Tuesday, December 26, 2017

Polyhedral Duality

Blender is 3D modeling software. It has an operation called bevel which can be used to convert a cube into an octahedron. When you are in Edit Mode, type w to bring up the Specials menu.

The operation converts each face of the cube into a vertex and each vertex of the cube into a face. Edges become edges, although the orientation of each edge rotates by 90ยบ.

The f-vector of a polyhedron is a triple containing the number of vertices, edges, and faces. The f-vector of the cube is (8, 12, 6) and the f-vector of the octahedron is (6, 12, 8). If a polyhedron can be converted to another polyhedron by the bevel operation, the f-vector of the first must be the same as the f-vector of the second reversed.

An icosahedron has f-vector (12, 30, 20) . The bevel operation converts it to a dodecahedron with f-vector (20, 30, 12).

The face lattice of a polyhedron shows which edges belong to which faces, and which vertices belong to which edges. Since the cube and the octahedrons are dual polyhedra, the face lattice of an octahedron
is an upside down version of the face lattice of a cube.
For the labels to agree we must map the faces, edges, and vertices of the first lattice to the vertices, edges, and faces second respectively.  The mapping should match the way the bevel operation maps them.

The bevel operation converts a tetrahedron into tetrahedron. The f-vector of the tetrahedron is a palindrome: (4, 6, 4). The tetrahedron is said to be self-dual.

One last note about duality. Dual polyhedra have isomorphic symmetry groups and isomorphic alternating groups.