Vertex figure Guide, Meaning , Facts, Information and Description
In geometry, a vertex figure is the polygon resulting when, taking a single vertex of a polyhedron, lines are drawn between the midpoints of all the edges through the vertex.Taking the cube as an example, there are three lines through any one vertex, and the vertex figure is therefore a triangle. As the angles of the three faces meeting at a vertex are equal, the triangle is equilateral.
Vertex figures can be generalised to higher dimensional polytopes, simply by noting that the vertex figure of an N-dimensional polytope is of dimension N-1. Thus, the vertex figure of both the hypercube and the 4-simplex are tetrahedra.
The vertex figure of a regular polytope is itself regular, and this is one of the two rules needed to categorise a regular polytope, the other being that it must have regular cells. This is easiest to see when writing Schläfli symbols describing the polytope, as for a polytope described by {j,k,...,x,y}, both {j,k,...,x} and {k,...,x,y} must be regular polytopes.
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