Uniform polyhedron

A uniform polyhedron has regular polygonal faces and a symmetry group acting transitively on its vertices. Thus every vertex has the same surrounding face pattern, but that pattern may use more than one regular polygon type. The definition includes both convex examples and, under the broader convention, star-faced or self-intersecting examples.

A cube is uniform: each vertex meets three congruent squares, and its rotational symmetries carry any vertex to any other. A cuboctahedron has triangles and squares, alternating around every vertex, and is also uniform. Uniformity does not require every pair of faces to be related by a symmetry; that stronger face condition distinguishes regular solids.

The requirement is geometric. Merely drawing a polyhedron with the same number of incident edges at every vertex does not ensure regular faces or a vertex-transitive symmetry group. Regular prisms and regular antiprisms supply infinite families of convex uniform polyhedra.

A cuboctahedron is a convex uniform polyhedron with eight equilateral triangular faces and six square faces. Around each of its twelve vertices the faces alternate triangle, square, triangle, square. It has twenty-four edges, and the Euler characteristic check is $12-24+14=2$.

A concrete model is the convex hull of all coordinate permutations of $(\pm 1,\pm 1,0)$. The six planes $x=\pm 1$, $y=\pm 1$, $z=\pm 1$ cut out the square faces; the eight planes $\pm x\pm y\pm z=2$ cut out the triangular faces. All edges have length $\sqrt{2}$. Equivalently, join the midpoints of neighbouring edges of a cube: its vertices become triangular faces, and its faces become squares. The construction makes the name and the shared symmetries with the cube and octahedron visible.