Schlegel Polyhedra for Regular Polytopes

We can create Schlegel polyhedra for the regular four-dimensional polytopes by means of central projection from four-space to three-space, the analogue of central projection from three-space to the plane. The Schlegel polyhedron of the hypercube is the cube within a cube, with corresponding vertices connected. Just as the Schlegel diagram of the tetrahedron is a triangle with its vertices connected to a central point, the Schlegel polyhedron of the four-simplex is a tetrahedron with its vertices connected to a central point. The six triangles joining this central point to the edges of the large tetrahedron divide the interior of the tetrahedron into four somewhat flattened triangular pyramids.

The Schlegel polyhedron for the 16-cell dual to the hypercube, composed of 16 tetrahedra, is similar in form to the Schlegel diagram of the octahedron dual to a cube. Instead of a triangle within a triangle, the Schlegel polyhedron of the 16-cell is a tetrahedron inside another rotated tetrahedron. Every vertex of the inner tetrahedron is connected to the three closest vertices on the outside tetrahedron, thus giving four more of the tetrahedra in the 16-cell, and four additional tetrahedra come from joining a vertex of the outer tetrahedron to the closest triangle of the inner tetrahedron. The remaining six faces are obtained by joining an edge of the inner tetrahedron to the closest edge on the outer one.

In the Schlegel diagram of the self-dual 24-cell, the vertices are arranged in three nested polyhedra: a large octahedron corresponding to the face closest to the viewing point, a small octahedron inside, and between the two a polyhedron called a cuboctahedron. The eight triangular faces and six square faces of the cuboctahedron are obtained by cutting off the corners from a cube all the way to the midpoint of each of the cube's 12 edges. Each of the eight triangles is a face of two octahedra, one with a triangular face on the outside octahedron and one with a triangular face on the inside octahedron. That accounts for 18 of the octahedra in the 24-cell. The remaining 6 octahedra are determined by joining a square face of the cuboctahedron to one vertex on the outer octahedron and one vertex on the inner octahedron. Similar presentations are possible for the 120-cell and the 600-cell, but their many vertices make the diagrams difficult to interpret. It is possible to construct stick models of the Schlegel polyhedra of the regular polytopes and to investigate them by turning them around in three-space. The models of Paul R. Donchian are the most famous physical constructions of such objects.

A single photograph of a Schlegel polyhedron can be quite confusing. As far back as the last century, mathematicians experimented with stereoscopic pairs of geometric objects, so that the left eye received one perspective image and the right eye received an image from a slightly different viewpoint, creating a three-dimensional effect. Although this technique is still useful in studying complicated configurations, the most effective way to see objects in three-space is to walk around them and record the separate images to form an animated film.

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