On Consistent and Inconsistent Combining of Chiralities, Using Polyhedral Augmentation

For any given chiral polyhedron, a way already exists to combine it with its own mirror-image — by creating a compound. However, using augmentation, rather than compounding, opens up new possibilities.

The most well-known chiral polyhedron is the snub cube. This reflection of it will be referred to here using the letter “A.”

Snub Cube ATo avoid unnecessary confusion, the same direction of rotation is used throughout this post. Apart from that, though, the image below, “Snub Cube B,” is the reflection of the first snub cube shown.

Snub Cube B

There are many ways to modify polyhedra, and one of them is augmentation. One way to augment a snub cube is to attach additional snub cubes to each square face of a central snub cube, creating a cluster of seven snub cubes. In the next image, all seven are of the “A” variety.

Snub Cube seven of them  AA

If one examines the reflection of this cluster of seven “A” snub cubes, all seven, in the reflection, are of the “B” variety, as shown here:

Snub Cube seven of them BB

Even though one is the reflection of the other, both clusters of seven snub cubes above have something in common: consistent chirality. As the next image shows, inconsistent chirality is also possible.

Snub Cube A augmented with B

The cluster shown immediately above has a central snub cube of the “A” variety, but is augmented with six “B”-variety snub cubes. It therefore exhibits inconsistent chirality, as does its reflection, a “B” snub cube augmented with “A” snub cubes:

Snub Cube B augmented with A

With simple seven-part snub-cube  clusters formed by augmentation of a central snub cube’s square faces by six snub cubes of identical chirality to each other, this exhausts the four possibilities. However, multiplying the possibilities would be easy, by adding more components, using other polyhedra, mixing chiralities within the set of polyhedra added during an augmentation, and/or mixing consistent and inconsistent chirality, at different stages of the growth of a polyhedral cluster formed via repeated augmentation.

All the images in this post were created using Stella 4d, which you can try for yourself at this website.

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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