Schlegel diagram

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

A tesseract 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.

In geometry, a Schlegel diagram is a projection of a polytope from R^d into R^{d − 1} through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in R^{d − 1} that is combinatorially equivalent to the original polytope. At the beginning of the 20th century, Schlegel diagrams were an important tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedra into a plane figure and polychora to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.

Construction

A Schlegel diagram 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 that 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.

The easiest way of drawing a Schlegel Diagram is to project the d-skeleton of the polytope into one of its facets.

Examples

A dodecahedron can project into the plane either with linear or curved edges.

A dodecaplex can project into 3-space either with linear or curved edges and faces.

Projection	Dodecahedron	Dodecaplex
Schlegel diagram (As a polytope)	12 pentagon faces in the plane	120 dodecahedral cells in 3-space
Stereographic projection (As a spherical tessellation)		With transparent faces

See also

• Net (polyhedron) - 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.

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
• Grünbaum, Branko (2003), Kaibel, Volker; Klee, Victor; Ziegler, Günter M., eds., Convex polytopes (2nd ed.), New York & London: Springer-Verlag, ISBN 0-387-00424-6 .

External links

• Weisstein, Eric W., "Schlegel graph" from MathWorld.
  • Weisstein, Eric W., "Skeleton" from MathWorld.
• Schlegel diagrams
• George W. Hart: 4D Polytope Projection Models by 3D Printing
• Four Dimensional Polytopes
• Nrich maths - for the teenager. Also useful for teachers.

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