Vertex configuration - AbsoluteAstronomy.com

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 (or vertex type, or vertex description) is a short-hand notation for representing the 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.-Definitions - theme and variations:...

of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra

Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

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 vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

s and pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

s. This vertex configuration defines the vertex-uniform icosidodecahedron

Icosidodecahedron

In geometry, 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

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 polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

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 object in two dimensions. It is a form of parallel projection, where all the projection lines are 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: http://commons.wikimedia.org/wiki/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 for regular polyhedra

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

. {p,q} means q p-agons around each vertex. So this can be written as p.p.p... (q times). For example an icosahedron is {3,5} = 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 precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

forms. For example, the snub cube

Snub cube

In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

has a clockwise and counterclockwise form which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration.

Star polygons

The notation also applies for nonconvex regular faces, the star polygons. For example a pentagram

Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes...

has 5/2 sides meaning 5 vertex going around the vertex twice. For example, the nonconvex regular polyhedron small stellated dodecahedron has a vertex configuration of Schläfli symbol of {⁵/₂,5} 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 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

The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...

can be enumerated by looking at their vertex configuration and the angle defect

Defect (geometry)

In geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...

: 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*π 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

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

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 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

Quadruples

Platonic solid 3.3.3.3
Octahedron
In geometry, 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 vertex....

(6)
antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two 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. As such it is a quasiregular polyhedron,...

(12), 3.5.3.5
Icosidodecahedron
In geometry, 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
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

(24), 3.4.5.4
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....

(60)
regular tiling 4.4.4.4
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

Quintuples
Finally configurations with five and six faces meeting at each vertex:

Platonic solid 3.3.3.3.3
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

(12)
Archimedean solids 3.3.3.3.4
Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...

(24), 3.3.3.3.5
Snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....

(60) (both chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

)
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 semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex.Conway calls it a isosnub quadrille....

, 3.3.4.3.4 (note that the two different orders of the same numbers give two different patterns)

Sextuples

regular tiling 3.3.3.3.3.3

Face configuration for duals

The dual polyhedron

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

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, one can tell a face configuration of V3.4.3.4 represents a rhombus

Rhombus

In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...

face since every edge is a V3-V4 type, and V3.4.5.4 will be a kite

Kite (geometry)

In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

with two types of edges: V3-V4 and V4-V5.

Notation used in articles

Uniform polyhedron
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
- Platonic solid
  Platonic solid
  In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
- Semiregular polyhedron
  Semiregular polyhedron
  The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
  - Archimedean solid
    Archimedean solid
    In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
  - Prism (geometry)
    Prism (geometry)
    In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
  - Antiprism
    Antiprism
    In geometry, an n-sided antiprism is a polyhedron composed of two 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 polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
  
  s
- Near-miss Johnson solid
  Near-miss Johnson solid
  In geometry, a near-miss Johnson solid is a strictly convex 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 prisms, the infinite set of antiprisms, and the 92 Johnson solids.The set of...
  
  s
- List of uniform polyhedra by vertex figure
- List of Wenninger polyhedron models
- List of Uniform Polyhedra
- List of uniform planar tilings
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 Belgian mathematician, Eugène Catalan, who first described them in 1865....
  
  - Archimedean duals
- Bipyramid
  Bipyramid
  An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
  
  - 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 kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...
  
  - antiprism duals

External links

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