Vertex configuration

Star polygons

The notation also applies for nonconvex regular faces, the star polygons. For example, a pentagram has the symbol {5/2}, meaning it has 5 sides going around the centre twice.

For example, there are 4 regular star polyhedra with regular polygon or star polygon vertex figures. The small stellated dodecahedron has the Schläfli symbol of {5/2,5} which expands to an explicit vertex configuration 5/2.5/2.5/2.5/2.5/2 or combined as (5/2)⁵. The great stellated dodecahedron, {5/2,3} has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2)³. The great dodecahedron, {5,5/2} has a pentagrammic vertex figure, with vertex configuration is (5.5.5.5.5)/2 or (5⁵)/2. A great icosahedron, {3,5/2} also has a pentagrammic vertex figure, with vertex configuration (3.3.3.3.3)/2 or (3⁵)/2.

Inverted polygons

Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the star polygon notation of sides p/q such that p<2q, where p is the number of sides and q the number of turns around a circle. For example, "3/2" means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly "5/3" is a backwards pentagram 5/2.

All uniform vertex configurations of regular convex polygons

Semiregular polyhedra have vertex configurations with positive angle defect.

NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if its defect is zero. It can represent a tiling of the hyperbolic plane if its defect is negative.

For uniform polyhedra, the angle defect can be used to compute the number of vertices. Descartes' theorem states that all the angle defects in a topological sphere must sum to 4π radians or 720 degrees.

Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices, which is 4π/defect or 720/defect.

Example: A truncated cube 3.8.8 has an angle defect of 30 degrees. Therefore, it has 720/30 = 24 vertices.

In particular it follows that {a,b} has the following number of vertices: $${\displaystyle \frac{4}{2-b(1-2/a)}}$$

Every enumerated vertex configuration potentially uniquely defines a semiregular polyhedron. However, not all configurations are possible.

Topological requirements limit existence. Specifically p.q.r implies that a p-gon is surrounded by alternating q-gons and r-gons, so either p is even or q equals r. Similarly q is even or p equals r, and r is even or p equals q. Therefore, potentially possible triples are:

In fact, all these configurations with three faces meeting at each vertex turn out to exist.

The number in parentheses is the number of vertices, determined by the angle defect.

Triples

• Platonic solids:
  • Tetrahedron: 3.3.3 (4)
  • Cube: 4.4.4 (8)
  • Dodecahedron: 5.5.5 (20)
• Prisms:
  • n-gonal prism: 4.4.n (2n)
• Archimedean solids:
  • Truncated tetrahedron: 3.6.6 (12)
  • Truncated cube: 3.8.8 (24)
  • Truncated dodecahedron: 3.10.10 (60)
  • Truncated octahedron: 4.6.6 (24)
  • Truncated cuboctahedron: 4.6.8 (48)
  • Truncated icosidodecahedron: 4.6.10 (120)
  • Truncated icosahedron: 5.6.6 (60)
• Regular tilings:
  • Hexagonal tiling: 6.6.6
• Semiregular tilings:
  • Truncated hexagonal tiling: 3.12.12
  • Truncated trihexagonal tiling: 4.6.12
  • Truncated square tiling: 4.8.8

Quadruples

• Platonic solids:
  • Octahedron: 3.3.3.3 (6)
• Antiprisms:
  • n-gonal antiprism: 3.3.3.n (2n)
• Archimedean solids:
  • Cuboctahedron: 3.4.3.4 (12)
  • Icosidodecahedron: 3.5.3.5 (30)
  • Rhombicuboctahedron: 3.4.4.4 (24)
  • Rhombicosidodecahedron: 3.4.5.4 (60)
• Regular tilings:
  • Square tiling: 4.4.4.4
• Semiregular tilings:
  • Trihexagonal tiling: 3.6.3.6
  • Rhombitrihexagonal tiling: 3.4.6.4

Quintuples

• Platonic solids:
  • icosahedron: 3.3.3.3.3 (12)
• Archimedean solids (both chiral):
  • Snub cube: 3.3.3.3.4 (24)
  • Snub dodecahedron: 3.3.3.3.5 (60)
• Semiregular tilings:
  • Snub hexagonal tiling: 3.3.3.3.6 (chiral)
  • Elongated triangular tiling: 3.3.3.4.4
  • Snub square tiling: 3.3.4.3.4 (note that the two different orders of the same numbers give two different patterns)

Sextuples

• Regular tilings:
  • Triangular tiling: 3.3.3.3.3.3

Face configuration

The uniform dual or Catalan solids, including the bipyramids and trapezohedra, are vertically-regular (face-transitive) and so they can be identified by a similar notation which is sometimes called face configuration. Cundy and Rollett prefixed these dual symbols by a V. In contrast, Tilings and patterns uses square brackets around the symbol for isohedral tilings.

This notation represents a sequential count of the number of faces that exist at each vertex around a face. For example, V3.4.3.4 or V(3.4)² represents the rhombic dodecahedron which is face-transitive: every face is a rhombus, and alternating vertices of the rhombus contain 3 or 4 faces each.