Vertex configuration

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Template:Short description Template:Multiple image In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or tiling as the sequence of faces around a vertex. It has variously been called a vertex description,[1][2][3] vertex type,[4][5] vertex symbol,[6][7] vertex arrangement,[8] vertex pattern,[9] face-vector,[10] vertex sequence.[11] It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models.Template:R[12][13] For uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration.)

For example, "Template:Math" indicates a vertex belonging to 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-transitive icosidodecahedron. The notation is cyclic and therefore is equivalent with different starting points, so Template:Math is the same as Template:Math The order is important, so Template:Math is different from Template:Math (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as Template:Math.

Notation

Regular vertex figure nets, Template:Math

Tetrahedron
Template:Math
Defect 180°
Octahedron
Template:Math
Defect 120°
Icosahedron
Template:Math
Defect 60°
Triangular tiling
Template:Math
Defect 0°

Cube
Template:Math
Defect 90°
Square tiling
Template:Math
Defect 0°
Dodecahedron
Template:Math
Defect 36°
Hexagonal tiling
Template:Math
Defect 0°
A vertex needs at least 3 faces, and an angle defect.
A 0° angle defect will fill the Euclidean plane with regular tiling.
By Descartes' theorem, the number of vertices is 720°/defect (4π radians/defect).

A vertex configuration is written as one or more numbers separated by either dots or commas. Each number represents the number of sides in each face that meets at each vertex.Template:R An icosidodecahedron is denoted as Template:Math because there are four faces at each vertex, alternating between triangles (with 3 sides) and pentagons (with 5 sides). This can also be written as Template:Math.

The vertex configuration can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. The Schläfli notation has the form Template:Math, where Template:Mvar is the number of sides in each face and Template:Mvar is the number of faces that meet at each vertex. Hence, the Schläfli notation Template:Math can be written as Template:Math (where Template:Mvar appears Template:Mvar times), or simply Template:Mvar.Template:R

This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex configuration denotes a uniform polyhedron.

The notation is ambiguous for chiral forms. For example, the snub cube has clockwise and counterclockwise forms which are identical across mirror images. Both have a Template:Math 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.

{5/2,5} = (5/2)5 {5/2,3} = (5/2)3 34.5/2 34.5/3 (34.5/2)/2
{5,5/2} = (55)/2 {3,5/2} = (35)/2 V.34.5/2 V34.5/3 V(34.5/2)/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

Template:See also 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 Template:Math has an angle defect of 30 degrees. Therefore, it has Template:Math vertices.

In particular it follows that Template:Math has the following number of vertices: 42−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 Template:Math implies that a Template:Mvar-gon is surrounded by alternating Template:Mvar-gons and Template:Mvar-gons, so either Template:Mvar is even or Template:Mvar equals Template:Mvar. Similarly Template:Mvar is even or Template:Mvar equals Template:Mvar, and Template:Mvar is even or Template:Mvar equals Template:Mvar. Therefore, potentially possible triples are:

Template:Math Template:Math
(∴ Template:Math) 		Template:Math Template:Math
(∴ Template:Math) 		Template:Math
Template:Math (∴ Template:Math) Template:Math
Template:Math Template:Math Template:Math (Template:Math)
Template:Math (∴ Template:Math) Template:Math
Template:Math Template:Math Template:Math Template:Math Template:Math
Template:Math Template:Math Template:Math
Template:Math Template:Math
Template:Math Template:Math

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
Quadruples
Quintuples
Sextuples

Face configuration

Rhombic dodecahedron

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.[14] 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.Template:R 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. Template:Clear

References

  1. ↑ Template:Cite book See 3.7 The Archimedean Polyhedra, pp. 101–115, pp. 118–119 in Table I, Nets of Archimedean Duals, V.a,b,c,… as vertically-regular symbols. Cite error: <ref> tag with name "cundy-rollett" defined in <references> is not used in prior text.
  2. ↑ Template:Cite book Cite error: <ref> tag with name "wd" defined in <references> is not used in prior text.
  3. ↑ Archimedean Polyhedra Template:Webarchive Steven Dutch
  4. ↑ Uniform Polyhedra Jim McNeill
  5. ↑ Uniform Polyhedra and their Duals Robert Webb
  6. ↑ Symmetry-type graphs of Platonic and Archimedean solids, Jurij Kovič, (2011)
  7. ↑ 3. General Theorems: Regular and Semi-Regular Tilings Kevin Mitchell, 1995
  8. ↑ Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins
  9. ↑ Vertex Symbol Robert Whittaker
  10. ↑ Structure and Form in Design: Critical Ideas for Creative Practice By Michael Hann
  11. ↑ Symmetry-type graphs of Platonic and Archimedean solids Jurij Kovič
  12. ↑ Template:Citation
  13. ↑ Template:Cite journal
  14. ↑ Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere 6.4.1 Cundy-Rollett symbol, p. 164
  15. ↑ Laughlin (2014), p. 16
  16. ↑ Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53
  • Template:The Geometrical Foundation of Natural Structure (book) Uses Cundy-Rollett symbol.
  • Template:Cite book Pp. 58–64, Tilings of regular polygons a.b.c.... (Tilings by regular polygons and star polygons) pp. 95–97, 176, 283, 614–620, Monohedral tiling symbol [v1.v2. ... .vr]. pp. 632–642 hollow tilings.
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, Template:Isbn (p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).
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