A Great Stellated Dodecahedron, Augmented With More Great Stellated Dodecahedra

Posted on 4 May 2025 by RobertLovesPi

I made this using Stella 4d, which you can try for free here.

A Cluster of a Dozen Icosidodecahedra Surrounding a Dodecahedron

Posted on 25 November 2024 by RobertLovesPi

I made this using Stella 4d: Polyhedron Navigator. If you’d like to try this program yourself, you can get a free trial download at http://www.software3d.com/Stella.php.

A Non-Convex Polyhedral Cluster Featuring Regular Heptagons, With a Puzzle

Posted on 10 February 2024 by RobertLovesPi

I stumbled upon this while playing with Stella 4d: Polyhedron Navigator, a program you can try for free at https://www.software3d.com/Stella.php. It’s difficult to herd regular heptagons.

There’s an obvious question raised by this image: what are the components of the polyhedral cluster, especially the one with heptagonal faces? Comments on this would be appreciated, especially ones with images of the components.

Clusters of Polyhedra Surrounding a Dodecahedron

Posted on 4 November 2023 by RobertLovesPi

The cluster below is formed by augmenting each face of the dodecahedron above with an icosidodecahedron.

Next, each triangular face is augmented with an icosahedron.

Now every pentagonal face is augmented with a dodecahedron.

Finally, the pentagonal faces in this cluster of polyhedra are augmented with icosidodecahedra. The resulting, new cluster has an overall icosahedral shape.

I did all of this using Stella 4d: Polyhedron Navigator. You can try this program for free at http://www.software3d.com/Stella.php.

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.

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.

Repeated Augmentations of a Dodecahedron With More and More Dodecahedra

Posted on 20 April 2022 by RobertLovesPi

Here’s a single dodecahedron.

A new “cluster polyhedron” can be made by augmenting each pentagonal face with another dodecahedron.

If you can do it once, you can do it again, augmenting each pentagon with a new dodecahedron.

Once more.

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

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.

Spectral Icosahedra

Posted on 2 February 2020 by RobertLovesPi

Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 61 icosahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.

Spectral Dodecahedra

Posted on 2 February 2020 by RobertLovesPi

Stella 4d: Polyhedron Navigator has a “put models on vertices” function which I used to build this cluster of 101 dodecahedra. If you’d like to try this software for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.