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In polyhedral geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 a vertex configuration is a short-hand notation for representing a polyhedron vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
 as the sequence of faces around a vertex. For uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 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.






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In polyhedral geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 a vertex configuration is a short-hand notation for representing a polyhedron vertex figure
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
 as the sequence of faces around a vertex. For uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 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. A a.b.c means a vertex has 3 faces around it, with a, b, and c sides.

For example 3.5.3.5 means a vertex has 4 faces, alternating triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s and pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s. This vertex configuration defines the vertex-uniform icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
 polyhedron.

Vertex Figures

Icosidodecahedron Vertfig
A vertex configuration can also be represented graphically as vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
 all the neighboring vertices are in the same plane and so this plane projection
Orthographic projection

Orthographic projection is a means of representing a Three-dimensional space object in 2D.It is a form of parallel projection, where the view direction is orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface....
 can be used to visually represent the vertex configuration.

See image category: :Category:Polyhedra-vf image

Variations and uses

Different notations are used, sometimes with a comma (,), and sometimes a period (.) separator. The period operator is useful because it looks like a product and an exponent notation can be used. For example 3.5.3.5 is sometimes written as (3.5)^2 or (3.5)2.

The order is important and so 3.3.5.5 is different from 3.5.3.5. The first has two triangles followed by two pentagons.

The notation can also be considered an expansive form of the simple Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 for regular polyhedra
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
. means q p-agons around each vertex. So this can be written as p.p.p... (q times). For example an icosahedron is = 3.3.3.3.3 = 3^ 5= 35.

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. To be unique, usually the smallest face (or sequence of smallest faces) are listed first.

This notation applies to polygon tiles as well as polyhedra. A planar vertex configuration can imply a uniform tiling just like a nonplanar vertex configuration can imply a uniform polyhedron.

The notation is ambiguous for chiral
Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
 forms. For example, the snub cube
Snub cube

The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles....
 has a clockwise and counterclockwise form which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.

Star polygons

Great Dirhombicosidodecahedron Vertfig
The notation also applies for nonconvex regular faces, the star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s. For example a pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
 has 5/2 sides meaning 5 vertex going around the vertex twice. For example, the nonconvex regular polyhedron small stellated dodecahedron
Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
 has a vertex configuration of Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 of which expands to an explicit vertex configuration as 5/2.5/2.5/2.5/2.5/2.

The last, U75, nonconvex uniform polyhedron great dirhombicosidodecahedron
Great dirhombicosidodecahedron

In geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex....
 has a vertex figure of 4.5/3.4.3.4.5/2.4.3/2. This complex vertex figure has 8 faces that pass around the vertex twice.

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
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 notation of sides p/q as an improper fraction (greater than one), 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


The existence of semiregular polyhedra
Semiregular polyhedron

A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
 can be enumerated by looking at their vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
 and the angle defect
Defect (geometry)

In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle....
: A set of regular faces must have internal angles less than 360 degees.

NOTE: The vertex figure can represent a regular or semiregular tiling on the plane if equal to 360. It can represent a tiling of the hyperbolic plane if greater than 360 degrees.

For uniform polyhedra, the angle defect can be used to compute the number of vertices. (The angle defect is defined as 360 degrees minus the sum of all the internal angles of the polygons that meet at the vertex.) Descartes' theorem states that the sum of all the angle defects in a topological sphere must add to 4*p radians or 720 degrees.

Since uniform polyhedra have all identical vertices, this relation allows us to compute the number of vertices: Vertices = 720/(angle-defect).

Example: A truncated cube
Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
 3.8.8 has an angle defect of 30 degrees. Therefore it has 720/30=24 vertices.

In particular it follows that has 4/(2-b(1-2/a)) vertices.

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 a q-gons and r-gons, so either p is even or q=r. Similarly q is even or p=r. Therefore potentially possible triples are 3.3.3, 3.4.4, 3.6.6, 3.8.8, 3.10.10, 3.12.12, 4.4.n (for any n>2), 4.6.6, 4.6.8, 4.6.10, 4.6.12, 4.8.8, 5.5.5, 5.6.6, 6.6.6. In fact, all these configurations with three faces meeting at each vertex turn out to exist.

Similarly when four faces meet at each vertex, p.q.r.s, if one number is odd its neighbors must be equal.

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

Triples
  • Platonic solids 3.3.3
    Tetrahedron

    A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
     (4), 4.4.4
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
     (8), 5.5.5
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
     (20)
  • prism
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    s 3.4.4 (6), 4.4.4 (8; also listed above), 4.4.n (2n)
  • Archimedean solids 3.6.6
    Truncated tetrahedron

    The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....
     (12), 3.8.8
    Truncated cube

    The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
     (24), 3.10.10
    Truncated dodecahedron

    In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....
     (60), 4.6.6
    Truncated octahedron

    The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
     (24), 4.6.8
    Truncated cuboctahedron

    The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....
     (48), 4.6.10
    Truncated icosidodecahedron

    The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....
     (120), 5.6.6
    Truncated icosahedron

    The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
     (60).
  • regular tiling 6.6.6
    Hexagonal tiling

    In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....
  • semiregular tilings 3.12.12
    Truncated hexagonal tiling

    In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....
    , 4.6.12
    Great rhombitrihexagonal tiling

    In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex ....
    , 4.8.8
    Truncated square tiling

    In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....


Quadruples
  • Platonic solid 3.3.3.3
    Octahedron

    An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
     (6)
  • antiprism
    Antiprism

    An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
    s 3.3.3.3 (6; also listed above), 3.3.3.n (2n)
  • Archimedean solids 3.4.3.4
    Cuboctahedron

    In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
     (12), 3.5.3.5
    Icosidodecahedron

    An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
     (30), 3.4.4.4
    Rhombicuboctahedron

    The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....
     (24), 3.4.5.4
    Rhombicosidodecahedron

    The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....
     (60)
  • regular tiling 4.4.4.4
    Square tiling

    In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
  • semiregular tilings 3.6.3.6
    Trihexagonal tiling

    In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....
    , 3.4.6.4
    Small rhombitrihexagonal tiling

    In geometry, the small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two Square s, and one hexagon on each vertex ....


Quintuples Finally configurations with five and six faces meeting at each vertex:
  • Platonic solid 3.3.3.3.3
    Icosahedron

    In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
     (12)
  • Archimedean solids 3.3.3.3.4
    Snub cube

    The snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are square s and the other 32 are equilateral triangles....
     (24), 3.3.3.3.5
    Snub dodecahedron

    The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles....
     (60) (both chiral
    Chirality (mathematics)

    In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
    )
  • semiregular tilings 3.3.3.3.6
    Snub hexagonal tiling

    In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....
     (chiral), 3.3.3.4.4
    Elongated triangular tiling

    In geometry, the elongated triangular tiling is a Tiling by regular polygons of the Euclidean plane. There are three triangles and two squares on each vertex ....
    , 3.3.4.3.4
    Snub square tiling

    In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....
     (note that the two different orders of the same numbers give two different patterns)


Sextuples
  • regular tiling 3.3.3.3.3.3
    Triangular tiling

    In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....


Face Configuration for duals


The dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 are can also be listed by this notation, but prefixed by a V. See face configuration
Face configuration

In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face ....
.

The faces of semiregular polyhedral duals are not regular polygons, but edges vary in length in relation regular polygons in the dual. For example, you can tell a face configuration of V3.4.3.4 represents a rhombus
Rhombus

In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
 face since every edge is a V3-V4 type, and V3.4.5.4 will be a kite
Kite (geometry)

In geometry a kite, or deltoid, is a quadrilateral with two disjoint sets pairs of congruent adjacent sides, in contrast to a parallelogram, where the congruent sides are opposite....
 with two types of edges: V3-V4 and V4-V5.

Notation used in articles

  • Uniform polyhedron
    Uniform polyhedron

    A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
    • Platonic solid
      Platonic solid

      In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
    • Semiregular polyhedron
      Semiregular polyhedron

      A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
      • Archimedean solid
        Archimedean solid

        In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
      • Prism (geometry)
        Prism (geometry)

        In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
      • Antiprism
        Antiprism

        An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
    • Johnson solid
      Johnson solid

      In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
      s
    • Near-miss Johnson solid
      Near-miss Johnson solid

      In geometry, a near-miss Johnson solid is a strictly convex set polyhedron, where every face is a regular or nearly regular polygon, and excluding the 5 Platonic solids, the 13 Archimedean solids, the infinite set of prism s, the infinite set of antiprisms, and the 92 Johnson solids....
      s
    • List of uniform polyhedra by vertex figure
      List of uniform polyhedra by vertex figure

      There are many relations among the uniform polyhedron.Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.Others share the same vertices and edges as other polyhedron....
    • List of Wenninger polyhedron models
      List of Wenninger polyhedron models

      This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
    • List of Uniform Polyhedra
      List of uniform polyhedra

      Uniform polyhedra and tilings form a well studied group. They are listed here for quick comparison of their properties and varied naming schemes and symbols....
    • List of uniform planar tilings
      List of uniform planar tilings

      This table shows the 11 convex Uniform tessellations of the Euclidean geometry, and their dual tilings.There are three regular, and eight semiregular, Tiling by regular polygons in the plane....
  • Face configuration
    Face configuration

    In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face ....
    • Catalan solid
      Catalan solid

      In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgium mathematician, Eug?ne Catalan, who first described them in 1865....
       - Archimedean duals
    • Bipyramid
      Bipyramid

      An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
       - prism duals
    • Trapezohedron
      Trapezohedron

      The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
       - antiprism duals