A Thought Experiment - XVII:

Divergence and convergence in mandalic geometry:

5- Restitution of mandalic coordinates

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We’ve noted previously how Cartesian coordinates are derived from mandalic coordinates. Here we will consider briefly the manner in which mandalic coordinates can be restored from Cartesian coordinates. Unlike the earlier situation where each 6-dimensional mandalic coordinate or hexagram yielded in 3-dimensional space a single Cartesian coordinate triplet*, each Cartesian coordinate triplet can reconstitute 1, 2, 4, or 8 hexagrams depending upon the number of zeros** it contains.

To reaffirm, this asymmetric relation arises because all 3-dimensional Cartesian coordinates are degenerate forms of the 6-dimensional mandalic coordinates described by the corresponding hexagrams. Put another way, the sharing hexagrams together contain more information than the single Cartesian point to which they refer. There is an inevitable information loss in translation from mandalic to Cartesian coordinates. This important fact has far-reaching consequences in any attempt to understand quantum structures and processes.

In order to reinstate the hexagrams or mandalic coordinates from their Cartesian counterparts use must be made of combinatorial mathematics to fill in the informational gaps.*** The formula is as follows: Any x-, y- or z-coordinate that is +1 becomes a solid line and any x-, y- or z-coordinate that is -1 becomes a broken line. In both cases doubling of the resulting line occurs in the hexagram. The x-coordinate becomes lines 1 and 4; the y-coordinate, lines 2 and 5; and the z-coordinate, lines 3 and 6. If on the other hand a zero coordinate is encountered, whether in the x-, y-, or z-coordinate or any combination of these it is translated into two lines in the hexagram, one solid, the second broken and in all combinatorial possibilities (two for one zero, four for two zeroes, eight for three zeros.) 

Image: The 64 hexagrams of the I Ching (King Wen arrangement). By TarcísioTS at pt.wikipedia [Public domain], from Wikimedia Commons.

Note that this is a 2-dimensional presentation of the I Ching hexagrams not the 6-dimensional arrangement applied in mandalic geometry. For the 6-dimensional patterning and a description of how it is generated see the series of blog posts beginning here.

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*In 3-dimensional Cartesian space every point is uniquely defined by a triplet of Cartesian coordinates. From the standpoint of 6-dimensional mandalic geometry though the same point more often than not cannot be uniquely defined by the Cartesian triplet, as the triplet is the degenerate form of a mandalic sextuplet or hexagram, so comprising less information.

**In situations where a Cartesian triplet contains one or more zeros the corresponding Cartesian coordinate location will be shared by two or more hexagrams, the total number of sharing hexagrams being equal to 2 raised to the nth power where n equals the number of zeros in the 3-D Cartesian triplet. In other words, since a Cartesian triplet may contain 0, 1, 2, or 3 zeros it can translate to 1, 2, 4, or 8 hexagrams in accordance with a specific and unique distribution pattern which itself constitutes the mandalic form and which has been described previously in this blog. That distribution pattern can be likened to a probability wave in six dimensions and would have implications important to quantum theory.

***Though only making use of an elementary range of combinatorial members, numbering 1, 2, 4, or 8 depending as noted on the number of zeros present in the specific Cartesian triplet in question.

© 2014 Martin Hauser