Filling Space with Cuboctahedra and Octahedra

Posted on 18 March 2020 by RobertLovesPi

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

Posted on 15 March 2020 by RobertLovesPi

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

Posted on 14 March 2020 by RobertLovesPi

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

Posted on 13 March 2020 by RobertLovesPi

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

Posted on 8 March 2020 by RobertLovesPi

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.

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.

Twelve Pentagrammic Prisms

Posted on 23 February 2020 by RobertLovesPi

I made this by augmentation (with prisms) of the faces of a small stellated dodecahedron, using Stella 4d, which you can try for free at this website.

A Symmetrohedron Featuring Twelve Regular Pentagons, Thirty Regular Octagons, and Twenty Equilateral Triangles

Posted on 18 February 2020 by RobertLovesPi

In addition to the sixty-two regular faces, this polyhedron also has two sets of sixty isosceles trapezoids each, shown in different shades of blue. That’s 182 faces in all. I made it using Stella 4d, which you can try for free at www.software3d.com/Stella.php.