Rhombi, Double Rhombi, and Triple Rhombi

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The Icosidodecahedron, as Well as Two Other Polyhedra Made by Augmenting It

Posted on 1 November 2023 by RobertLovesPi

Here’s the icosidodecahedron — one of the Archimedean solids.

Next, here’s what this solid looks like, if you augment each pentagonal face with a dodecahedron, and each triangular face with an icosahedron.

Finally, here’s what happens if you augment each pentagonal face of this second solid with an icosidodecahedron.

I used Stella 4d: Polyhedron Navigator to create these images. You can try this program, for free, at http://www.software3d.com/Stella.php.

A Zonohedron Based on the Faces, Edges, and Vertices of a Truncated Tetrahedron, as Well as Its Two- and Three-Fold Symmetry Axes, and its XYZ Axes

Posted on 29 October 2023 by RobertLovesPi

I made this polyhedron, which has 170 faces, using Stella 4d: Polyhedron Navigator. You may try this program, for free, at http://www.software3d.com/Stella.php.

A Zonohedron Based on the Faces, Edges, and Vertices of a Truncated Tetrahedron

Posted on 27 October 2023 by RobertLovesPi

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

A Three-Component Compound

Posted on 26 October 2023 by RobertLovesPi

I made this compound using Stella 4d, which you can try for yourself right here.

A Polyhedron With 152 Faces, and Icosidodecahedral Symmetry

Posted on 25 October 2023 by RobertLovesPi

I made this solid using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.softwared.com/Stella.php.

The Octahemioctahedron, and Three Views of Its First Stellation

Posted on 18 October 2023 by RobertLovesPi

This is the octahemioctahedron, which is one of the uniform polyhedra. It has twelve faces: eight equilateral triangles, and four regular hexagons. The hexagons pass through the center of the polyhedron.

Here’s the first stellation of this polyhedron, colored by face type.

Here’s a second view of this first stellation, colored by face, with parallel faces given the same color.

Finally, here’s this first stellation in “rainbow color mode.”

I used Stella 4d: Polyhedron Navigator to make these images. You can try this program for free at http://www.software3d.com/Stella.php.

The Great Dodecahemiicosahedron, With Two of Its Stellations

Posted on 17 October 2023 by RobertLovesPi

Recently (here and here), I blogged about the small dodecahemiicosahedron, along with several of its stellations. Today, I got curious about it’s “big brother,” the great dodecahemiicosahedron, so I looked up the latter with Stella 4d (free trial download available here), examined some of its stellations, and made virtual models of what I found.

First, the great dodecahemiicosahedron itself. The 22 faces of this uniform polyhedron are twelve regular pentagons (shown in yellow) and twenty regular hexagons (shown in red).

Next, its 30th stellation:

Lastly, the 88th stellation:

When creating polyhedral models, I usually use gold spheres for vertices, and silver cylinders for edges. However, in this case, that decoration is getting in the way — so here’s the 88th stellation again, without the spheres and cylinders. This way, it really looks like what it is: a dozen tall pentagonal pyramids meeting only at their common vertex.

More Stellations of the Small Dodecahemiicosahedron

Posted on 11 October 2023 by RobertLovesPi

In the last post here, I showed off the small dodecahemiicosahedron, along with three of its stellations. I’ve now had time to take a deeper dive into this stellation-series, and I found several more stellations to share. First, the 32nd stellation.

Next, the 38th stellation.

The 40th stellation also caught my eye.

Finally, here’s the 51st stellation of this uniform solid.

I made these models using Stella 4d: Polyhedron Navigator, and you can try this program for yourself, free, at http://www.software3d.com/Stella.php.

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.