Cyclic Polytope

This represents a 3-dimensional shadow of the dual of the 4-dimensional cyclic polytope with 11 vertices.  The cyclic polytope is neighborly, implying that the 1-dimensional skeleton is simply the complete graph.  Thus, in the interests of aesthetics, here for your consideration is the dual of the cyclic polytope.

The cyclic polytope is usually defined as the convex hull of a set of points on the moment curve.  In this realization, however, the vertices are taken as uniformly spaced on a pair of unit circles in 4-space.  In so doing, we obtain a shadow which has 11-fold ghost symmetry.  Thus, if one takes this model outside on a bright sunny day and holds it at the right angle, one should be able to see a 2-dimensional shadow with 11-fold symmetry.

In this model, one should be able to identify 11 facets of dimension 3.  Each facet has 2 nonogons, 2 triangles, and 6 quadrilaterals.  Due to the manner of construction, these facets in 4-dimensional space are congruent; however, they are distorted by the projection map.  Since the cyclic polytope is neighborly, every facet of this model is adjacent to every other facet. The face vector of this polytope is (44,88,55,11).