Monthly Archives: March 2020

Filling Space with Cuboctahedra and Octahedra

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To get started packing space with cuboctahedra and octahedra, I started with a single octahedron, then augmented its square faces with additional cuboctahedra.

Next, I augmented each triangular face with a blue octahedron.

Next, I augmented each square face with a cuboctahedron.

Next, I added still more cuboctahedra.

The next step was to augment the yellow triangular faces with blue octahedra.

I next added more cuboctahedra.

This process may be continued without limit. I used a program called Stella 4d to make these models, and you can try this software yourself, for free, at this website.

The 15th Stellation of the Dodecahedron/Icosahedron Compound

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I made this using Stella 4d, software you can try for free at this website.

Golden Triangles and Golden Gnomons

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A Compound of Three Elongated Octahedra

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This compound is the 16th stellation of the tetrakis hexahedron, the Catalan solid which is the dual of the Archimedean truncated octahedron. I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

An Icosahedron, Augmented with Twenty Triangular Cupolae

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To make the polyhedron shown above, I started with an icosahedron . . .

. . . and then I augmented each face with a triangular cupola, with this polyhedron’s hexagonal faces pointed outward.

I made these images using Stella 4d, which you can try for free at this website.

A Rhombic Triacontahedron, Vertices Surrounded By Smaller Rhombic Triacontahedra, and Its Interesting Dual

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The first image shows a central yellow rhombic triacontahedron, with smaller, blue rhombic triacontahedra attached to each of its thirty-two vertices. The second polyhedron shown is the dual of the first one, with colors chosen by the number of sides per face in the second image — pentagons red, and triangles yellow. The convex hull of this second polyhedral complex shown would be an icosidodecahedron, itself the dual of the rhombic triacontahedron.

I use software called Stella 4d: Polyhedron Navigator to make the rotating polyhedral images on this blog. You can try Stella for yourself, for free, at http://www.software3d.com/Stella.php.

My Primary Vote, 2020

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I wonder how many of the other 6,202 people who voted for Bill Weld in the Arkansas Republican primary were doing so simply because they’re allergic to Trump and Trumpism? That reason was good enough for me to proudly say “Republican” when asked which ballot I wanted, even though I more often vote for Democrats in recent years. I don’t pass up a chance to vote against Trump, as directly as possible.

If anyone ever tries to tell me that one vote makes no difference, I’ll refer them to this statistic, along with my previous such vote in 2016. I know exactly how much difference my vote makes.

[Image source: here.]

Tessellation Featuring Rhombi and Concave, Equilateral Dodecagons

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This particular tessellation is full of angles measuring 20 degrees, 40 degrees, and other angles which are not constructable using the traditional rules of Euclidean constructions. This is because this tessellation is based on a matrix which includes regular enneagons.