D(4) Orthotope

An "orthotope" is a union of finitely many closed boxes whose edges are alligned with the usual coordinate axes.  This shape is designed to represent a realization of the Coxeter complex of type D(4) as an orthotope.  This model works well because D(4) has triality.  Due to triality, one may realize a 3-dimensional shadow of the D(4) orthotope as a subdivision of a cube into 23 smaller boxes.  Together, these 24(=1+23) boxes represent the cubes found in the Cayley graph of the D(4) Coxeter group, using the usual set of generators.

The shape also realizes a configuration of 96 points and 96 lines, where each line contains 3 points and each point has 3 lines passing through it.  The points are represented by balls and the lines are represented by bars.