Schlegel diagrams

fundamental theorem

Schlegel diagrams are planar graphs (graphs that can be drawn in the plan without edges crossing) associated with polyhedra: the nodes and lines of the graph correspond to the vertices and edges of the polyhedron.
Theorem: Any 3-connected graph can be realized as a convex polyhedron, and a graph associated with a polyhedron is 3-connected.
A simple description of 3-connectedness is: any pair of vertices is connected by three different paths; this is equivalent to the fact that any two vertices can be removed, and all the edges using them, without disturbing the connectedness of the graph.
An elementary way to draw the Schlegel diagram of a polyhedron is to "project" the polyhedron's skeleton on one of its faces (the nodes corresponding to the vertices which not belong to that face must lie inside the face). shadow of a cube

a few examples

The Schlegel diagrams of the five platonical polyhedra (tetrahedron, cube, octahedron, dodecahedron and icosahedron): Schlegel platonic

The diagrams of the two types of pentahedra (pyramids and "roofs"): Schlegel penta├Ędres

The diagrams of the seven types of convex hexahedra: Schlegel pentahedra

The hermaphrodites are assemblings of a half prism and a half diamond.
The antihermaphrodites are assemblings of a half antiprism and a half antidiamond.
Here are the diagrams of the pentagonal forms of this two polyhedra whose canonical forms are self-dual (11 faces and 11 vertices).

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Schlegel (anti)hermaphrodites

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Diagrams of two curious polyhedra: the simplest with an odd number of n-faces (9 quadrilaterals) and a topological dual.

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Schlegel (9 quadrilaterals and dual)

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a proof of Euler's formula:  f+v = e+2

We move from the polyhedron to its Schlegel diagram by removing a face: the formula becomes  polygons + vertices = sides + 1
Then we apply to the diagram transformations which don't modify this relation  p+v=s+1 :
  •   we triangulate all the polygons using diagonals: for each diagonal we add one side and one polygon
  •   we suppress the triangles starting at the border of the diagram:
      - if only one triangle's side belongs to the diagram border we suppress this side and the triangle,
      - if two sides belong to the border we suppress these two sides, their common vertex and the triangle.
When only one triangle is left we have  1+3=3+1  which corresponds to  p+v=s+1  for all diagrams, and to  f+v=e+2  for the polyhedron. Euler proof

On this example (hexahedron with 6+8=12+2) we have successively 5+8=12+1, 9+8=16+1, 8+8=15+1, 7+8=14+1, 6+7=12+1 ... and finally 1+3=3+1.

January 2004
updated 15-03-2007