#3-cube
Quantum Naughts and Crosses Revisited - VII
The Cube Sliced and Diced
Cartesian Series: Sections SH, SHE, SE
(continued from here)
Below are the three sagittal sections of the Cartesian 3-cube. All three have nine distinct ordered triads, one located at each discretized Cartesian spatial locus. SH and SE sections have four vertices, four edge centers and one face center as did the FH, FE, TH, and TE sections seen earlier. The SHE section contains four edge centers and four face centers and also, as its central point, the single cube center, as did the FHE and THEsections.[1]
These are all Cartesian yz-planes, seen in three-dimensional context at different x-values. For SH, x = +1. For SHE, x = 0. For SE, x = -1.
The key to labeling of points in these sections and all those to follow can be found here.
Section SH
Section SHE
Section SE
Having completed our survey of sections of the Cartesian 3- cube, we are now ready to view the Taoist transliteration equivalents. Almost. But first …
(continuedhere)
Notes
[1] See here for comment regarding the spurious appearance of edge and face centers in section FHE. That comment applies to the SHE and the THE sections as well. The deceptive appearance can be described as being an initially misleading artifact of the sectioning methodology.
© 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
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Quantum Naughts and Crosses Revisited - VI
The Cube Sliced and Diced
Cartesian Series: Sections TH, THE, TE
(continued from here)
We see grouped together below the three transverse sections of the Cartesian 3-cube. All three have nine distinct ordered triads, one located at each discretized Cartesian spatial locus. Both TH and TE sections have four vertices, four edge centers, and one face center as did the FH and FE sections seen earlier. The THE section contains four edge centers and four face centers, and also, as its central point, the single cube center.[1] These are all Cartesian xz-planes, seen in three-dimensional context at different y-values. For TH, y = +1. For THE, y = 0. For TE, y = -1.
The key to labeling of points in these sections and all those to follow can be found here.
Section TH
Section THE
Section TE
In the next post we’ll look at the three sagittal sections of the Cartesian 3-cube.
(continuedhere)
Notes
[1] See here for comment regarding the spurious appearance of edge and face centers in section FHE. That comment applies to the THE section as well and also to the SHE section we will view in the next post.
© 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 291-
Quantum Naughts and Crosses Revisited - V
The Cube Sliced and Diced
Cartesian Series: Section FE
(continued from here)
We see below the third and final frontal section of the 3-cube. The FE section is composed of nine distinct Cartesian ordered triads, with four vertices, four edge centers, and one face center like the FH section seen earlier. All x- and y-coordinates are identical to those in the FH section and in identical relative position. All the z-coordinates here have a vector direction of -1 instead of +1. In other words, this is a Cartesian xy-plane placed in three-dimensional context with z value of -1.
The key to labeling of points in this section and all those to follow can be found here.
Section FE
Next up, we begin a survey of transverse sections of the Cartesian 3- cube.
(continuedhere)
© 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 290-
Quantum Naughts and Crosses Revisited - IV
The Cube Sliced and Diced
Cartesian Series: Section FHE
(continued from here)
Below we have the second of three frontal sections through the 3-cube, labeled with the Cartesian coordinates of each point. This “slice” is through a plane that lies between an identity face, which contains the trigram HEAVEN, and an inversion face with the trigram EARTH. As such it does not belong fully to either the one or the other, but it shares some characteristics of both. It is a plane, then, of mediation. Again we see here nine Cartesian ordered triads. Due to an artifact of the “slicing” procedure, the four edge centers deceptively appear as though vertices, and the four face centers could be taken as edge centers. Make note that these appearances are illusory. At the center of this section we have the origin point of the cube, Cartesian (0,0,0).[1]
The key to labeling of points in this section[2] and all those to follow can be found here.
Section FHE
(continuedhere)
Notes
[1] It might be well to note here that the origin point of the coordinate system never appears in either an identity plane or an inversion plane of any of the three section types. All of the planes in which it appears are mediation planes of three dimensions in the case of the Cartesian 3-cube, or of six dimensions in the case of the hybrid mandalic 6D/3D hypercube. This is likely the rationale for why in the I Ching a change involving passage through this central point is referred to as "crossing the Great Water.“ There must be more than coincidence in the fact that Western thought refers to this point as the "origin” and Taoist thought views it as the source and beginning of all things. It’s not that something important was lost in translation. The two notions arose independently, from two very different worldviews. Somehow in the scheme of things, the West came to equate “origin” with "zero" whereas the East came to equate "origin" with “the beginning and end of all things.” Taoism, in particular, sees in this a focus of both creation and dissolution. As we shall soon enough discover, this alternative perspective leads to a different species of arithmetic, one of great antiquity though long lost in the sands of time. Mandalic geometry has unearthed it and will reveal it here, in this blog, for the first time in millennia. As a teaser, it involves a different treatment of what the West calls “zero”. It is an arithmetic more in line with Boolean logic.
[2] The 2-dimensional version of this section is obtained from the x and y coordinates, which by convention are the first and second, respectively, in the Cartesian ordered triads seen here. So the only difference between this section and the FHsectionpreviously viewed is the fact that the z-coordinates here are all zero (0) instead of +1. In our next section, FE, the x and y coordinates will again be as seen here but all z-coordinates will be -1. I believe I detect a trend developing here.
© 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 289-
