Wednesday, March 7, 2012

Weaving binary distortion isomers from accented undip words

If a flexeez basket is made of Y-shaped joins, its graph is trivalent. If furthermore, it does not contain self-loops, then its graph is a trivalent multigraph. Most trivalent multigraphs are hamiltonian, that is, they have a cycle of edges that visits every vertex without retracing any edge. If the basket is trivalent, hamiltonian, and has spherical topology (genus zero), then it can be represented by at least one undip word (usually many.) If it can be represented by an undip word, then, taking the "all-long" version of that basket as the primal isomer, all of its binary distortion isomers can be represented as an accented undip word in the following way.

Represent a flexeez played "long" by a prime ('), and one played "short" with a star (*). A convention that comes naturally to twog weaving (but is hardly necessary in weaving flexeez) is to consistently take the units added at a given vertex in a counterclockwise, right-to-left order. Notice that at a vertex described by an open letter, two units are added to the work; at a vertex described by a closed letter, one unit is added to the work. After each letter in the undip word, append primes and stars, one for each unit added at that vertex, taken in the counterclockwise, right-to-left order. The first and last letters are anomalous: the first letter adds three units to the work, the last letter adds none. A convenient solution is to append the extra prime or star (needed at the first letter) onto the last letter (where none is needed.) Then we gain the hard and fast rule that open letters have two accents, closed letters have one. (The weaver must look ahead at the accent on the last letter in order to orient the very first piece in the weaving.)

The color convention in the weaving shown in the following photographs is that pink is the first energy color, purple is the second; green is for photons.

Here we are weaving:


A big help in weaving distortion isomers is observing the convention that long plays are played right-side-up, and short plays are played upside-down. With the flexeez brought in in the right order and flipped in the right way, rather less thinking is involved in getting it properly into the work.

Here is the work after the first letter, u'*:

Notice that we read ahead to the asterisk on the last letter, p*, to learn that the first flexeez is played short. Here is the work after the second letter, n'* :

After the third, d' :

After the fourth and final letter, p* :

And the completed binary distortion isomer, u'*n'*d'p* :

The following eight accented undip words suffice for all eight distortion isomers of the tetrahedron:









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