The Small Ditrigonal Icosidodecahedron, Together with Its Fifth Stellation

Faceted Dodeca

I made the polyhedron above by performing a faceting of the dodecahedron, and only realized, after the fact, that I had stumbled upon one of the uniform polyhedra, a set of polyhedra I have not yet studied extensively. It is called the small ditrigonal icosidodecahedron, and its faces are twelve star pentagons and twenty equilateral triangles, with the triangles intersecting each other. Below is its fifth stellation, which appears to be a compound of a yellow dodecahedron and a red polyhedron which I do not (yet) recognize, although it does look quite familiar.

5th stellation of the Small Ditrigonal Icosidodeca

Both images were created using Stella 4d, software you can try right here.

Uniform Polyhedra: A Study, Beginning with the Small Ditrigonal Icosidodecahedron

A set of polyhedra which I have not (yet) studied much are the uniform polyhedra. The uniform polyhedra do, however, include some sets of polyhedra which I have studied extensively:

  • The Five Platonic Solids
  • The Four Kepler-Poinsot Solids
  • The Thirteen Archimedean Solids

Subtracting these 22 polyhedra (and the infinite sets of prisms and antiprisms), from the uniform polyhedra, leaves 53 uniform star polyhedra, of which 5 are quasiregular and 48 are semiregular. There’s also one other star polyhedron, only counted sometimes, which is different from the others in that it has pairs of edges that coincide. Discovered by John Skilling, it is often simply called Skilling’s figure. There are also 40 “degenerate” uniform polyhedra; these are generally not counted toward the total. I’ve been aware that these 54 polyhedra existed for years, but was preoccupied with the others. Now, it’s time to fix that.

There is a listing of all 75 (or 76) uniform polyhedra at https://en.wikipedia.org/wiki/List_of_uniform_polyhedra, for those who’d like to examine them as a group. My approach will be different: I’m going to study the ones I don’t already know one at a time, starting with one I picked on the basis of aesthetics alone: the small ditrigonal icosidodecahedron. To be a uniform polyhedron, all vertices must be the same (in other words, it is vertex-transitive), and all faces must be regular, with regular star polygons allowed. In this figure, each vertex has three equilateral triangles meet, as well as three star pentagons, with these figures alternating as one moves around the vertex, examining them.

Small Ditri Icosidodeca

Here are just the twelve star pentagons, with only parallel faces having the same color.

12 STAR PENTAGONS

Here are only the twenty equilateral triangles, with only parallel triangles having the same color. As you can see, the triangles interpenetrate.

20 EQUILATERAL TRIANGLES

At least for me, the reason I had trouble understanding this figure, for so long, was that I mistook the small triangular “facelets” (the visible parts of the faces) for the triangular faces, themselves. In reality, the edges of the triangles are just as long as the star polygon edges. Because it has exactly two face-types which alternate around a vertex, it is edge-transitive (not all uniform polyhedra are), and so this polyhedron is part of smaller subset of uniform polyhedron called the quasiregular polyhedra.

Stella 4d, a program I use to study polyhedra, and make these images, will be the primary tool I use to investigate these uniform polyhedra with which I am not already familiar. It is available at http://www.software3d.com/Stella.php.