as the reverse order, are where the aforewritten reiterates place, and as a developmental dynamic, the sequencing written of prior also holds in the same developmental context. Two-dimensional frames have the ability to articulate structural resonance within a three-dimensional scape, due to a two axial configuration serving as the architectural necessitants of appropriate allotment. The quanta found associated with this supposition are fixed within this provisioning as elements to a larger framework that assists in granting multifaceted dimensionality through the use of dual-axial imagery conducted in the throe of summating the above through the idea of interpretable geometric depiction.
Three Dimensional Frames
Three-dimensional frames are extremely unique, because they are either stilled or motioned occurrences held within what could be described as a dimensional context of interpretable configuration contingent upon depth for measure. If one were to use a camera to take a picture of one’s bedroom, one would have a two-dimensional window of the room and its content. Hypothetically, stepping into the window would reveal a three-dimensional scape that, by some measure, would be framed within a light of respective form. The unique compositional geometric matrix that is found throughout the whole of the picture’s transitional complexity would serve as the principled construction to be had in interpreting the concept of a three-dimensional frame. As the science goes further, there is a continued backdrop that moves into how the different environments where three-dimensional frames are found utilize a fundamentally based variance that spans beyond physical dimensions. Three-dimensional frames that are situated within one’s own imaginative processes would find orchestration along the respective axes of non-physical geometricism. Frames that find housing within this aforewritten scape can be viewed as part-visuals and part-intangentials, due to their theoretical base being in a suppositionally recreated depiction of wrought tangible existentialism by metric of coordinated composition. Both interpretable and, even more so, discernable geometry situates frames in the same capacitative measure of universally interpretable environments wherein the constituents of their scape are viewed from the perspective of actionable visceral consequence. All of the above progresses by disciplined and nuanced accordance to propagable fieldings of dimensional contextualization that reside within three axes of orientation. As the frames take on serial standardization, the uniformity which can be seen within their articulation, be it of abstract sort or elsewise, imparts a gradated fundamentality unto the viewer that inevitably manifests at levels of complexity conducive to integral compartmentalized arts. From this notion, there then spans the added aspect of increased planarity which is giving of multifaceted frameworks cultivate of the same resonant depictions seen across all scapes of standardized mathematical fielding.
Mapping
Along the whole of geometry’s fielding of three-dimensional morphological axes, there is a suppositionally modular set of intangible webworks that aid in articulating its symmetrical and asymmetrical forms with nodes and the subsequent regular and irregular geometries that nodes compose. As the nodes find placing along the body’s internal cavities and external structures, the variant linear measures which see them connected gradate an all-encompassing scape purposed with mapping the human body as a collection of gathered configurations which inevitably find their most minute partings as pointilized bundles of spectral thermodynamics. Mapping these bundles is key, so that in knowing how it is that the serially definable architecture holds, the different elements that serve within their make, as subsequently integral constituents, are ones which can come to be of interpretable sort, thereby granting articulable capacity and effective interaction. One of the markedly profound uniquenesses of the above described dynamics is found in the integrity of mapping as a conceptual term granting of both tangible and intangible articulations. The mathematical morphologies that fall in line with the above described spatial environments are ones that analogously processuate spatial environments by variant and differing degrees of fluctuating interpretation. If one were to think of one’s right hand, then its form, feel, texture, shape, tone, et cetera may come to be of mind in the same way that actually viewing the hand would more than likely grant. Mapping the tangible hand, then, would more than likely grant the exact same patterning if one were to visualize the extremity within a cognitive scape. Physical mappings may be as rudimentary as markers and soft pen sketches along the abstract axes of the hand, and intangible mappings may hold within actively thinking of and conceptualizing the full outer appearance of not only the hand's morphology but also its physiological capacities. From this narrative, there then stems mapping as an integral serialism that functions across and within all environments as an instrument in extracting variant measures associated with defining, constructing, deconstructing, and fundamentally evolving the mathematical fundamentalities of gradated measure and subsequently complex design.
Two Dimensional Mapping
Considering two-dimensional planarity, mapping, along a two axial orientation, is typically characterized by geometric configurations that span across shape making nodes. The nodes may hold a pattern that is symmetrical, asymmetrical, or both, but the central principle to consider is one that provisions a specific region within the measurable finity of shaped enclosures. Over the span of a plane that is set within the suppositional median of an origin point, this type of architectural hierarchy places among the form of three-dimensional morphology as an integral component conducive of holistic structural conveyance. This is seen in the morphological dynamics of an x- and y-axis arrangement serving as the backdrop for shapes that reside within ratioed, flat portions which are the building blocks for larger geometric environments. As the different transfixed situants pervade across the whole of the implicated shape oriented progression and construction, the linear configurations that compose their morphology contribute an added dynamic of measurable gauge, so as to operate within the calculable frame that is their respective mathematical function. The uniform standardization and fielding seeded in the overarching science of mathematics, as well as geometry, covers the full range of three-dimensional spatial environments but in such a light that two-dimensional planarity reigns as the theme of delineative brace. Moving from the finite and infinitesimal scapes which articulate the intercombinative array of spatialist narratives grants room for the successive planes that compose the necessary sequences of gradated serialist prose to be framed within layers that move the whole of their incorporant dynamics into mathematical morphologies by whatever depth or thickness resides as relevant in metric and measure. A supplementary additive develops the precept of considering dimensions and layers within the exemplary arena of where they find definite, finite mathematical measure, though their fieldings remain as theoretically infinite. Mapping arrangements layer their subsequent accentuations over this capacity, and the developmental focus they maintain throughout this axial operacy is one that broadens only by the defined or undefined finities of their associative compartmental form.
Three Dimensional Mapping
Three-dimensional mapping is the next phasal arrangement in mapping instrumentation, as the different compartmentalized compositries that make clear the evident linear prose composing of depthwise geometric constructions are responsible for drafting the context for three-dimensional environments. Those environments are what give form for the whole of mathematics’ philosophical bedrock to orchestrate what inevitably arises as the naturally developed field of tactile dynamism. Spatial environments - liken unto what is typically seen in mapping - are what give form to the infinitesimally variant spectra of three-dimensional configurations. Considering the different contexts wherein three-dimensional mappings hold as the implicated technique of approach, there arises a clear and direct pathology toward understanding three-dimensional mapping as an integral method in morphological geometric comprehension. A wrought mathematical example can be had in the use of three-axial orientations being spread over what can be described as the whole of geometry’s pontentiacy of conceptuate forming. From the macroscopic dimensions of discernable spatial articulation to their corresponding microscopic particulates, the diverse arrangements that field the whole of geometry’s structural situants are respective of three-dimensional contexts as modular hierarchies dependent on the greater whole of mathematical conceptualization for stability. “The whole of what is standardized in geometric fielding is greater than the sum of its parts”, but the centering dynamic principally holds to the accord of mapping as a technique being appropriate for making sense of the variant serial constituents of the visually quantifying sciences’ organic narrative. Focusing on serialism holds the above in the light of a rudimentary technique that is integral for not only base serialism but also the more advanced gradations that span from the complexity associated with the dynamic, textured dimensional contextualizations prescribed strictly to mapping within geometry’s throes. All uses, from cross-sections to simple drafts and designs, orient the minute individuations granting of processuate propagants into fundamentalist bases of compounded delineative brace. Mathematics’ philosophical rudimentations then go on to core the analytics of universally interpretable and implicate serialisms along the matrices moving of its developmental accentuations in kind.
Progression
Progression is the conceptual occurrence of an interpretable event moving by some measured dimension. Time is usually the medium by which this dimension is measured, but as the components to its traced movement shift, the linearity may come to be of a spatial orientation wherein a combinative of tempospatial gauge would be the most appropriate dimension of rule. If the different constituents of a progressional matrix, then, were to be serially defined, the all-encompassing pattern of trace modularity would be fitted within a delineative narrative of time based, scape based, or ‘timescape’ based root. Tempospatial mathematics falls in step with the above fielding as an infinitesimally dynamic landscape granting of perpetually relevant planar hierarchy, and as the continuate construction of the aforewritten concepts describing the incorporant axial measure and spatial geometry propagates, the seeded proponents of effectively interpreting time and its movement come to be found present in the full and broad reach of how it is that progression fundamentally places context and narrative in all of the above.
Dynamic or static movement, by any metric, is forded by the progressional matrices that move from not only geometric articulations but also the full scape of what houses tempospatial integrities rooted directly in the field of physics. The two intercombinative elements of connective dimensionality aid in gradating the theoretically abstract constituents which give rise to the inevitable standardizations of uniform growth and development. Progression is defined by the processual sequencing of interpretable form and function. The discerned context need not be articulable within visual depictions, given that the fundamental concept remains as a discernable metric which aids in imparting how it is that the linearity - oft written strictly as the stringent axial continuate of forthward or regressional movement - is granting of the rudimentary all-encompassing metrics stapled in philosophical progressional denotation and fundamentalism
Linear Progression
Describing linear progression along the conceptualized lines of temporality is a feat demanding of sequenced patterning, given the ordered succession that is a natural element to patterns fundamentally organized by some dimension of step by step congruency. The linear component of any progressional matrix is found in how one perceives the sequence as it moves in its throes. The scientific supposition for linearity finds definition as the recognizable changes in immunological disposition, or any analogous laywork, that are tracked by a respective entity of interpretation. From one moment to the next, the different thermodynamic components to an environment would be changing to a degree measured by the unique shifts or alterations in the scape’s principled thermodynamic arrangement. Considering the prior concept further, if one were to have a complete and all-encompassing measure of a finite locality within serial arrangement, the whole of that region’s organic and inorganic compositry would be held within linear progression as the result of successive moments that are completely different in thermal arrangement, and by base rudimentary extraction, immunological disposition the same. Because the principles of hot and cold vary from moment to moment, the utilitarian extraction which is then to follow comes to be one that evolves the developmental mechanics of an environment that is made sense of through gradated disposition. As the unique gradated dispositions go on to find ordered arrangement, the linear matrices of ‘part-symmetrical’ ‘part-asymmetrical’ array grow from their points of integral significance and move on to be integral machinations of wrought serialist form. The temporal measure of progressional order is one that follows along in suit as well, so that in utilizing the different spectra necessary to properly quantify the variant gauges of a dynamic environment, the graduated morphologies find form upon a foundation that is designed to construct a plausible laywork for its throes. The environment would be one of relative finity, but within the larger breadth of its serial dimensions, combinatives of finity and infinity would move into positions of furthered serialism purposed with maintaining hierarchical organization by way of not only temporal construction but also geometric fielding and any associated mathematics granting of relevant provisioning.
Spatial Progression
Spatial progression operates along the same principled dimensions as linear progression, but the order, or sequencing, comes to be less important when considering the resulting culminations which arise from wrought formulable action. Serialist prose would situate the above in chemical variation that is characterized by quantants and suppositional constants that are necessary to not only define the reactions which take place within a spatial environment but also order them within plausible formulants. The dimensional range, or planar fielding, also operates along axes of integral arrays, considering the above, given the propensity to associate different levels of mathematical mechanics with the constituents necessary to properly articulate their conduction. A two-dimensional spatial progression would stand to be a slight more fundamental than a three-dimensional spatial progression, but from a position of cross-sectional complexity, any degree of planarity would find suiting as a matrix of fitting composition. Rudimentary mention spans along the fundamentality of the latter by way of conformatory sort, due to the inherent knowledge of fewer lanes meaning fewer angles, lines, shapes, et cetera, but the resurrecting developmental fielding is provisioned by way of unique environments being composed of unique scapes of processual dynamics. One of the greatest degrees of comparative contrast between linear progression and its spatial counterpart would then be the extent to which immunological disposition and thermodynamic constituency can find delineative brace among the folds of an analytical dichotomy granting of structurally integral architecture. As the hierarchical componentry molds to progress a spatial landscape that is holistically defined within perpetually gradated serialism, the corresponding suppositional contrast posed by a linear pathology would reticulate along an axis of interconnected symmetries. The implicated matrix, though, would be purposed with combinative deconstruction wherein the lacings meted by some sustainable degree of relation would still be divisible by means of comparative serial separation. Spatial progression would be relegated to scaped thermodynamic variance, and its linear counterpart would contribute to the dynamics of the sequencing by way of a comparative shift from one line of perspective to another.
Branes
Viewing branes from a lens that initially utilizes two-dimensional planes places the idea of a minute, or potentially expansive, three-dimensional context within the scope of a particulate fashioning, given the depth of axial gauge that would be associated with the transition from two-dimensional form to a three-dimensional orientation. A brane, itself, is akin to a perceivable fielding that houses all of the interpretable matter and energy within a spatial ‘fold’ that articulates a specific environment by either finite, infinite, or both finite and infinite means. A two-dimensional brane, then, would be a two axial plane that either communicates its compositry by way of depth oriented constituents or simple reflectivity indicative of existential measures. One would be able to see the scape no matter what it housed, but the brane which would house the scape would be two-dimensions no matter what depth may be portrayed. A painting expounds upon this with its frame being of infinite capacity in portrayal but still residing within a two-dimensional context. The three-dimensionality that interestingly develops to be an integral intrinsincy for the organic interpretations which give rise to the natural articulations imparting of the proper and appropriate instrumentations of discernment are where the factual tangentialisms that hold as applicable formulate the role of branes within physics, geometry, and mathematics in general. A three-dimensional example would then place one within a room where the chairs, carpet flooring, wooden shelves, and stained glass would all be at different levels of shade, value, depth, geometric compositry, and modular design, but as a holistic depiction, the physical room would be set within a wroughtly, stagnant three-dimensions. In continuation, a three-dimensional brane would be characterized by a transition from a framed, two-dimensional painting to a three-dimensional model pointedly held as the cognitive interpretation of the physical environment in which one would find the compounding interpretation of perceivable and definable space. A key point in situating the integrity of the above within mathematics shifts in the direction of basing analytical fundamentals in the seat of inclusivitous metrics imparting of a foundational frame of understanding.
The Final Concept of Mathematics: The Measuring of Curved Lines and Curved Surfaces
When a curve is measured, the standing means of doing so within analog instrumentation seats the origins of the measuring art in understanding the full circle of the curve, or ‘arc’, before the individuated arc or curve can be drawn. The curve’s dimensions are based off of the whole measurements which compose the circle wherein the metrics which aid in conducting the process are utilized along standard formalities. The diameter of the circles (otherwise understood as the distance present between two points that cut a circle directly in half) which compose the different dimensions are the central means by which to understand the ratios which make sense of the metrics and measurements surrounding their form. A curved line that spans outside of an ‘archetypal’ circular dimensionality is one that would house many circles, and from the different diameters which compose those circles, the distances measured from those curves would be formulated from how the individual diameters would fit around the circumference of those circles and their lines.
Outside of what may be viewed within the two-dimensional contexts described above, three-dimensional surfaces are of the same caliber. The diameters of the circles which are found within spheroid environments build from the larger circumferences which serve an integral role in measuring wherein the individuated distances and areas that compose those aforewritten dynamics come to be ones that are built from ratioed proportions granting of interpretable serial arrangements which naturally gravitate toward mathematical articulation.
On the consideration of what is found as integral within the above, as a means of determining value, the rallying principle of conclusion for the concept of measuring curved lines and surfaces arrives at what symbols are used to discern the aforewritten term of value.
and .… have been used in part throughout the duration of this text, and by comparison 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 have been used to articulate different principles, concepts, and knowledges the same. The two each have a role in making whole the practicable measures that aid in compounding how mathematics is built as an art form which makes whole the fundamental expressions of existence through different interpretable philosophies.
The first set of 'angular numerics' helps to make sense of how one can begin to initially engage a number. A number is a symbol which is purposed with articulating a certain 'amount' or 'value' wherein the idea of having to do so begins to formulate a unique philosophical endeavor which is completed only upon agreement. The concept, though, of a symbol containing '1' junction or angle helps in this light, because the value is visceral, it is no longer 'spoken and agreed'. It can be extracted from what is present, be it visual, kinesthetic, auditory, or a combination of the aforescribed triumvirate. 'Value' can be principally discerned with the above, and from this delineation, a strong, resonant emphasis on effective philosophy helps to build the fundamentalism naturally present in the integrity of mathematics.
The second set of numbers is what grows the idea of having to determine value into an actionable arena. When measuring a curved line or surface, it is written that the variant ratios which define the necessary metrics are borne out of extracting the diametric determinants that go on to make whole what would otherwise be left to standardization or even inconclusive interpretation. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 form an integral dichotomy in this array from the different systems of mathematical measurement and standardization which utilize their componentry. The suppositional modularity found rooted in determining a curved distance is based out of unique classes of standardized uniformity, and from this relationship, the role of the above set of numbers aids in compounding the fundamental purpose of symbols in mathematics. Two known systems of measurement - termed as the 'customary system' and the 'metric system' - articulate different standardized measurements found within principled fields such as distance, weight, and energy, and from their respective standardizations a unique determinant can be extracted from their uniformity.
The metric system is a system of numbers that translates along the axes of compounding zeroes within different places, so as to engage different suppositional amounts within different fieldings like distance, weight, and energy with varied degrees of efficacy. 'Distance', for example, houses terms such as nanometer, millimeter, centimeter, decimeter, meter, and kilometer wherein the only factor of transition from one termed amount to the next is knowing how many of each unit 'fits' into the next or prior. 100 centimeters are found in a meter, 10,000 decimeters are found in a kilometer, and 1,000,000 nanometers are found in a millimeter. Each amount factors into the other along the base principle of a 'square' or 'rounded uniform' fitting. Measuring curved lines and surfaces, then, grows this dynamic by way of maintaining a wrought uniformity within the different matrices which would be found among the ratios determined from the standard formulae seeded in composing curved metrics. The metric system aids in interpreting curved measurements by way of utilizing wroughtly uniform measures of distance for the infinitesimal shapes and sizes that compose the dynamic field of curvature within mathematics.
The customary system does this as well, but its conversion principles are based in a different light. The terms used are different, and the values ascribed to those terms are different as well. Termed standardizations like inch, foot, yard, and mile serve as examples by way of their ascribed values being outside of the 'zeroed' modularity described earlier. 12 inches are found in a foot. 3 feet are found in a yard. 1,760 yards are found in a mile, and each term transitions from one to the other along those principles of standardization. When determining the distance of a curve, the dynamic variance of valued measurement compounds the difference that one would find in a 'metric system' setting by way of the figured amounts used configuring to a greater degree of difference by the simple principle of the standardized amounts not aligning to the same uniform dynamics seen in the metric system. The 'tens of tens' found in the metric system do not exist in the customary system, so the different configurative ratios are able to be viewed at a greater degree of contrast. A circle with a diameter of three inches will not translate by the same token as a circle with a diameter of ten centimeters. The two different amounts translate within their respective systems by different principled arrangements, and from this dynamic the customary system and the metric system find a strong connection of fundamental comparison and contrast. One is useful for certain tasks, and the other operates in the same light. The fundamental philosophy of this mathematical duality is what builds the ability to gauge and engage curved measurements at variant levels of capacity.
From the above, the beauty of different symbols being used for different purposes 'comes full circle' wherein the cyclicality in life and death that brings balance and wholeness to all things blooms as a natural instrumentation which grants universal meaning and expression to creation. Art, history, science, literature, culture, tradition, heritage, and even the abstract fielding of comprehending and determining the value and worth of different virtues all find a means of articulation in the above, and from their respective interpretation, greater knowledges can be shared, cultivated, and known.
This concept is what finishes mathematics, and from its prose, any and all things found anywhere can be grasped, defined, and known.
Afterword
With this text being based in the ancient art and discipline of mathematics, it is one that has come to serve, and will continue to serve, in the way of imparting the profundity of philosophical and doctoral study through conceptual principle and wrought scientific fielding. The developmental propagants of the narrative written within its passages are based on the integrity of discipline being the key arraignment for inevitable growth, due to the natural dichotomy of holism granting of such by humanity’s throes. Mathematics is the key instrumentation within this conversation, and as its prose works further into what is to be strictly gauged as a fundamental philosophy, the whole of its delineative functional complexity is rightly housed along the fundamental propagacy of all that may be found articulated through some degree of interpretable and progressional language. Defined as a language earlier in the text, mathematics is vital in understanding the world in whatever way it may appear to one’s person, and as the compartmentalized list of works grows from this crystalline outline of what is known sacredly as integral, the different roles they perform come to be building blocks for continued advancement and study. Those works list as follows and can be read as such.
- Fundamental Philosophy Art Book
- The Fundamental Philosophy of Mathematics
- The Fundamental Philosophy of Mathematics Workbook
- Fundamental Philosophy Surgery Series | Introduction
- The Fundamental Philosophy Preparatory Text
- Fundamental Philosophy Encyclopedia Series
- The Fundamental Philosophy Supplemental Nutritional Formulas
- The Fundamental Philosophical Premise for Murderously Correct Surgery
- Fundamental Philosophy Surgery Series | Head and Neck
- Fundamental Philosophy Surgery Series | Back and Spinal Cord
- Fundamental Philosophy Surgery Series | Thorax
- Fundamental Philosophy Surgery Series | Abdomen
- Fundamental Philosophy Surgery Series | Pelvis and Perineum
- Fundamental Philosophy Surgery Series | Upper Limb
- Fundamental Philosophy Surgery Series | Lower Limb
- Fundamental Philosophy Surgery Series | Cross-Sectional Anatomy
- Fundamental Philosophy Encyclopedia Rainbow Drumtones
- Lunar Charts
- War
Conclusive Prayer
Dear Lord,
Joy
Amen.
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