Discover Top Posts Tagged with #cartesian geometry

Solution #3

Solution behind the “Keep reading” thingo!

#puzzle #maths #geometry #coordinate geometry #cartesian geometry

Puzzle #3

Okay, this is where the “for a given definition of puzzle” starts to come in, because this one’s less of a puzzle per se and more of an “I invented an A-Level C1 exam question with a neat trick at the end”. It’s certainly much less open-ended than Puzzle #2, and you’ll need to know how coordinate geometry works to actually solve it. But I’m still fond of it, mostly for the trick. Tell you what, if you can work out what the trick is without doing the nitty-gritty of the algebra, that’s also fine.

Triangle OAB is defined by the origin, O, and the points A(14,2) and B(2,11).

Diagram not to scale

a) Show that line AB has the equation: 3x + 4y = 50

b) Find (i) the equation of a line perpendicular to AB that passes through O. (ii) the point C where this line intersects AB

c) By considering the geometry of what you found in part b) and the length of the lines AB and OC, find the area of triangle OAB.

Solution tomorrow!

(post edited because I made some big mistakes with my numbers!)

#puzzle #maths #geometry #coordinate geometry #cartesian geometry

Mandalic Line Segments, Entanglement and Quantum Gravity Part I

(continued from here)

We are going to consider once again now geometric line segments of mandalic geometry and their relation to Cartesian line segments and the Western number line. Yes, this is sort of a detour from what I stated we would look at next. But this is not unrelated and lies at the very heart of mandalic geometry, and I’m not yet ready to address what I projected in the last remark of my previous post.

I keep returning to this subject because of its extreme importance. Beyond its significance to understanding the logic encoded in mandalic geometry and the I Ching, I believe it may also hold the key to quantum entanglement and quantum gravity. Despite the fact that mandalic line segments are really fundamentally mental constructs, a fiction of sorts, it is still important to understand how they are composed and how their components interact. Though they may themselves be fictions, the line segments and the points that compose them do in fact map a number of physical entities, realities that may be related to quantum numbers and quantum particles and states.

When Descartes invented his coordinate system, with its points and line segments, he based his system on the number line extended to two or three dimensions. In modeling it on the number line the space he described was imagined to bear a necessary one to one correspondence to the real numbers.[1] However this 1:1 mapping of geometric space to the real numbers was a premise implicitly assumed by Descartes. It was in fact axiomatic,[2] but apparently not stated as such.[3] As a result, the presumed relation has become a blind spot[4] in Western thought, never proved nor disproved, at least not at subatomic scales.[5]

Neither mandalic geometry nor the primal I Ching make such an assumption. In place of Descartes' 1:1 correspondence of geometric space and the numbers on the number line, we find a mandalic arrangement in which there are different categories of spatial location which can host one or more discrete numbers in a probabilistic manner. This creates various dimensional amplitudes and a multidimensional waveform of component entities.[6]

To my mind these characteristics of the mandalic coordinate system in combination with others described elsewhere make it more relevant to investigation and interpretation of many quantum phenomena which are as yet poorly understood than Cartesian coordinate dynamics may be and without need for recourse to imaginary numbers and complex plane.

(continued here)

Image: 6 steps of the Sierpinski carpet, animated. By KarocksOrkav (Own work) [CC BY-SA 3.0], via Wikimedia Commons

Notes

[1] Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers.

[2] That is to say, taken for granted as self-evident.

[3] See Note [4] here.

[4] We have lived with this unproved premise so long that we no longer even question it, or imagine that there might be an alternative which conforms better to reality at certain scales, notably subatomic scales. The I Ching also seems to suggest that a complete true description of complex relationships that involve a large number of dimensions, including complex societal relationships, requires more than a simple 1:1 correspondence between the notational symbols involved and the realities they represent.

[5] And from what I can see, no one seems to have much interest in proving or disproving this assumption.

[6] When speaking about hexagrams the number of dimensions involved is six as each Line of the hexagram encodes a value for a single distinct dimension in a 6-dimensional space. In a hybrid 6D/3D compositing of dimensions though, two such Lines in relation reference a single Cartesian dimension in 2- or 3-space. A concept not to be missed here is that interactions of quantum particles may well involve such integration of dimension, of dimensions we are not even aware of beyond the unsettling fact they upset the neat applecart of customary conceptual categories.

Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)

Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn't make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)

-Page 301-

Beyond Taoism - Part 3 A Multidimensional Number System

(continued from here)

Descartes modeled his coordinate system on the Western number line, itself an extension of the decimal number system to include the new negative numbers, and upon the Euclidean notion of a three-dimensional geometry. All these events took place in historical times. In approaching the I Ching and its number system we are dealing mostly with events that took place before recorded history so it is impossible to say with certainty how anything involved came about. We can't so much as be sure whether the I Ching was based on an antecedent number system, or predated and foreshadowed a subsequent number system of Chinese antiquity possibly contingent on it. We view all such things as through a glass, darkly.[1]

It is clear, though, that the number system of the I Ching is one far more complex than that of Western mathematics. The number system of the West is unidimensional (linear). Descartes, in his coordinate system, extends it for use in three dimensions. The number system of the I Ching, on the other hand, is in origin multidimensional. It is mandalic as well, which is to say it consists of multiple dimensions interwoven in a specific manner which can best be characterized as mandalic in form, possessing a number of interlaced and interlinked concentric shells or orbitals about a unifying center.

At the important origin of Descartes' coordinate system is found his triple zero ordered triad (0,0,0). Descartes views this point,[2] as all his points, primarily in terms of location, not relationship. The matter of relationship is left to analytic geometry, the geometry Descartes codified based on his coordinate system.[3] The coordinate system itself seems not to care how points are formed or related beyond the most elementary and trivial operations of addition and subtraction throughout what essentially remain predominantly isolated dimensions.[4] In the end this becomes an effective and prodigious mind snare.[5]

In contrast to the Cartesian approach, the I Ching offers a unified coordinate system and geometry in a single entity which emphasizes the relationship of "points" and other "parts" (e.g., lines, faces) as much, if not more, than location, beginning with wholeness and ending with the same. In between, all sorts of complex and interesting interactions and changes take place. In analyzing these, it is best to begin at the origin of the coordinate system of the I Ching, the unceasing wellspring of being that supplants the triple vacuity of Descartes and Western mathematics.

Section FH(n)[6]

(continued here)

Notes

[1] My thinking is that the I Ching was originally primarily a non-numerical relationship system that subsequently was repurposed to include, as one of its more important contextual capacities, numerical relationships. That said, from a contemporary perspective, rooted in a comprehensive awareness that spans combinatorics, Boolean algebra, particle physics, and the elusive but alluring logic of quantum mechanics and the Standard Model, it would seem that this relationship system is an exemplary candidate for an altogether natural number system, one that a self-organizing reality could readily manage.

[2] As do most geometers who follow after Descartes.

[3] Strictly speaking, this approach is not in error, though it does seem a somniferous misdirection. Due to the specific focus and emphasis enfolded in Descartes' system, certain essential aspects of mathematical and physical reality tend to be overlooked. These are important relational aspects, highly significant to particle physicists among others. These remarks are in no way intended to denigrate Cartesian coordinates and geometry, but to motivate physicists and all freethinkers to investigate further in their explorations of reality.

[4] The Cartesian system neglects, for instance, to express anywhere that the fact the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom, which in mathematical logic

has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one to one correspondence between real numbers and points on a line. This axiom is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes explicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend. [Wikipedia]

Neither mandalic geometry nor the I Ching, upon which it is based, accept this axiom as true in circumstances other than those restrictive settings, such as Cartesian geometry, where it is explicitly demanded as axiomatic to the system. In other words, they do not recognize the described one to one correspondence between number and geometric space as something that reality is contingent on. The assumption contained in this axiom, however, has been with us so long that we tend to see it as a necessary part of nature. Use of the stated correspondence may indeed be expedient in everyday macro-circumstances but continued use in other situations, particularly to describe subatomic spatial relations, is illogical and counterproductive, to paraphrase a certain Vulcan science officer.

[5] For an interesting take on the grounding metaphors at the basis of the real number line and neurological conflation see The Importance of Deconstructing the Real Number Line. Also on my reading list regarding this subject matter is Where Mathematics Comes From:How the Embodied Mind Brings Mathematics into Being (1,2,3) by George Lakoff and Rafael Nuñez. Neither of the authors is a mathematician, but sometimes it is good to get an outside perspective on what is in the box.

[6] This is the closest frontal section to the viewer through the 3-dimensional cube using Taoist notation. See here for further explanation. Keep in mind this graph barely hints at the complexity of relationships found in the 6-dimensional hypercube which has in total 4096 distinct changing and unchanging hexagrams in contrast to the 16 changing and unchanging trigrams we see here. Simple by comparison though this model may be it will nevertheless serve us well as a key to deciphering the line derived from the mandala of I Ching hexagrams, and we will be referring back to this figure for that purpose in the near future.

-Page 298-

Beyond Descartes - Part 10 Taoism Meets Boolean Logic: Introduction

Logic gate symbols

(continued from here)

Before we can hope to comprehend Taoist arithmetic and geometry we need to take a short detour through Boolean logic. First and foremost, we need to see how Boolean logic[1] relates to Cartesian coordinates. That will provide what may be the best foundation available for understanding the Taoist approach to mapping of spacetime and the methodology which mandalic geometry derived from it.[2]

For Descartes, his coordinate system is one thing, his coordinate geometry another. For Taoism, the coordinate system is the geometry.[3] Boolean logic helps to explain how the two perspectives are similar, how different. Cartesian coordinates are static and passive. Taoist coordinates and the derivative mandalic coordinates are active and dynamic. In brief, the latter are changeable and self-changeable, a feat carried out by means of a brand of Boolean logic intrinsic to the system. Although it is true that Descartes' coordinates do encode much the same information, that is not where their focus of interest lies. Accordingly they turn our own attention elsewhere and we overlook those inherent possibilities.[4]

Descartes' geometric system is one based on vectors, that is, on both magnitude and direction. But in the scheme of things, the former has somehow eclipsed the preeminence of the latter in the Western hive mind. The opposite is true of Taoist thought and of mandalic geometry. Direction is uniformly revered as primary and prepotent. Magnitude, or scale, is viewed as secondary and subordinate. This mindset allows the Boolean nuances inherent in the system to come to the fore, where they are more easily recognized and deployed.

From such small and seemingly insignificant differences ensue entirely disparate worldviews.

(continued here)

Notes

[1] George Boole's monumental contribution to symbolic logic was published in 1854 but was viewed as only an interesting academic novelty until the second decade of the twentieth century, when it was at last exhumed as a mathematical masterpiece by Whitehead and Russell in their Principia Mathematica.

[2] In Boolean logic (Boolean algebra) logical propositions are represented by algebraic equations in which multiplication and addition (and negation) are replaced with 'and' and 'or' (and 'not'), and where the numbers '0' and '1' represent 'false' and 'true' respectively. Boolean logic has played a significant role in the development of computer programming and continues to do so.

[3] This is true also of mandalic geometry in its current formulation.

[4] This might be a proper place to proclaim that nature has little use for Descartes' breed of coordinates, finding them far too stagnant and limiting for her purposes. Fortuitously, she devised her own choice coordinate stock long before Descartes thought to invent his.

-Page 293-

Quantum Naughts and Crosses Revisited - II

(continued from here)

This post builds on orientational material offered in the previous post. An explanation of the procedural method of graphic demonstration used in this post and those following can be found there, and it would be helpful to review that earlier post, if not already done, before proceeding further.

Due in part to the challenging subject matter, in part to arduous graphic demonstration, we'll approach this investigation in three stages of progressive difficulty. In the first stage we'll just dangle our feet in the water by looking at how the "slicing methodology" works with ordinary three-dimensional Cartesian coordinates. In the second stage, we'll go waist-deep, and consider the same Cartesian coordinates in their Taoist notation transliteration equivalents. And in the final stage, we'll go for full immersion, with graphic representation of true mandalic geometry, that is, plotting all 64 hexagrams in a hybrid 6D/3D coordinate system using the methodology of composite dimension which, of course, has no analogue in purely Cartesian terms.

At each stage - Cartesian, Taoist transliteration, and mandalic - we'll look at the respective cube in frontal, transverse, and sagittal slices, always in that order and always progressing from identity face containing Cartesian (1,1,1), trigram HEAVEN, or hexagram HEAVEN to inversion face, containing Cartesian (-1,-1,-1), trigram EARTH, or hexagram Earth, as the case may be.

To accomplish our purpose we will require an effective, consistent way to refer to the individual "slices" and each of the 27 Cartesian points. There are three "slices" for each type of sectioning of the "cube", so a total of nine. I propose that we uniquely identify each "slice" by labeling it with the first letter of the section type (frontal, transverse, or sagittal) and the subscript letters "H" for planes containing trigram or hexagram HEAVEN but not Earth, "E" for planes containing trigram or hexagram EARTH but not HEAVEN, and "HE" for planes containing both trigram forms.[1]

The labels of the sections, then, will be:

FH frontal section containing HEAVEN but not EARTH

FHE frontal section containing both HEAVEN and EARTH

FE frontal section containing EARTH but not HEAVEN

TH transverse section containing HEAVEN but not EARTH

THE transverse section containing both HEAVEN and EARTH

TE transverse section containing EARTH but not HEAVEN

SH sagittal section containing HEAVEN but not EARTH

SHE sagittal section containing both HEAVEN and EARTH

SE sagittal section containing EARTH but not HEAVEN

For the 27 individual discretized Cartesian points, I propose the following labeling convention:

Each point is to be first identified as to type. There are four point types: vertex(V), edge center(E), face center(F), and cube center(O). The cube center corresponds to the Cartesian triad (0,0,0), the origin point of the Cartesian coordinate system. In the Cartesian/Euclidean cube there are 8 vertices, 12 edge centers, 6 face centers, and a single cube center. The higher dimensional mandalic cube has many more of each of these.

Vertices

Having identified the point type, each point is then further identified by a subscript consisting of the first letter of the name of trigram or hexagram that is resident at the point. The single exception to this will be WATER. To differentiate between WATER and WIND, I propose using the letter "A" (first letter of "aqua", Latin for "water") to specify WATER. This plan allows us, then, to discriminate among the various vertex points, and also to distinguish them from the other point types. Accordingly, we arrive at these labels for the 8 vertex points:

VH HEAVEN

VE EARTH

VT THUNDER

VW WIND

VA WATER

VF FIRE

VM MOUNTAIN

VL LAKE

Edge centers

Edge centers will be labeled "E" along with a subscript consisting of the first letter of its two vertices, "A" being used instead of "W" for WATER. Though this may initially seem excessively complicated, the reasons for setting things up this way will soon be made clear, and it will all become second nature. The 12 edge centers will be labeled as below:

EHW

EHF

EHL

EET

EEA

EEM

ETF

ETL

EAW

EAL

EMW

EMF

Face centers

There are six face centers. Three occur in identity faces of the cube that contain the trigram or hexagram HEAVEN; three, in inversion faces that contain the trigram or hexagram EARTH. Labeling will be with the letter "F" and a subscript consisting of either "E" for EARTH along with one of its companion diagonal vertices, "W" for WIND, "F", FIRE, "L", LAKE or "H" for HEAVEN, along with one of its companion diagonal vertices, "T" for THUNDER, "A", WATER, "M", MOUNTAIN. So these six face center labels are:

FEW

FEF

FEL

FHT

FHA

FHM

Cube center

The cube center, which is singular in Cartesian terms but a multiple composite in terms of mandalic geometry, will be labeled as:

identifying it as the origin of the coordinate system, that is to say, of both the Cartesian coordinate system and the mandalic coordinate system.

With that, let the games begin!

(continued here)

Notes

[1] There are no sections among those described that include both the hexagram HEAVEN and the hexagram EARTH.

-Page 287-

Quantum Naughts and Crosses Revisited - I

(continued from here)

Because nature is ever playful, grokking mandalic geometry is much like a game. We view it as a largely serious one, though, one that involves combinatorics, Boolean logic, and magic squares and cubes. Groundwork for what appears in this post, and several to follow, was laid in May, 2014 in a series titled "Quantum Naughts and Crosses" which began here.

The game is played on a board or field made of three-dimensional coordinates of the Cartesian variety upon which are superimposed the six additional extraordinary dimensions unique to mandalic coordinates. For convenience and ease of representation, the board will be displayed here in two dimensional sections abstracted from the Cartesian cube and from the superimposed mandalic hypercube in a manner analogous to the way computed tomography renders sections of the human body.

The sections commonly used[1] in computed tomography and magnetic resonance imaging (MRI) are

Frontal

Transverse

Sagittal

For our purposes here, these can be thought of as

Planes perpendicular to the z-axis viewed from front to back of cube

Planes perpendicular to the y-axis viewed from top to bottom of cube

Planes perpendicular to the x-axis viewed from side right to left of cube

These “cuts” will produce square sections through xy-, xz-, and yz-planes, respectively, of the Cartesian cube and, in the case of the mandalic cube, analogous sections of higher dimension.

These choices of sections are made largely for convenience and ease of communicability. They are mainly of a conventional nature.[2] On the other hand, there is special significance in the fact that all three section types progress from identity faces of the cube, containing the trigram or hexagram HEAVEN, to inversion faces, containing trigram or hexagram EARTH. Some manner of consistency of this sort is necessary. The one chosen here will make things easier as we progress.

Our gameboard has 27 discretized Cartesian points, centered in 3 amplitude levels about the Cartesian origin (0,0,0).[3] Each point in the figure on the right above is represented by a single small cube, but in the two-dimensional sections we'll be using for elaboration, they will appear as small squares. So the gameboard is "composed of" 27 cubes arranged in a 3x3x3 pattern. But in descriptions of sections, we will view 9 squares in a 3x3 pattern. This configuration will appear as

But keep in mind each small square in this figure is actually a small cube representing one of the 27 discretized Cartesian points we've described.

Until next time, then.

(continued here)

Notes

[1] The origin of the word "tomography" is from the Greek word "tomos" meaning "slice" or "section" and "graphe" meaning "drawing." A CT imaging system produces cross-sectional images or "slices" of anatomy, like the slices in a loaf of bread. The "slices" made are transverse (cross-sections from head to toes or, more often, a portion thereof), but reconstructions of the other types of sections described above are sometimes made, and MRI generates all three types natively.

[2] Admittedly, I've chosen the convention here myself and to date it is shared by no one else. Perhaps at some future time it will be a shared convention. One can only hope.

[3] These three discrete amplitude levels of potentiality in the mandalic 9-cube correspond geometrically to face centers, edge centers and vertices of the 3-cube of Cartesian coordinates. They are encoded by the six new potential dimensions interacting with the three ordinary Cartesian dimensions in context of the hybrid 6D/3D mandalic cube. They are a feature of the manner of interaction of all nine temporospatial dimensions acting together in holistic fashion. This should begin to give an idea why there is no Taoist line that can generate a 9-cube in a fashion analogous to the way the Western number line is used to generate the Cartesian / Euclidean 3-cube. The 9-dimensional entity is primeval and a variety of different types of "line" can be derived from it. Similarly, the mandala of the I Ching hexagrams cannot be derived from the logic encoded in any linear structure. An overarching perspective is required to derive first the mandala of hexagrams and then from it, a variety of Taoist line types. Nature may be playful, but it is not nearly as simplistic as our Western science, mathematics, and philosophy would have it.

-Page 286-

Beyond Descartes - Part 8: A Good Convention Gone Bad, An Opportunity Missed

Composite Dimension and Amplitudes of Potentiality Episode 2

(continued from here)

We cannot blame Descartes for imaginary numbers. It was he, after all, who christened these numbers "imaginary" due to his disdain for them. We can, however, fault him for his lack of insight into how his coordinate system could be extended to create a viable substitute to show that imaginary numbers and the complex plane were nonsensical and make them unnecessary. Alas, that was not to be. Certain powerful forces of history decreed that imaginary numbers were here to stay and we seem stuck with them still, nearly five centuries later.

Not all would agree that imaginary numbers are a bad convention. We should all, however, be able to agree that they are a convention and nothing more. They were invented by humanity.[1] Mathematics may not have taken to them at first - but did eventually welcome them into its fold for better or worse. The real damage was done when physics did the same without first subjecting the mathematical concepts involved to the kind of scrutiny and empirical review it demands of its own theories.

Where is the proof that imaginary numbers and complex plane in fact apply to the real world and particularly to the subatomic realm? It is lacking in the main, and though the geometric concepts have indeed been successfully applied to a number of branches of physics and explanations of a variety of physical phenomena, the reconciliation is incomplete, the fit an uncomfortable one, and too many mysteries remain unexplained.

The term imaginary unit refers to a solution to the equation x2 = -1. By convention, the solution is usually denoted i. As no real number exists with this property, the imaginary number i extends the real numbers and creates an entirely new and different category of numbers. And crucially, at this point an assumption is made, a rather sweeping assumption. It is assumed that the properties of addition and multiplication we're familiar with - (closure, associativity, commutativity and distributivity) - continue to hold true for this new species of number, or I should say, for this newly derived artificial species of number. That may fly in the ivory tower[2] of pure mathematics, but it lacks the wings and propelling force required to maneuver effectively in the real world that physics investigates. Still, the complex plane, generated by mathematically motivated minds, was soon adopted by physicists the world over.[3]

Mandalic geometry offers an alternative solution in the effective combination of dimensional numbers, composite dimension, and plane of potentiality. We'll take a close look at potential numbers first. Let's see how they stack up against the imaginary numbers, how and where they differ. Distinctions between complex plane and potential plane are subtle but they make for a world - a universe, actually - of difference. When next we meet, kindly check all preconceptions at the door. Entirely untrodden paths await.

(continued here)

Image: (lower left) Imaginary unit i in the complex or Cartesian plane. Real numbers lie on horizontal axis, imaginary numbers on the vertical axis. By Loadmaster (David R. Tribble), (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons; (lower right) A diagram of the complex plane. The imaginary numbers are on the vertical axis, the real numbers on the horizontal axis. By Oleg Alexandrov [GFDL or CC-BY-SA-3.0], via Wikimedia Commons

Notes

[1] Let those who suppose differently, who believe them to be an indelible part of nature itself, prove their case. Until they do, I will see fit to call such numbers manmade inventions.

[2] I use the term ivory tower without malice of any kind in this context, rather judiciously, because mathematics demands no more than internal consistency for its particular brand of truth. It is not much interested in examining its definitions and axioms to determine how they shape up against hard reality. Mathematicians leave that "sordid work" to physicists and philosophers, both of whom are more willing to dig in the mire of nature, seeking its actual relics. Enthusiastically to persist in such a real world-oblivious manner as pure mathematicians do, I think, requires a very special type of mind, one I don't fully understand myself.

[3] In some circles this would be considered no less than a monumental leap of faith, particularly in view of the many unproved assumptions made in creation of imaginary and complex numbers. This was, in fact, the New Faith promulgated by Descartes' contemporaries, the rationalists of the Age of Reason, to supplant the Old Faiths of Religion and Scholasticism.

-Page 284-