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The vertex figure at a polygon vertex of a polygon is the line segment joining the midpoints of the two adjacent sides meeting at . For a regular -gon with edge length , the length of the vertex figure is

The vertex figure at a polyhedron vertex of a polyhedron is the polygon whose sides are the vertex figures of the faces surrounding . The faces that join at a polyhedron vertex form a solid angle whose section by the plane is the vertex figure, as illustrated above for one polyhedron vertex of the cube.

For a -dimensional convex polytope with , a vertex figure is a -dimensional polytope formed by intersecting the convex polytope with a hyperplane that separates one polytope vertex from all the others. The convex polytope is a simple polytope iff every vertex figure is a -simplex (Renner 2009). In three dimensions, a convex polyhedron is therefore a simple polyhedron in the combinatorial sense iff all its vertex figures are triangles, which are 2-simplices.

This simplex condition on vertex figures is distinct from the condition defining a simplicial polytope, whose facets are simplices. Every simplex satisfies both conditions (Fukuda 2004).

The vertex figures of the Platonic solids yield the polyhedra (with holes centered on the centroids of the original faces) whose convex hulls are illustrated above and summarized in the following table.

The illustrations above show the Archimedean solids, their vertex figures, and the solids obtained by taking the convex hulls of the vertex figures.

See also

Dual Polyhedron, Midpoint, Rectification, Simple Polyhedron, Simple Polytope, Simplex, Simplicial Polytope, Truncation

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References

Coxeter, H. S. M. "The Polytopes with Regular-Prismatic Vertex Figures." Phil. Trans. Roy. Soc. 229, 330-425, 1930.Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 16, 1973.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 76, 1989.Fukuda, K. "What Is Simple/Simplicial Polytope?" Aug. 26, 2004. https://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node10.html.Renner, L. E. "Descent Systems for Bruhat Posets." J. Algebraic Combin. 29, 413-435, 2009. https://doi.org/10.1007/s10801-008-0141-4.

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Vertex Figure

Cite this as:

Weisstein, Eric W. "Vertex Figure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VertexFigure.html

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