After building the model I used to make the Polyhedral Waltz animation, I wanted to use it for other things. I was particularly interested in trying the scaling feature of the Keyframe Animation plugin that I had not yet used.

One of my colleagues commented on seeing *Polyhedral Waltz* that the video showed dual polyhedra in a way that was more helpful than a static model, and I thought that with scaling, videos could show duality even better.

A *dual* is formed from a polyhedron by creating a vertex at the center of each face, and then connecting vertices on adjacent faces. This process is hard for most people to visualize without a model, and even with a static model, it can still seem abstract.

Here are three videos, in the order I made them: cube and octahedron, icosahedron and dodecahedron, and tetrahedra. These videos show pairs of dual polyedra, growing or shrinking one-at-a-time to fit inside or to surround the other.

Making the videos, I was surprised by the differences in scale between the three pairs of dual polyhedra. In the past, when thinking about duals, I usually focused on combinatorial properties — that the cube has 8 vertices and 6 faces, while the octahedron has 8 faces and 6 vertices, etc. — rather than on their geometric properties, such as how the size of a polyhedron compares to the size of its dual.

For the cube and octahedron, the scale factor is \( \dfrac{1}{\sqrt{3}} \approx 0.577\). At the start of the video, the cube and octahedron have the same radius, and then the octahedron shrinks to have a radius about 57.7% as large, until it fits inside the cube. Then the cube’s radius shrinks to about 57.7% of its previous value, and the two have the same radius again. Then the cube’s radius shrinks again to about 57.7% of its value, and the cube is inside the octahedron. Later in the video, when the radii start growing, the two steps are combined, so the octahedron first grows by a factor of \((\sqrt{3})^2=3\), and goes from being inside the cube to surrounding it.

For the icosahedron and dodecahedron, the scale factor is approximately 0.79465; the exact scale factor is a complicated expression involving the golden ratio. Thus, the figures shrink by to be about 79.5% as large at each step, making it possible for more steps to fit in the video than in the cube/octahedron video. When the figures grow from inside to surrounding each other, the scale factor is about 1.58.

For the two tetrahedra, the scaling is much sharper. Shrinking at the beginning, the factor is \(\dfrac{1}{3}\) or to about 33.3% of the current radius. When the figures grow from inside to surrounding each other, the scale factor is 9. Thus, the tetrahedra video is the shortest of the three, as the figures quickly become too small or too large to see on the same screen.

Volume scales as the cube of the scale factor for the radius. Thus, when the video shows one tetrahedron inside the other, the volume of the outer tetrahedron is 27 times the volume of the inner tetrahedron.

I used coordinates to figure out the scale factors. This is a nice 3D geometry exercise; it’s good to be able to use SketchUp for immediate feedback on the calculations.

For my next projects, I’ve been exploring the duals of some Archimedean Solids, which are made from two or more different types of regular polygons.