Quantum Naughts and Crosses 12

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Here we have the xz-plane with y = +1 with all its resident hexagrams. This is the face the mandalic cube presents when we view it from directly above, with all of the planar Cartesian coordinate conventions maintained. The z-axis is positive toward the viewer and the x-axis positive toward the viewer’s right. (1)

The xz-plane with y = -1, which is to say the opposite or complementary face of the mandalic cube, could be viewed by simply lifting the roof face above off of the cube and then looking down at what has now become the floor of the cube. Once again, this ploy preserves the Cartesian coordinate conventions. Also, this is the only way to view this opposite face of the cube in such a manner that its tetragrams are all congruent to those in the hexagram patterning pictured above. (2)

We could always view this lower face from a vantage point outside the cube, as we did the upper face, but not without disregarding one of the Cartesian coordinate conventions, that of either the x-axis or the z-axis. We wouldn’t be breaking any laws of nature were we to do that, but some of us would, initially at least, be somewhat confused. My suggestion in that case would be, “Brush up on your Lewis Carroll (3) and in particular on his Looking Glass House.”

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(1) It is good always to keep in mind that nature has neither respect for nor allegiance toward human convention. These artificially fabricated coordinate conventions play no fundamental role in any of the geometric descriptions found in this blog, but the pretense that they somehow do really matter must still be maintained to make consistent communication between human minds possible. I’m betting that more highly advanced alien civilizations and reality itself would view this whole formulation of things in a conventional manner as somewhat quaint.

(2) As used here the term “congruent” refers to the situation in which all the lines of the figures of concern (here all the tetragrams positioned opposite in the vertical y-dimension) are identical. It is the fact that the hexagrams in the opposite planes differ only in the value of y, which is to say in lines 2 and 5, that gives rise to this congruence. The hexagrams of the lower plane can be generated by substituting yin lines for the yang lines at positions 2 and 5 of the hexagrams shown in the diagram above.

(3) In addition to his literary works Lewis Carroll penned a good number of mathematical works under his real name Charles Lutwidge Dodgson as well, mainly in the fields of mathematical logic, geometry, linear and matrix algebra, and recreational mathematics.

© 2014 Martin Hauser