A Gallery of Fourteen Polyhedra with Cuboctahedral Symmetry

Any of these rotating polyhedra may be made larger with a single click. All were created using Stella 4d, a program you may try for free at this website: 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.

Icosa showing tet symm

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.

Icosa showing tet symm stellation 1

The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.

Icosa showing tet symm stellation 2 pyritohedral dodecahedron

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.

Icosa showing tet symm stellation 3

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.

Icosa showing tet symm stellation 4 an octahedron with 2 face types

Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.

Icosa showing tet symm stellation 5

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:

Icosa tet sym stellation 5's 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.

Icosa tet sym stellation 5's Convex hull ist stellation

Eleven Convex, Non-Chiral Polyhedra Featuring Cuboctahedral Symmetry

To enlarge any of these images, simply click on the ones you choose.

All of these images were created using Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.

87 Rotating Non-Convex, Non-Chiral Polyhedral Images Featuring Icosidodecahedral Symmetry, Plus Four Which Snuck In with Cuboctahedral Symmetry — Can You Find All the Intruders?

To see larger versions of any of these, simply click on the images.

24 to this point….

That’s 40 so far…

Now the count is at four dozen.

That was 26 more, so there are 48 + 26 = 74 so far.

Now the count is up to 83.

So there were 91 of these stored on my hard drive, from all my “hard work” playing with polyhedra using Stella 4d: Polyhedron Navigator. (It will be good for my computer to get all that hard drive space back!) If you’d like to try playing with the same program — for free — just try the free download at http://www.software3d.com/Stella.php.

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

Compound of Icosidodeca and Trunc Cube

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.

Compound of RTC and Triakis octahedron also pyritohedral

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.