The Compound of Three 16-Cells

Introduction. This is a Zome model of a shadow of the regular compound of three 16-cells. In order to better understand this model, it is suggested that the reader first construct both the Zome models of the triality of 16-cells and the 24-cell which has trihedral symmetry. In building this model, it helps to have the triality of 16-cells on hand for study. It also helps to remember that this compound is the first stellation of the 24-cell, somewhat analogous to the compound of two tetrahedra being the first stellation of the regular octahedron.

Crossed-eye and parallel stereograms

Constructing the Model. The pictured model uses 36 long blue struts, 36 long yellow struts, 36 long red struts, 36 medium red struts, 8 orange connectors, 8 green connectors, 8 purple connectors, and 24 white connectors. The colors of the connectors are used for distinguishing the 16-cells. One of the 16-cells has 8 green connectors as vertices, etc.

Start by building an edge-doubled version of one of the 16-cells. It does not matter which of the three you choose. You will eventually make all three. In order to make an edge-doubled 16-cell, you need 8 connectors for vertices and 24 connectors for the midpoints of the 24 edges, 12 long blue struts, 12 long yellow struts, and 12 long red struts, 12 medium red struts. In the model pictured, the 24 midpoints are represented by the 24 white connectors.

Once you have completed an edge-doubled version of one of the 16-cells, mount the other two 16-cells in such a way that the 24 midpoints already present also serve as midpoints for the other two 16-cells. Remember that these 24 connectors coincide with the connectors in the model of the 24-cell with trihedral symmetry. Each of these 24 connectors serve as midpoints for precisely three edges of the 16-cell, one from each.

As you proceed, you will notice some false vertices. In the pictured model, you might be able to see that some of the struts are forced to bend around each other. If you have some extremely small red struts and a few more connectors, then you might choose to represent these intersections by connectors.

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