The 24-cell is the convex hull of the D(4) root system, which coincides with
the units in the ring of Hurwitz integers. It is also the only convex,
self-dual, Euclidean polytope which is neither a polygon nor a simplex.
It is therefore a singular example of a regular polytope.

Ghost Symmetry.

This is the first balls-and-straws model I made. One may find the coordinates
of the vertices in Coxeter's Regular Polytopes.

The 24-Cell, with Dodecagonal Ghost Symmetry.

A Zome-Inspired Model.

One may have noticed by now that these models resemble Zome models. In fact,
this project was motivated by a desire to make Zome-like models which are not
Zomeable. One can certainly make this projection of the 24-cell using Zome,
but one cannot arbitrarily choose the colors. The colors where chosen in order
to emphasize 8 of the octahedral cells which correspond to the cubical cells
of the hypercube.

Trihedrally-Symmetric Projection of the 24-Cell.

A Schlegel Diagram.

Suppose P is a point and H is a hyperplane not containing P. Then, given any
vertex V of the 24-cell, the line joining V to P intersects H in a unique point
f(V). Mapping the higher-dimensional faces in the same way, this yields
a Schlegel diagram of the 24-cell. In this type of projection, each
k-dimensional face of the 24-cell is represented uniquely by a k-dimensional
polytope in 3-dimensional space: