57-cell

Abstract: Both the 11-Cell and the 57-Cell are abstract 4-D polychora (multi-cells) bounded by all identical 3-D “surface” cells. Most people in the computer-graphics community are familiar with the 5 completely regular 3-D polyhedra: the Platonic solids. It is also fairly well known that by using these solids as boundary elements, one can construct 6 different, completely regular polychora in 4-D space (the hypercube, and the 5-, 16-, 24-, 120-, and 600-Cell). Most of these objects have been known for more than a century. The 11-Cell [Grunbaum 1976] and the 57-Cell [Coxeter 1982] are much more recent discoveries. They have received very little exposure outside the hardcore mathematics community ‐ mostly because these two objects cannot be represented in 3-D space in any reasonable way. These objects are highly self-intersecting even in 4-D space, because their boundary cells are single-sided manifolds like a Moebius band or a Klein bottle. [Sequin and Lanier 2007] have tried to make the 11-Cell accessible to nonmathematicians. Here we focus on the even more complicated 57cell and try to foster an understanding of this fascinating object. We start by presenting a visualization of the building blocks that comprise the 3-D “surface” or “crust” of this object.