Vertex figure

In geometry, a vertex figure is the figure that results when the area around a vertex (corner) of a polyhedron or other kind of polytope is sliced off. The vertex figure of a polytope looks like a shape from the previous dimension. For example, the vertex figure of a cube is a triangle.

The shape of a vertex figure on a polyhedron comes from how many of its faces are connected at one vertex. For example, the octahedron has a vertex figure shaped like a square (a four-sided shape) because its vertices each have four triangles connecting together.

Besides its number of sides, the exact shape of a vertex figure is different if the faces are meet in a different way. For example, each vertex of a cuboctahedron has four faces meeting: two triangles and two squares. The polygon faces’ different numbers of sides means that the vertex figure, a rectangle, has two different lengths for its sides.

When the vertex figures of one polyhedron have exactly the same shape as the faces of another polyhedron, and the other polyhedron’s vertex figures match the faces of the first polyhedron, the two polyhedra are dual to each other.