In geometry, a vertex configuration[1][2][3][4] is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. 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.)
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with a, b, and c sides.
For example, "3.5.3.5" 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 3.5.3.5 is the same as 5.3.5.3. The order is important, so 3.3.5.5 is different from 3.5.3.5 (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as (3.5)2.
It has variously been called a vertex description,[5][6][7]vertex type,[8][9]vertex symbol,[10][11]vertex arrangement,[12]vertex pattern,[13]face-vector.[14] It is also called a Cundy and Rollett symbol for its usage for the Archimedean solids in their 1952 book Mathematical Models.[15][16][17]
^Uniform Solution for Uniform Polyhedra Archived 2015-11-27 at the Wayback Machine (1993)
^The Uniform Polyhedra Roman E. Maeder (1995)
^Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53
^Physical Metallurgy: 3-Volume Set, Volume 1 edited by David E. Laughlin, (2014) pp. 16–20
^Archimedean Polyhedra Archived 2017-07-05 at the Wayback Machine Steven Dutch
^Uniform Polyhedra Jim McNeill
^Uniform Polyhedra and their Duals Robert Webb
^Symmetry-type graphs of Platonic and Archimedean solids, Jurij Kovič, (2011)
^3. General Theorems: Regular and Semi-Regular Tilings Kevin Mitchell, 1995
^Resources for Teaching Discrete Mathematics: Classroom Projects, History, modules, and articles, edited by Brian Hopkins
^Vertex Symbol Robert Whittaker
^Structure and Form in Design: Critical Ideas for Creative Practice
By Michael Hann
^Symmetry-type graphs of Platonic and Archimedean solids Jurij Kovič
^Deza, Michel; Shtogrin, Mikhail (2000), "Uniform partitions of 3-space, their relatives and embedding", European Journal of Combinatorics, 21 (6): 807–814, arXiv:math/9906034, doi:10.1006/eujc.1999.0385, MR 1791208
^Weisstein, Eric W., "Archimedean solid", MathWorld
^Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere 6.4.1 Cundy-Rollett symbol, p. 164
^Laughlin (2014), p. 16
and 25 Related for: Vertex configuration information
a vertexconfiguration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For...
symbol will have a vertexconfiguration p.q.p.q (or (p.q)2). More generally, a quasiregular figure can have a vertexconfiguration (p.q)r, representing...
is a quasiregular tiling, alternating two types of polygons, with vertexconfiguration (3.6)2. It is also a uniform tiling, one of eight derived from the...
be represented in vertexconfiguration notation, by listing the faces in sequence around the vertex. For example 3.4.4.4 is a vertex with one triangle...
of each vertex (in G) specified. For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled...
"Exception": a retrograde star antiprism with equilateral triangle bases (vertexconfiguration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an...
the vertexconfiguration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertexconfiguration of...
tiling can be described by its vertex configuration: the (identical) sequence of polygons around each (equivalent) vertex. All uniform tilings can be constructed...
center of the base, so that the apex, the center of the base and any other vertex form a right triangle. A hexagonal pyramid of edge length 1 has the following...
separately). Here the vertexconfiguration refers to the type of regular polygons that meet at any given vertex. For example, a vertexconfiguration of 4.6.8 means...
congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic...
by their vertexconfiguration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex. These 11...
pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertexconfiguration, with the regular dodecahedron...
solid because it lacks a set of global symmetries that map every vertex to every other vertex, unlike the 13 Archimedean solids. It is also a canonical polyhedron...
cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with...
edges are known as the lateral edges of the pyramid; they meet at the fifth vertex, called the apex. If the pyramid's apex lies on a line erected perpendicularly...
t{6}, or a twice-truncated triangle, tt{3}. The internal angle at each vertex of a regular dodecagon is 150°. The area of a regular dodecagon of side...
and are therefore prismatoids. Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group. The difference...
rhombicuboctahedron that, while some of the faces are not regular polygons, are still vertex-uniform. Some of these can be made by taking a cube or octahedron and cutting...
is also a part of a sequence of cantic polyhedra and tilings with vertexconfiguration 3.6.n.6. In this wythoff construction the edges between the hexagons...
heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}. Compare to trihexagonal tiling with vertexconfiguration 3.6.3.6. In geometry, the...
forms a gyroelongated pentagonal pyramid, J11. More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base...
self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra...
related as a part of sequence of uniform truncated polyhedra with vertexconfigurations (3.2n.2n), and [n,3] Coxeter group symmetry. Two 2-uniform tilings...
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