Schlegel diagram

In geometry, a Schlegel diagram is a special projection of a polytope down one dimension. It projects polyhedra to a plane figure, and polychora 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 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 skeleton of the shape into one side.

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. [http://www.citr.auckland.ac.nz/dgt/Publications.php?id=544]
* 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
*.

External links

*
** mathworld | urlname = Skeleton | title = Skeleton
* [http://www.ac-noumea.nc/maths/amc/polyhedr/Schlegel_.htm Schlegel diagrams]
* [http://www.georgehart.com/hyperspace/hart-120-cell.html George W. Hart: 4D Polytope Projection Models by 3D Printing]
* [http://www.cs.sunysb.edu/~cse125/notes/08-4D-Forms.ppt Four Dimensional Polytopes]
* [http://www.nrich.maths.org/public/viewer.php?obj_id=897 Nrich maths - for the teenager. Also useful for teachers.]