Platonic Solids and Duality

I recently came across a post on odForce about hexagonal meshes. After searching info on this subject, I came across the concept of polyhedrons. This stuff is pretty interesting. Polyhedron in Greek means "many faces", so its just a 3-dimensional shape with many faces. There are special polyhedrons called Platonic Solids.


The image above are the 5 Platonic Solids and their "duals". Only 5 of them exist. All angles in these special polyhedrons have equal faces, angles, and edge lengths.
The dual is when you change each vertex to a face. And when this happens on a platonic solid, the number of points and primitives gets swapped.

In Houdini, just use a Platonic Solids SOP to get them.


The last two types, Soccer Ball and Utah Teapot, are technically not platonic, but they have been used so much in computer graphics, that they have become adopted!

To get the dual, use a Divide SOP with "Convex Polygons" unchecked and "Compute Dual" checked.

If you go to Wikipedia, and search polyhedron or platonic solids, you can get alot deeper than this.

Anyway, to the hexagonal mesh thing, here is a way to do it. Take a grid, then use a PolyBevel SOP with "Relative Inset" set to 1, then the Divide SOP with Compute Dual and keep Convex Polygons checked.

You end up with a nice honeycomb shape! You can then go further and use PolyWire, PolyExtrude, etc. and do some neat things with it.