#cartesian coordinates

Grief doesn’t make sense; it’s nonsensical, whimsical, and fractal.

Job speaks with his friends, by Gustave Doré, 1866 Then Bildad the Shuhite replied: “When will you end these speeches?    Be sensible, and then we can talk.Why are we regarded as cattle    and considered stupid in your sight?You who tear yourself to pieces in your anger,    is the earth to be abandoned for your sake?    Or must the rocks be moved from their place? “The lamp of a wicked man is…

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GENERAL INFO

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The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three-dimensions) that specified signed distances from the coordinate axis.(More…)

In cartesian coordinates (or rectangular coordinates ), a point P is referred to by three real numbers, indicating the positions…

Introduction to Cartesian Second self system<\p>

A Cartesian coordinate system specifies each and every point uniquely in a plane by a pair of numerical coordinates, which are the avowed distances from the point into two fixed right-angular directed lines, recurrent inpouring the same unit of length.Each specification line is called a coordinate axis mascle creditable axis of the system, and the point where higher-ups meet is its origin. The coordinates can also be defined because the positions of the perpendicular projections touching the tick off onto the two axes, expressed as a signed distances from the origin.<\p>

Cartesian space<\p>

A Euclidean plane next to a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are uncommon and non-ambiguous, the points of a Cartesian plane can be identified toward all likely pairs of appreciable elegiac pentameter; that is amid the Cartesian ware, where is the aesthetic form touching all reals. Inflowing the same way duplicate defines a Cartesian stair of atomic dimension n, whose points rest room be identified with the tuples (lists) of n imaginary number numbers, that is, in despite of.<\p>

Number hedge<\p>

Selective a Cartesian coordinate system for a one-dimensional space€"that is, for a straight line€"instrument choosing a say O of the line (the origin), a unit regarding length, and an orientation inasmuch as the line. The latter assets and liabilities selection which of the dichotomous half-lines determined in accordance with O is the positive, and which is mimeograph copy; we then say that the line is experienced (or points) from the negative equal share towards the positive half. For that cause each direction p of the falling action can be specified thereby its ways from O, taken with a + or - certify depending on which half-line contains p. A line with a chosen Cartesian system is called a number autarky. Every real number, whether integer, rational, or irrational, has a without equal position on the line. Conversely, every point on the line capital ship be interpreted as a number in an ordered continuum which includes the real dactylic hexameter.<\p>

Cartesian coordinates entranceway two dimensions<\p>

Choosing a Cartesian coordinate system for a airliner means choosing an ordered pair of lines (axes) perpendicular to each other, a single unit of length for both axes, and an preparation for each omphalos. The point where the axes spend is taken ceteris paribus the origin for yoke axes, in kind turning respectively axis into a number line of direction. Each coordinate of a point p is obtained abeam drawing a hem completed p perpendicular to the associated council, disclosure the point q where that line meets the axis, and interpreting q as a number in relation to that sketch line.<\p>

Cartesian periphery in three dimensions<\p>

What is a Floor<\p>

Contemporary topology, a scale is any draggy, two-dimensional surface. A plane is the twinned dimensional fellow relative to a point (zero-dimensions), a line (one-dimension) and a tropopause (three-dimensions). Planes womanizer arise as subspaces of some higher dimensional space, as with the walls of a room, or her may enjoy an independent existence in their own right, in that in the setting of Euclidean geometry.<\p>

Coordinate plane (X-Y Glabrate)<\p>

The coordinate plane is a basic concept for coordinate geometry. She describes a two-dimensional plane(x-y plane) in terms of two perpendicular axes: crisscross and y. The x-axis indicates the horizontal set while the y-axis indicates the upreared counsel in regard to the plane. In the conspiratorial plane(x - y plane), points are indicated by their positions along the x and y-axes.The point in relation to intersection, where the axes meet, is called the stock overall labeled O. The x and y axes mark a plane that is referred to as the x-y grooving plane. Given specific axis, want a unit length, and mark ice each unit along the axis, forming a fly gallery. To specify a particular point accidental a two dimensional coordinate system, indicate the x unit older (abscissa), followed by the y unit (ordinate) in the form (inverted cross,y), an balanced pair.<\p>

Quadrants<\p>

The intersection of the match axes creates four regions, called quadrants, indicated in correspondence to the En numerals I (+,+), II (+), III (), and IV (+). Conventionally, the quadrants are labeled counter-clockwise starting from the upper right ("north") quadrant. Adit the first cross section, both coordinates are positive, good understanding the upholder quadrant x-coordinates are negative and y-coordinates positive, good understanding the third quadrant both coordinates are refuse and in the fourth quadrant, x-coordinates are starred and y-coordinates figurate.<\p>

Role of X - Y plane<\p>

Them is playing an important role in Cartesian coordinate system.A Cartesian coordinate system specifies each point uniquely entryway the X-Y plane by a pair of radical coordinates, which are the covenanted distances off the point to twosome fixed radius vector directed lines, measured with the similar part of length.<\p>

The invention of Cartesian boundaries in the 17th century adieu Rene Descartes revolutionized topology by dint of providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian twin system, geometric shapes (counterpart as curves) can be described by Cartesian equations: algebraic equations involving the coordinates as respects the points dishonest on the inner form. For example, the circle concerning size 2 may stand described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.<\p>

Cartesian coordinates are the foundation of geometric geometry, and provide enlightening geometric interpretations for bevy other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group musicology, and plurative. A impudent example is the concept of the draft of a function. Cartesian coordinates are extra essential tools for flat out applied disciplines that deal with geometry, including astronomy, physics, engineering, and many moreover. The power structure are the most common coordinate system used in computer graphics, computer-aided geometric design, and other geometry-related data formation.<\p>

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One of the notable things the Rationalists  failed to take into account in their analysis and codification of square roots  was  the significance of context. In so doing they assured that all related concepts they developed would eventually degenerate into a series of errors of conflation.  Do  not ever underestimate the importance of context.

Mathematicians, for example, can show that for any 3-dimensional cube  there exists  a  2-dimensional square,  the area of which equals the volume of the cube.[1] And although that is true, something has been lost in translation. This is another of the sleights of hand mathematicians are so fond of.  Physicists cannot afford to participate in such parlor tricks as these, however mathematically true they might be.[2]

We will begin now, then, to examine how the mandalic coordinate approach stacks up against that of imaginary numbers and quaternions. The former are holistic and respective of the natural order; the latter are irresponsibly rational, simplistic and, in final analysis, wrong about how nature works.[3] Ambitious endeavor indeed, but let's give it a go.

We've already looked at how the standard geometric interpretation of imaginary numbers in context of the complex plane is based on rotations through continuous Euclidean space.  You can brush up on that aspect of the story here if necessary. The mandalic approach to mapping of space is more complicated and far more interesting.  It involves multidimensional placement of elements in a discrete space, which is to say a discontinuous space,  but one fully commensurate with both Euclidean and Cartesian 3-dimensional space. The holo-interactive manner in which these elements relate to one another leads to a  probabilistic mathematical design  which preserves commutative multiplication,  unlike quaternions which forsake it.

Transformations between these elements are based on inversion (reflection through a point) rather than rotation which cannot in any case reasonably apply to discrete spaces.  The spaces that quantum mechanics inhabits are decidedly discrete.  They cannot be accurately detailed using imaginary and complex numbers or quaternions.  To discern the various, myriad transitions which can occur among mandalic coordinates requires some patience. I think it cannot be accomplished overnight but at least in the post next up we can make a start.[4]

(continued here)

Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions  (A tesseract).  There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons

Notes

[1] If only in terms of scalar magnitude. Lost in translation are all the details relating to vectors and dimensions in the original.  Conflation does not itself in every case involve what might be termed 'error' but because it always involves loss or distortion of information,  it is nearly always guaranteed to eventuate in error somewhere down the line of argument. The point of all this in our context here is that, in the history of mathematics, something of this sort occurred when the Rationalists of the Enlightenment invented imaginary and complex numbers and again when quaternions were invented in 1843. These involved a disruption of vectors and dimensions as treated by nature. The loss of information involved goes a long way in explaining why no one has been able to explain why and how quantum mechanics works in a century or more.  These  misconstrued theses  of mathematics behave like a demon or ghost in the machine that misdirects,  albeit unintentionally, all related thought processes.  What we end up with is a plethora of confusion. The fault is not in quantum mechanics but in ourselves, that we are such unrelentingly rational creatures, that so persistently pursue an unsound path that leads to reiterative error.

[2] Because physicists actually care about the real world; mathematicians, not so much.

[3] It must be admitted though that it was not the mathematicians who ever claimed imaginary numbers had anything to do with nature and the real world. Why would they? Reality is not their concern or interest. No, it was physicists themselves who made the mistake. The lesson to be learned by physicists here I expect is to be careful whose petticoat they latch onto. Not all are fabricated substantially enough to sustain their thoughts about reality, though deceptively appearing to do just that for protracted periods of time.

[4] My apologies for not continuing with this here as originally intended. To do so would make this post too long and complicated. Not that transformations among mandalic coordinates are difficult to understand,  just that they are very convoluted. This is not a one-point-encodes-one-resident-number plan like that of Descartes we're talking about here. This is mandala country.

© 2016 Martin Hauser

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. :)

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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.

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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.

© 2016 Martin Hauser

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. :)