Schlegel diagram

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Examples colored by the number of sides on each face. Yellow triangles, red squares, and green pentagons.

A hypercube projected into 3-space as a Schlegel diagram. There are 8 cubic cells visible, one in the center, one below each of the six exterior faces, and the last one is inside-out representing the space outside the cubic boundary.

A 120-cell with edges projected onto a hypersphere, stereographic projection to 3-space

In geometry, a Schlegel diagram is a special projection of a polytope down one dimension. It projects polyhedrons to a plane figure, and polychorons to 3-space. It is used as a visual aid in seeing the topological connectivity of the polytope edges.

It can be constructed by a perspective projection viewed from a point outside of the polytope, above the center of a facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.

The simplest way to guarantee this projection results in nonoverlapping edges on a general convex polytope is to first project all the vertices onto an n-sphere, and then perform a stereographic projection. The edges can appear curved in the final diagram if they are also mapped onto the n-sphere.

A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets, and unfolding until the facets can exist on a single hyperplane. This maintains the geometric scale and shape, but makes the topological connections harder to see.

[edit] References

Victor Schlegel (1843-1905), (German) Theorie der homogen zusammengesetzten Raumgebilde, Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden, 1883. [1]
Victor Schlegel, Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper, Waren, 1886.
Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948). (p. 242)
- Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
Branko Grünbaum, Grünbaum, Branko (2003). Convex polytopes. New York: Springer. ISBN 0-387-00424-6. (Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler).