Vertex figure
Channel: all the knowledge of the universe PRINCIPIA
Date: 2022-01-25

Vertex figure, by Wikipedia https://en.wikipedia.org/wiki?curid=1005719 / CC BY SA 3.0

#Polyhedra
#Polytopes
"Half-edge" vertex figure of the cube In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
"Whole-edge" vertex figure of the cube Spherical vertex figure of the cube Point-set vertex figure of the cube Take some corner or vertex of a polyhedron.
Mark a point somewhere along each connected edge.
Draw lines across the connected faces, joining adjacent points around the face.
When done, these lines form a complete circuit, i.
e.
a polygon, around the vertex.
This polygon is the vertex figure.
More precise formal definitions can vary quite widely, according to circumstance.
For example Coxeter （e.g. 1948, 1954） varies his definition as convenient for the current area of discussion.
Most of the following definitions of a vertex figure apply equally well to infinite tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes.
Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex.
The cut surface is the vertex figure.
This is perhaps the most common approach, and the most easily understood.
Different authors make the slice in different places.
Wenninger （2003） cuts each edge a unit distance from the vertex, as does Coxeter （1948）.
For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint.
Other authors make the cut through the vertex at the other end of each edge.
For an irregular polyhedron, cutting all edges incident to a given vertex at equal distances from the vertex may produce a figure that does not lie in a plane.
A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary.
This construction determines the combinatorial str...