geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a vertex configurationCrystallography of Quasicrystals: Concepts, Methods and Structures
by Walter Steurer, Sofia Deloudi, (2009) pp. 18–20 and 51–53Physical Metallurgy: 3-Volume Set, Volume 1
edited by David E. Laughlin, (2014) pp. 16–20 is a shorthand notation for representing the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

tiling Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...

as the sequence of

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

around a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

. For

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (

Chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

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 "" describes a vertex that has 3 faces around it, faces with , , and sides. For example, "" indicates a vertex belonging to 4 faces, alternating

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

s and

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

s. This vertex configuration defines the

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...

. The notation is cyclic and therefore is equivalent with different starting points, so is the same as The order is important, so is different from (the first has two triangles followed by two pentagons). Repeated elements can be collected as exponents so this example is also represented as . It has variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector. It is also called a Cundy and Rollett symbol for its usage for the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s in their 1952 book ''

Mathematical Models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

''.

Vertex figures

A ''vertex configuration'' can also be represented as a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

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 all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration.

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

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

for

regular polyhedra A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

. The Schläfli notation means ''q'' ''p''-gons around each vertex. So can be written as ''p.p.p...'' (''q'' times) or ''p^q''. For example, an icosahedron is = 3.3.3.3.3 or 3⁵. 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 Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

forms. For example, the

snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron; that is, it has two distinct forms, which are mirr ...

has clockwise and counterclockwise forms 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 polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

s. For example, a

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

has the symbol , 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 In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...

has the

of 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 In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

, has a triangular vertex figure and configuration (5/2.5/2.5/2) or (5/2)³. The

great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...

, has a pentagrammic vertex figure, with ''vertex configuration'' is (5.5.5.5.5)/2 or (5⁵)/2. A

great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...

, 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

notation of sides ''p/q'' such that ''p''<2''q'', 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 In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

have vertex configurations with positive

angle defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the de ...

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

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

3.8.8 has an angle defect of 30 degrees. Therefore, it has vertices. In particular it follows that has 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 ''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 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. The number in parentheses is the number of vertices, determined by the angle defect. ;Triples * Platonic solids 3.3.3 (4), 4.4.4 (8), 5.5.5 (20) *

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

s 3.4.4 (6), 4.4.4 (8; also listed above), 4.4.''n'' (2''n'') * Archimedean solids 3.6.6 (12), 3.8.8 (24), 3.10.10 (60), 4.6.6 (24), 4.6.8 (48), 4.6.10 (120), 5.6.6 (60). * regular tiling 6.6.6 * semiregular tilings 3.12.12, 4.6.12, 4.8.8 ;Quadruples * Platonic solid 3.3.3.3 (6) *

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s 3.3.3.3 (6; also listed above), 3.3.3.''n'' (2''n'') * Archimedean solids 3.4.3.4 (12), 3.5.3.5 (30), 3.4.4.4 (24), 3.4.5.4 (60) * regular tiling 4.4.4.4 * semiregular tilings 3.6.3.6, 3.4.6.4 ;Quintuples * Platonic solid 3.3.3.3.3 (12) * Archimedean solids 3.3.3.3.4 (24), 3.3.3.3.5 (60) (both

) * semiregular tilings 3.3.3.3.6 (chiral), 3.3.3.4.4, 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

The uniform dual or

Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...

s, including the bipyramids and

trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

, are ''vertically-regular'' (

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

) 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

around a

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

.Cundy and Rollett (1952) For example, V3.4.3.4 or V(3.4)² represents the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

which is face-transitive: every face is a

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

, and alternating vertices of the rhombus contain 3 or 4 faces each.

Notes

References

* Cundy, H. and Rollett, A., ''

'' (1952), (3rd edition, 1989, Stradbroke, England: Tarquin Pub.), 3.7 ''The Archimedean Polyhedra''. Pp. 101–115, pp. 118–119 Table I, Nets of Archimedean Duals, V.''a''.''b''.''c''... as ''vertically-regular'' symbols. * Peter Cromwell, ''

Polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

'', Cambridge University Press (1977) The Archimedean solids. Pp. 156–167. * Uses Cundy-Rollett symbol. * 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 ₁.v₂. ... .v_r pp. 632–642 hollow tilings. * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, {{isbn, 978-1-56881-220-5 (p. 289 Vertex figures, uses comma separator, for Archimedean solids and tilings).

External links

Consistent Vertex Descriptions
Stella (software), Robert Webb Polyhedra Mathematical notation