A Pair of Symmetrohedra Derived From the Rhombicosidodecahedron

Posted on 7 October 2023 by RobertLovesPi

To make this first symmetrohedron, I started with the rhombicosidodecahedron, augmented its thirty square faces with antiprisms, and then formed the convex hull of that solid. (I did all of this using Stella 34d: Polyhedron Navigator, which you can try for free right here.) The resulting polyhedron contains, as faces, twelve regular pentagons, twenty equilateral triangles, thirty squares, and sixty isosceles trapezoids, or 122 faces in all.

Next, I applied Stella’s “try to make faces regular” function, which produced the solid seen below. This had the effect of transforming the squares into rhombi, and changing the trapezoids so that, while still isosceles trapezoids, they closely resemble squares.

A Pair of Polyhedra Derived From the Snub Archimedeans

Posted on 4 October 2023 by RobertLovesPi

To make the polyhedron shown above, I started with a snub cube, then augmented all the triangular faces with prisms. Next, I formed this augmented solid’s convex hull, and, finally, I used the “try to make faces regular” function of the software I use to manipulate polyhedra (Stella 4d, which you can try here). The regular faces are the octagons and the small triangles.

I used the same process to make the polyhedron below, except that I started with the snub dodecahedron.

Clusters of Rhombic Triacontahedra

Posted on 2 October 2023 by RobertLovesPi

Here’s a rhombic triacontahedron, possibly the most well-known of the Catalan solids. It’s thirty faces are all golden rhombi (rhombi with their diagonals in the golden ratio).

Here’s what you get if you augment each face with another rhombic triacontahedron:

Coloring this cluster-polyhedron by face type has this result:

So what happens if each face of this cluster-polyhedron is augmented by another rhombic triacontahedron? This does:

Here’s another view of that, in “rainbow color mode”:

I made these virtual models with Stella 4d: Polyhedron Navigator. If you’d like to try this software for free, the website to visit is http://www.software3d.com/Stella.php.

Four Interpenetrating Regular Hexagons

Posted on 22 September 2023 by RobertLovesPi

To make the image above, using Stella 4d (available here), I started with a uniform polyhedron called the octahemioctahedron, and then made the triangular faces of it invisible. Here’s what this polyhedron looks like without its triangles hidden:

The convex hull of either of these figures is a cuboctahedron, which is shown below.

Six Interpenetrating Regular Decagons

Posted on 21 September 2023 by RobertLovesPi

To make the image above, using Stella 4d (available here), I started with a uniform polyhedron called the small icosihemidodecahedron, and then made the triangular faces of it invisible. Here’s what this polyhedron looks like without its triangles hidden:

The convex hull of either of these figures is an icosidodecahedron, which is shown below.

A Zonohedron Derived From the Truncated Octahedron

Posted on 20 September 2023 by RobertLovesPi

This zonohedron is made from the truncated octahedron’s vertices, edges, and faces. I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

A Cluster of Fifteen Cuboctahedra

Posted on 14 September 2023 by RobertLovesPi

There’s one cuboctahedron in the center of this cluster, with each of its fourteen faces augmented by another cuboctahedron. The image below uses coloring of each face by its number of sides, while the next one is in “rainbow color mode.”

I made these images using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

A Compound of a Strombic Icositetrahedron and the Rhombic Dodecahedron

Posted on 13 September 2023 by RobertLovesPi

I made this using Stella 4d, which you can try for yourself, free, at this website: http://www.software3d.com/Stella.php.

A Four-Part Polyhedral Compound Derived From the Small Cubicuboctahedron

Posted on 12 September 2023 by RobertLovesPi

I found this compound by repeatedly stellating the small hexacronic icositetrahedron, which is the dual of the small cubicuboctahedron, one of the uniform solids. I used Stella 4d: Polyhedron Navigator to do this, and you can try this program, as a free trial download, at http://www.software3d.com/Stella.php.

A Truncated Cube, Decorated With Tessellations

Posted on 5 September 2023 by RobertLovesPi

Using Stella 4d (available here), I put this tessellation on the octagonal faces of a truncated octahedron, while hiding the triangular faces. The tessellation itself was made using Geometer’s Sketchpad.