A Compound of the Platonic Octahedron and a Pyritohedral Dodecahedron

The red component of this compound is one of an infinite number of possible pyritohedral dodecahedra. It’s shown by itself in the image below.

I made this compound using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.

A Zonohedron Based on the Vertices of a Snub Cube

This zonohedron has 432 faces, and is shown here with two different coloring-schemes — coloring faces by number of sides (above) and rainbow color mode (below). I made these using Stella 4d, which you can try for yourself, for free, at this website: http://www.software3d.com/Stella.php.

A Zome Rhombicosidodecahedron, and Some of Its Variations

Here’s a Zome rhombicosidodecahedron, made up of white Zomeballs and short blue struts.

If you replace each of the rhombicosidodecahedron’s thirty squares with golden rectangles in a certain way, by replacing certain short struts with medium struts along the triangle/rectangle edges, you get a Zomeball made of Zome — what some have called a “metazomeball.” It has enlarged triangles, compared to the pentagons.

It’s almost possible to augment each of the Zomeball’s 62 faces with all-blue pyramids. Here’s the attempt.

The Zomeball’s twelve pentagons are augmented with pyramids, using medium struts, so that each of these pyramids’ lateral faces is a golden isosceles triangle. The twenty triangles are also augmented by pyramids, with golden isosceles triangles as lateral faces. This requires the use of long blue struts. As for the augmentation performed on the thirty golden rectangles, that’s a bit more complicated.

Here’s what the golden rectangles are augmented with, with the new pieces all being short blue struts. It isn’t quite a pyramid, nor is it quite a prism. Its lateral faces are two equilateral triangles and two golden isosceles trapezoids. It’s a five-faced polyhedron in need of a good name. Until someone comes up with a better name, then, I’m going to call it a golden pentahedron.

Here’s a close-up of a golden isosceles trapezoid, by itself. It’s made of three short struts and one medium strut.

It’s also possible to do the first change in another way. When the rhombicosidodecahedron has its squares replaced with golden rectangles, rotate the golden rectangles 90 degrees, such that it increases the size of the pentagons, rather than the triangles. This has been called an “antizomeball.”

With the antizomeball, augmentation with pyramids is possible with all 62 faces.

Here’s what the three types of pyramid used in the augmentation of the antizomeball look like, separated from the main structure.

With the antizomeball, the pentagonal pyramids’ bases are made of medium struts, and long struts make the five lateral faces into golden isosceles triangles. The triangular faces of the antizomeball are made of short struts, so medium lateral edges for these pyramids make their lateral faces golden isosceles triangles. The golden rectangles in the antizomeball are different from the ones in the Zomeball, and the changes make it possible to augment the antizomeball’s golden rectangles using four medium lateral edges, forming two golden isosceles triangles and two equilateral triangles as the lateral faces of a true pyramid.

If you’d like to try Zome for yourself, which I strongly recommend, the site to visit is http://www.zometool.com.

A Symmetrohedron Featuring Four Regular Enneagons, Four Regular Hexagons, and Six Pairs of “Bowtie” Isosceles Trapezoids

I made this twenty-faced polyhedron using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.