Dual Compound

Posted on 13 December 2014 by RobertLovesPi

The snub cube and its dual make an attractive compound. Since the snub cube is chiral, its chirality is preserved in this compound.

If you examine the convex hull of this compound, you will find it to be chiral as well.

Here is the mirror image of that convex hull:

These two convex hulls, of course, have twin, chiral, duals:

The two chiral convex hulls above (the red, blue, and yellow ones), made an interesting compound, as well.

This is also true of their chiral duals:

I next stellated this last figure numerous times (I stopped counting at ~200), to obtain this polyhedron:

After seeing this, I wanted to know what its dual would look like — and it was a nice polyhedron on which to end this particular polyhedral journey.

I make these transformations of polyhedra, and create these virtual models, using a program called Stella 4d. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.

A Faceting of the Snub Dodecahedron

Posted on 6 August 2014 by RobertLovesPi

The snub dodecahedron is chiral, meaning it appears in left- and right-handed forms. This faceted version, where the same set of vertices is connected in different ways (compared to the original), possesses the same property.

Chiral polyhedra can always be tranformed into interesting polyhedral compounds by combining them with their own mirror-images. If this is done with the polyhedron above, you get this result, presented with a different coloring-scheme.

Both of these images were created using Stella 4d: Polyhedron Navigator, and you may try it at www.software3d.com/Stella.php.

Two Chiral Polyhedra

Image

Two Chiral Polyhedra

To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.

A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.

These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.