Schlegel diagram

The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes.

As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.

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

More generally, a polytope in n-dimensions has a Schlegel 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.

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: the outer cell into which the others are projected, one below each of the six exterior faces, and one in the center.