#magic square

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:

• F_H     frontal section containing HEAVEN but not EARTH
• F_HE   frontal section containing both HEAVEN and EARTH
• F_E     frontal section containing EARTH but not HEAVEN
• T_H    transverse section containing HEAVEN but not EARTH
• T_HE   transverse section containing both HEAVEN and EARTH
• T_E     transverse section containing EARTH but not HEAVEN
• S_H     sagittal section containing HEAVEN but not EARTH
• S_HE   sagittal section containing both HEAVEN and EARTH
• S_E      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:

• V_H  HEAVEN
• V_E   EARTH
• V_T  THUNDER
• V_W WIND
• V_A  WATER
• V_F   FIRE
• V_M  MOUNTAIN
• V_L   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:

• E_HW
• E_HF
• E_HL
• E_ET
• E_EA
• E_EM
• E_TF
• E_TL
• E_AW
• E_AL
• E_MW
• E_MF

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:

• F_EW
• F_EF
• F_EL
• F_HT
• F_HA
• F_HM

Cube center

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

• O

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!

(continuedhere)

Notes

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

© 2015 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. :)

-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 tomographyandmagnetic 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.

Ourgameboard 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.

(continuedhere)

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.

© 2015 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. :)

-Page 286-