Balls and Straws
The 24-Cell

Introduction.

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:


A Schlegel Diagram of the 24-Cell.









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