# Two Non-convex Polyhedra with Cuboctahedral Symmetry

I made both of these while playing around with Stella 4d: Polyhedron Navigator, a program you can try out for free at http://www.software3d.com/Stella.php.

# A Tetrahedral Exploration of the Icosahedron

Mathematicians have discovered more than one set of rules for polyhedral stellation. The software I use for rapidly manipulating polyhedra (Stella 4d, available here, including as a free trial download) lets the user choose between different sets of stellation criteria, but I generally favor what are called the “fully supported” stellation rules.

For this exercise, I still used the fully supported stellation rules, but set Stella to view these polyhedra as having only tetrahedral symmetry, rather than icosidodecahedral (or “icosahedral”) symmetry. For the icosahedron, this tetrahedral symmetry can be seen in this coloring-pattern. The next image shows what the icosahedron looks like after a single stellation, when performed through the “lens” of tetrahedral symmetry. This stellation extends the red triangles as kites, and hides the yellow triangles from view in the process. The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process. The third tetrahedral stellation of the icosahedron produces another pyritohedral figure, which further demonstrates that pyritohedral symmetry is related to both icosidodecahedral and tetrahedral symmetry. The fourth such stellation produces a Platonic octahedron, but one where the coloring-scheme makes it plain that Stella is still viewing this figure as having tetrahedral symmetry. Given that the octahedron itself has cuboctahedral (or “octahedral”) symmetry, this is an increase in the number of polyhedral symmetry-types which have appeared, so far, in this brief survey. Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found. While I was curious about what would happen if I continued stellating this polyhedron, I also wanted to see this fifth stellation’s convex hull, since I could already tell it would have only hexagons and triangles as faces. Here is that convex hull: For the last step in this survey, I performed one more tetrahedral stellation, this time on the convex hull I had just produced. # Open Octahedral Lattice of Cubes and Rhombicosidodecahedra

This pattern could be continued, indefinitely, into space. Here is a second view, in rainbow color mode, and with all the squares hidden. [These images were created with Stella 4d, software you may buy — or try for free — right here.]

# Six Non-Convex Polyhedra with Cuboctahedral Symmetry

Each of these polyhedral images (any of which may be enlarged with a click) was created using Stella 4d: Polyhedron Navigator, and this program may be tried for free at http://www.software3d.com/Stella.php.

Also, a question, for regular readers of my blog — you have probably noticed that this post has a different format, but it’s just an experimental thing I’m trying out.

Do you prefer this style of polyhedra-post, or the format I usually use?

# Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball. A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.

# Combining Octahedral and Icosahedral Symmetry to Form Pyritohedral Symmetry Pyritohedral symmetry, seen by example both above and below, is most often described at the symmetry of a volleyball: [Image of volleyball found here.]

To make the rotating polyhedral compound at the top, from an octahedron and an icosahedron, I simply combined these two polyhedra, using Stella 4d, which may be purchased (or tried for free) here.

In the process, I demonstrated that it is possible to combine a figure with octahedral (sometimes called cuboctahedral) symmetry, with a figure with icosahedral (sometimes called icosidodecahedral) symmetry, to produce a figure with pyritohedral symmetry.

Now I can continue with the rest of my day. No matter what happens, I’ll at least know I accomplished something.