Variations of the Snub Dodecahedron

Convex hull of a triangle-expansion of the snub dodecahedron

To make the first of these variations, above, I augmented each triangular face of a snub dodecahedron with an antiprism 2.618 times as tall as the triangles’ edge length, and then took the convex hull of the result. The other polyhedra shown, below, were obtained by various other manipulations of the snub dodecahedron, all performed using a program called Stella 4d: Polyhedron Navigator, which you can try right here.

expanded snub truncated dodecahedron

The variant above looked like it needed a name, so I called it an expanded snub truncated dodecahedron. As for the one below, it is one of many facetings of the snub dodecahedron.

Faceted snub dodecahedron

Finally, the last figure shown (stumbled upon during a “random walk” with Stella) is one of many possible figures which are non-convex relatives of the snub dodecahedron.

nco thing

A Gallery of Twenty-One Polyhedra with Icosidodecahedral Symmetry


Multiple Variants of the Icosidodecahedron

Click on the smaller pictures, if you wish to enlarge them, one at a time.

convex hull of prismaugmented RTCConvex hull of prismaugmented strombic hexacontahedronConvex hull of reaugmented convex hull of augmented RTCConvex hull qConvex hull z dualConvex hull z

Those last two were duals of each other. The next two are as well.

300-faced dual of 362-faced expanded snub dodecahedron convex hull augmented with 3x prisms362-faced expanded snub dodecahedron convex hull augmented with 3x prismsDual of Convex hullID variant

These next two are duals, as are the pair that follows them.

variant on the SSDdual of variant of SSDpolyhedron xpolyhedron x dual

regularized convex hull of prism-augmented RTCtwisted Convex hullStellated rainbow thingConvex hull

I’ll finish with one more dual pair.


All of these were made using Stella 4d:  Polyhedron Navigator, which is available at

A Bizarre Variant of the Stella Octangula


A Bizarre Variant of the Stella Octangula

The Stella Octangula is another name for the compound of two tetrahedra. In this variant, each triangular face is replaced by a panel of three irregular pentagons. I used Stella 4d to make it, and you can find that program at

An Icosahedron Variant Featuring Kite-Stars


An Icosahedron Variant Featuring Kite-Stars

This variant of the icosahedron has five kites meeting at each of its twelve vertices, forming what I call the twelve “kite-stars” of this polyhedron. Also, two kites meet at the midpoint of each of the icosahedron’s thirty edges. The emplacement of the kites changes the triangular faces of the icosahedron into equilateral, but non-equiangular, hexagons.

Software credit: see to try or buy Stella 4d, the software I used to create this image.