A Thought Experiment - XI:

Not just for Flatlanders

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The geometry of one composite dimension is in one sense built from the bigrams. Its four component points (three in ordinary 1-dimensional space) indeed look exactly like the bigrams. There is, however, at least one important difference. The bigrams as shown heretofore refer only to the four quadrants of the xy-plane in Cartesian geometry. The points of all the dimension-composite planes (xy, xz and yz) require an enhanced notation derived from the hexagram notation for consistency and clarity. The significant idea here is that the context of a bigram determines its unique signature. More on this presently.*

It is important to understand that as used above the bigrams refer only to the x-axis, not to the xy-plane. The second (upper) line here is not a y-axis coordinate but rather a second x-axis coordinate in a new kind of geometry derived from composite dimensions. At this point it would be a good idea to review the differences between the real number line and the mandalic number line.

The simple rule for translating the composite coordinates of mandalic geometry into the ordinary coordinates of Cartesian plane geometry is to add the two composite coordinates and divide by 2. This operation will yield one of only three possible results: +1, -1, or 0. The two bigrams that share the zero coordinate of Cartesian space both yield Cartesian x=0 when the stated operation is performed upon them.

Note that this mechanism obviates any need for the vacuous zero of Western mathematics, replacing it with the fully functional and capacious “zero” of Taoist notation. Although the two alternate forms of “zero” here are necessarily shown side by side it should be understood that they are in actuality superpositions of one another at the origin and may be properly shown with either to the right of the other.

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*For the time being, for simplicity and preciseness in introduction of the subject we will continue to use the bigram notation as before, here in reference to the x-axis rather than the xy-plane. We will not need to use the enhanced bigram notation until we arrive at the description of the geometry of two composite dimensions to follow in subsequent posts.

Nevertheless it would be good to keep in mind that every point in mandalic space consists of three ordinary dimensions determined by six extraordinary dimensions and therefore every point requires a six-line designation for full characterization.

Ergo, the four higher dimension points shown in the diagram above if placed in context of three ordinary (or six extraordinary) dimensions would require six lines for unique delineation and appear as hexagrams. Moreover, the bigrams shown refer to the xy-plane and are derived from the first and fourth lines of the hexagram, not from the first and second lines as might be initially intuitively thought proper.

The short explanation for this is that hexagrams are composed of two trigrams, an upper trigram and a lower trigram Each trigram is composed of three lines referring to three dimensions of space, the first to the x-axis dimension, the second to the y-axis dimension and the third to the z-axis dimension. Therefore the two x-axis coordinates of mandalic geometry notation come from the first (lowest) line of the lower trigram and the first (lowest) line of the upper trigram (or put another way as we have, from the first and fourth lines of the hexagram.)

© 2014 Martin Hauser