A Small Grid Embedding of the Dodecahedron

One of the main area of research are grid embeddings of convex 3d polytopes. By Steinitz' theorem every 3-connected planar graph can be embedded as a convex 3d polytope, even with integer coordinates. The question is, how large have the coordinates to be, in terms of the number of the vertices . Although there were a series of improvements over the grid size of Steinitz' construction (of course he was not interested in getting integer or even small coordinates) the current best algorithm uses coordinates of size . At least for triangulations (simplicial polytopes) I expect however that the necessary grid size is bounded by a polynomial. In fact, the current lower bound is only .

This post is about embedding a very special graph as a convex 3d polytope: the graph of the dodecahedron. In other words, what is the smallest grid in which one can embed a (nonregular) dodecahedron. The dodecahedron is interesting since it is the smallest example with no triangular and quadrilateral faces. This seems to be a more tricky case, at least for the current best algorithm that would embed the dodecahedron on the grid with z-coordinates ranging between zero and 11 083 163 098 782 678 334 820 352. Of course we can do better. Francisco Santos found an embedding that fits inside the 8 x 6 x 4 grid. Also, there is a very symmetric realization known as pyritohedron that needs a 12 x 12 x 12 grid.

Continue reading

Vertex-first projections of hypercubes

A few days ago I was answering a question posted on math.stackexchange.com. It was asked what would be the next polytope in the following sequence "Hexagon,Rhombic Dodecahedron,???"?

One possible answer for this question goes along the following lines: Both the hexagon and the rhombic dodecahedron are vertex-first projections of cubes. The hexagon is the projection of the 3-cube, and the rhombic dodecahedron is the projection of the 4-cube. So the next polytope in this sequence would be the vertex-first projection of the 5d cube. (I consider the vertex-first projection that aligns two opposing cube vertices along the normal vector of the projection hyperplane.)

Continue reading