An Icosahedron, Augmented with Twenty Triangular Cupolae, Together with Its Dual

Icosahedron Augmented with Triangular Cupolae

After making the above polyhedron using Stella 4d (a program you can try for free at, I checked its dual, which is shown below. I was surprised at its appearance, for it resembles a stellated polyhedron, even though it was created by a completely different process.

Icosahedron Augmented with Triangular Cupolae's Dual




Regions between close-packed circles of equal radius resemble triangles, but with 60 degree arcs replacing the sides. As these regions are the only things left of a plane after all such circles are sliced out, and they each are outside all the circles used, I’ve decided to name them “circumslices.” Interestingly, the three interior angles of a circumslice each asymptotically approach zero degrees, as one approaches circumslice-vertices, which are also the points of contact of the circles.

Why did I name these things “circumslices?” Because they needed a name, that’s why!