# 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.
Various visualizations of the icosahedron

2D
 Schlegel

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 or faces. The resulting entity is a polytopal subdivision of the facet in $R^{d-1}$ that is combinatorially equivalent to the original polytope. Named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimensions 3 and 4, a Schlegel diagram is a projection of a polyhedron into a plane figure and a projection of a polychoron to 3-space, respectively. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.

## Construction

The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows:[1]

A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a Schlegel diagram of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from a vertex; this vertex is projected to infinity and does not appear in the diagram, the edges through it are represented by lines drawn outwards.

Sommerville also considers the case of a simplex in four dimensions:[2] "The Schlegel diagram of simplex in S4 is a tetrahedron divided into four tetrahedral." More generally, a polytope in n-dimensions has a Schegel diagram 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.

## Examples

Dodecahedron Dodecaplex

12 pentagon faces in the plane

120 dodecahedral cells in 3-space