Enzymind Labs

This past spring Ansel Dow made a nice shape with wood scraps, in shop class after he had finished his main project. I modeled it in Space Software, which only took about 10 minutes -- most of which was starting over because of minor bugs in my software. Here's an animated rotation of the shape, six rectilinear "L's" attached to a central cube:

It has interesting symmetries. Along one of the main diagonals of the central (or bounding) cube there is a three-fold rotational symmetry. A cube has this symmetry about each of its three long diagonals. There is no more symmetric arrangement of these "L's".

In many ways it is the 3-D analogue of the 2-D swastika. Swastikas are interesting for many of the same reasons: they have similar symmetries. Swastikas naturally occur for a variety of reasons (e.g. basketweaving), and tesselate a plane. They are chiral -- their mirror image cannot be rotated to match the original -- so there is a "left" and "right" version of any swastika.

But the 3-D shape is not chiral! At first glance it looks chiral, and all of its 2-D projections are chiral, but the 3-D shape's mirror image is isomorphic to itself. Can it tesselate space? I don't think so. Can a 3-D "+" tesselate space?

[2008-09-05

I recently read Martin Gardener's "Ambitextrous Universe" (1967 edition, very good) that gave me an insight on this apparent strangeness.

Imagine any chiral shape in 2-D, and its enantiomorph (a copy that is mirrored), perhaps a left and right swastika. Stretch both in the third dimension, just a bit to give them depth. Now stack them and glue together. This new 3-D shape is not chiral, because a 180 degree rotation is a symmetry, bringing one half into the other. This works for any 2-D chiral object. A 180 degree rotation through a third dimension is equivalent to a mirror in 2-D.

The same trick can be played with n-D chiral shapes, rotated in (n+1)-D. In 1-D, an arrow (line segment with something distinguishing one end from the other) with one end at 0, is mirrored about 0 by a) 180 degree rotation about 0 through a second dimension, or equvalently b) multiplication by -1, resulting in its enantiomorph.

To Do: Illustrate this.]

Swastikas have come up across cultures through time a large number of ways. I wonder why I've never seen or noticed this 3-D shape. Someone must have used it for a logo. But I can't find a name for it, so I'm calling it an "Ansel star" unless I find an established name.

Its projections are nice too, so I couldn't pass up the chance to try some tilings. I made a cool animated hexagonal tiling (the file is a bit too large to post here) of this shape.