Wednesday, March 7, 2012

Binary distortion isomers of baskets

In chemistry, a distortion isomer is a chemical compound distinguished from another chemical compound solely by a difference in the length of one or more of its chemical bonds. The incredibly versatile Flexeez construction toy makes it practical to explore this topic as it relates to baskets and surface meshes.

Flexeez go together a zillion different ways, but only one will concern us here: three flexeez can be joined to form a rigid Y-shaped connection around an inner triangle that supplies the rigidity:

It is natural to form such a Y with long arms, but each of the three flexeez in the Y can be be independently flipped over and rotated 90° to become a short arm. This of course alters the shape of the inner triangle as well—thereby changing the relative angles of the arms—but topologically nothing changes:

It proves possible in practice to assemble any genus zero (sphere-like) basket with any or all of its flexeez flipped over from "long" to "short."

I call flexeez baskets that differ only in the way their flexeez are played (long or short) binary distortion isomers of each other. Binary because there are just two options for bond length when weaving with flexeez. Note that whether a given edge has been played "long" or "short" is a matter of perspective. For example, a basket in which all flexeez have been played short, could also be seen as the dual basket in which every flexeez has been played long. If some particular isomer is distinguished as the primal isomer the confusion is eliminated.

A fully unsymmetrical, or chiral, basket with n edges (i.e., using n flexeez) has 2^n distortion isomers (counting the original.) By that count, the tetrahedral basket, having 6 edges, would have 2^6 = 64 binary distortion isomers. But symmetry reduces the number of distortion isomers having distinct shapes. Being highly symmetrical, the tetrahedral basket has binary distortion isomers in only 8 (not 64) different shapes, as shown in the photo above.

Here are the four binary distortion isomers of the theta graph on the sphere:

Here are the six binary distortion isomers of the monogonal prism:

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