Insufficient space

Having established the equivalence of the three-dimensional Cartesian coordinate system and a similar coordinate system based upon the I Ching trigrams I felt confident that I could proceed with placement of all sixty-four hexagrams in a coherent three-dimensional pattern which would encompass all the symmetries that combinations of the eight trigrams allowed. Immediately I was met with difficulties which initially seemed insurmountable. I hadn’t foreseen the challenges imposed by quantization of a three-dimensional coordinate system. Then there was that other matter of dimension number that had nearly entirely escaped me.

Modeling the sought after pattern on the described coordinate systems I believed that each hexagram should be placeable adjacent to all others that differed from it by one line only and that no adjacent hexagrams should differ in more than a single line. It soon become clear that this is impossible in three dimensions if a point is to be occupied by one and only one hexagram analogous to the manner in which a single point in the three-dimensional coordinate systems is specified by a single identifying coordinate. There are incontrovertibly not enough points in a quantized three-dimensional coordinate system to accommodate 64 hexagrams in the manner described.

© 2012 Martin Hauser