List of regular polytopes

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List of regular polytopes

This page lists the regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
s in Euclidean

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, spherical

Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
and hyperbolic

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
spaces.

The 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....
notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...
. Infinite forms tessellate

Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
.

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Encyclopedia

This page lists the regular polytope

Regular polytope

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, spherical

Spherical geometry

Hyperbolic geometry

Schläfli symbol

Facet (mathematics)

Tessellation

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects

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....
, like making a vertex with 7 equilateral triangle

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

Regular polytope summary count by dimension

Dimension	Convex	Nonconvex	Convex Euclidean tessellations	Convex hyperbolic tessellations	Nonconvex hyperbolic tessellations	Abstract Polytopes
1	1 line segment List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	0	0	0	0	1
2	8 polygon List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.... s	8 star polygon List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.... s	1	1	0	8
3	5 Platonic solids List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	4 Kepler-Poinsot solid List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.... s	3 tiling List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.... s	8	8	8
4	6 convex polychora List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	10 Schläfli-Hess polychora List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	1 honeycomb List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	4	0	8
5	3 convex 5-polytopes List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	0 nonconvex 5-polytopes	3 tessellations List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....	5	4	8
6+	3	0	1	0	0	8

One-dimensional regular polytopes

There is only one polytope in 1 dimensions, whose boundaries are the two endpoints of a line segment

Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
, represented by the empty Schläfli symbol

Schläfli symbol

Two-dimensional regular polytopes

The two dimensional polytopes are called polygon

Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s. Regular polygons are equilateral

Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths....
and cyclic. A p-gonal regular polygon is represented by Schläfli symbol

Schläfli symbol

Convex polygon

In geometry, a polygon can be either convex or concave....
s are considered regular, but 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, like the 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"....
, can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.

Star polygons should be called nonconvex rather than concave because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.

Three-dimensional regular polytopes

In three dimensions, the regular polytopes are called polyhedra

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
:

A regular polyhedron with Schläfli symbol

Schläfli symbol

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
.

A 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....
(of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron is constrained by an inequality, related to the vertex figure's 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....
:

: Polyhedron (existing in Euclidean 3-space)

: Euclidean plane tiling

: Hyperbolic plane tiling

By enumerating the permutation

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
s, we find 5 convex forms, 4 nonconvex forms and 3 plane tilings, all with polygons and limited to: , , and .

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

Four-dimensional regular polytopes

Regular polychora with Schläfli symbol

Schläfli symbol

A 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....
(of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
An edge 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....
is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.

The existence of a regular polychoron is constrained by the existence of the regular polyhedra .

Each will exist in a space dependent upon this expression:

: Hyperspherical 3-space honeycomb or 4-space polychoron : Euclidean 3-space honeycomb : Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
for polychora is and is zero for all forms.

Five-dimensional regular polytopes

In five dimensions
Fifth dimension

In physics and mathematics, a tuple of N real numbers can be understood to represent a coordinate system in an N-dimensional Euclidean space. When N=5, the space consisting of all locations with a nonzero fifth number is called the fifth dimension....
, a regular polytope can be named as where is the hypercell (or teron) type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.

A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
).

A 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....
(of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.
An edge figure
Edge figure

In higher order polytopes, an edge figure of a polychoron or Honeycomb is a polygon representing the set of faces around an edge. For example the edge figure for a regular cubic honeycomb is a Square , and for a regular polychoron is the polygon ....
(of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular polytope exists only if and are regular polychora.

The space it fits in is based on the expression:

: Spherical 4-space tessellation or 5-space polytope : Euclidean 4-space tessellation : hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations.

Classical convex polytopes

Two dimensions

The Schläfli symbol represents a regular p-gon.

A regular henagon
Henagon

In geometry a henagon is a polygon with one Edge and one Vertex . It has Schl?fli symbol . Since a henagon has only one side and only one interior angle, every henagon is regular polygon by definition....
, and regular digon
Digon

In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....
, can be considered a degenerate regular polygon. They can exist nondegenerately in non-Euclidean spaces like on the surface of a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
or torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
.

The regular apeirogon
Apeirogon

An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides....
exists in the limit as , and can be considered as a tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
of 1-dimensional space.

Convex
Convex polygon

In geometry, a polygon can be either convex or concave....
regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s by name and 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....

henagon
Henagon

In geometry a henagon is a polygon with one Edge and one Vertex . It has Schl?fli symbol . Since a henagon has only one side and only one interior angle, every henagon is regular polygon by definition....

digon
Digon

In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....

triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....

(2-simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
)

square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....

(2-cube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
)
(2-orthoplex
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
)

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?....

hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....

heptagon
Heptagon

In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....

octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....

enneagon
Enneagon

In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a hybrid word, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century....

decagon
Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular polygon decagon, having all sides of equal length and all internal angles equal to 4π/5 ....

hendecagon
Hendecagon

In geometry, a hendecagon is an 11-sided polygon.The name "undecagon" is often seen as incorrect, but the matter is up for debate. The Greek language prefix 'hen', is preferable to the Latin 'uni' or 'un' ....

dodecagon
Dodecagon

In geometry, a dodecagon is any polygon with 12 sides and twelve angles....

triskaidecagon
Triskaidecagon

In geometry, a triskaidecagon is a polygon with 13 sides and angles.The measure of each internal angle of a Regular polygon triskaidecagon is approximately 152.308 degree s, and the area with side length a is given by...

tetradecagon
Tetradecagon

In geometry, a tetrakaidecagon is a polygon with 14 sides and angles.The area of a Regular polygon tetradecagon of side length a is given by...

pentadecagon
Pentadecagon

In geometry, a pentadecagon is any 15-sided, 15-angled, polygon.A Regular polygon pentadecagon has interior angles of 156?, and with a side length a, has an area given by...

hexadecagon
Hexadecagon

In mathematics, a hexadecagon is a polygon with 16 Edge and 16 Vertex .A regular hexadecagon is constructible polygon with a Compass and straightedge constructions....

heptadecagon
Heptadecagon

In geometry, a heptadecagon is a seventeen-sided polygon....

octadecagon
Octadecagon

An octadecagon is a polygon with 18 Edge and 18 Vertex . Another name for an octadecagon is octakaidecagon....

enneadecagon
Enneadecagon

In geometry, an enneadecagon is a polygon with 19 sides and angles. It is also known as an enneakaidecagon or a nonadecagon.The radius of the circumcircle of the regular enneadecagon with side length t is...

icosagon
Icosagon

In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.As a golygonal path, the swastika is considered to be an irregular icosagon....

n-gon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....

......
apeirogon
Apeirogon

An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides....

Three dimensions

The convex regular polyhedra
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
are called the 5 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....
s. (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....
is given with each vertex count.)
Name Schläfli
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....

Faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....

Edges
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
Vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Symmetry
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
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....
Tetrahedron
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....

(3-simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
) 4
6 4
2 T_d Self-dual
Cube
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....
(hexahedron)
(3-cube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
) 6
12 8
2 O_h Octahedron
Octahedron
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....

(3-orthoplex
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
) 8
12 6
2 O_h Cube
Dodecahedron
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....
12
30 20
2 I_h Icosahedron
Icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
20
30 12
2 I_h Dodecahedron

In spherical geometry
Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
, hosohedron
Hosohedron

An Polygon hosohedron is a tessellation of Lune s on a spherical surface, such that each lune shares the same two vertices. A regular n-gonal hosohedron has Schl?fli symbol ....
, and dihedron
Dihedron

A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....
can be considered regular polyhedra (tiling
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
s of the sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
).

Four dimensions

The 6 convex polychora
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
are as follows:

Name
Schläfli
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....

Cells
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....

Faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....

Edges
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

Vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Dual

5-cell
(pentachoron)
(4-simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
) 5
10
10
5
0 Self-dual
8-cell
(Tesseract)
(4-cube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
) 8
24
32
16
0 16-cell
16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
16-cell
16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....

(4-orthoplex
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
) 16
32
24
8
0 Tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
24
96
96
24
0 Self-dual
120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
120
720
1200
600
0 600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
600
1200
720
120
0 120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....

5-cell 8-cell 16-cell
16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
24-cell
24-cell

In geometry, the 24-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . It is also called an octaplex and polyoctahedron, being constructed of Octahedron Cell ....
120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....

Wireframe orthographic 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....
s

Solid orthographic 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....
s (cell-centered)

tetrahedral
envelope
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....

cubic envelope
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....

octahedral
envelope
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....

cuboctahedral
envelope
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....

truncated rhombic
triacontahedron
envelope
Truncated rhombic triacontahedron

The truncated rhombic triacontahedron is a convex set polyhedron constructed from the rhombic triacontahedron by truncating the twelve vertices where five faces meet at their acute corners....

pentakis dodecahedral
envelope
Pentakis dodecahedron

A pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face....
Wireframe Schlegel diagram
Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope....
s (Perspective projection)

(Cell-centered)
(Cell-centered)
(Cell-centered)
(Cell-centered)
(Cell-centered)
(Vertex-centered)
Wireframe stereographic projection
Stereographic projection

In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane . The projection is defined on the entire sphere, except at one point — the projection point....
s (Hyperspherical)

Five dimensions

There are three kinds of convex regular polytopes in five dimensions:

Name projective
graph
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....
Schläfli
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....

Symbol
Facets
Cells
Faces
Edges Vertices Face
figure
Edge
figure
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....

Dual
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....
5-simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....

(or hexateron
Hexateron

In Fifth dimension geometry, a hexateron, or hexa-5-tope, is a 5-simplex, a self-dual Regular polytope 5-polytope with 6 vertex , 15 Edge s, 20 triangle Face , 15 tetrahedral Cell , 6 5-cell hypercells....
)
6
15
20
15 6 Self-dual
5-hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....

(or decateron
or penteract
Penteract

In Fifth dimension geometry, a penteract is a name for a Fifth dimension hypercube with 32 Vertex , 80 Edge s, 80 square Face , 40 cubic Cell , and 10 tesseract hypercells....
) 10
40
80
80 32 pentacross
Pentacross

In fifth dimension geometry, a pentacross, also called a triacontakaiditeron, is a five-dimensional polytope with 10 Vertex , 40 Edge s, 80 triangle Face , 80 octahedron Cell , 32 5-cell hypercells....
5-orthoplex
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....

(or triacontakaiditeron
or pentacross
Pentacross

In fifth dimension geometry, a pentacross, also called a triacontakaiditeron, is a five-dimensional polytope with 10 Vertex , 40 Edge s, 80 triangle Face , 80 octahedron Cell , 32 5-cell hypercells....
) 32
80
80
40 10 penteract
Penteract

In Fifth dimension geometry, a penteract is a name for a Fifth dimension hypercube with 32 Vertex , 80 Edge s, 80 square Face , 40 cubic Cell , and 10 tesseract hypercells....

Higher dimensions

In dimensions 5 and higher, there are only three kinds of convex regular polytopes. [Coxeter, Regular Polytopes, Table I: Regular polytopes, (iii) The three regular polytopes in n dimensions (n>=5), pp. 294-295]

Name Schläfli
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....

Symbol
Facet
type 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....
Dual
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....
n-simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
Self-dual
n-cube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
n-orthoplex
n-orthoplex
Cross-polytope

In geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular polytope, convex polytope that exists in any number of dimensions....
n-cube

Finite non-convex polytopes - star-polytopes

Two dimensions
There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers . They are called 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.

In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols for all m such that m < n/2 (strictly speaking =) and m and n are coprime
Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common divisor other than 1 or, equivalently, if their greatest common divisor is 1....
.

Name Schläfli
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....

Symbol
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"....
heptagram
Heptagram

A heptagram or septegram is a seven-pointed Star drawn with seven straight strokes....
s ,
octagram
Octagram

In geometry, an octagram is an eight-sided star polygon....
enneagram
Enneagram

In geometry, an enneagram is a nine-pointed geometric figure. The term derives from two ancient Greek words: ennea and gramma ....
s ,
decagram
Decagram (geometry)

In geometry, a decagram is a 10-sided star polygon.There is one regular decagram star polygon, , containing the vertices of a regular decagon, but connected by every third point....
hendecagram
Hendecagram

A hendecagram is a star polygon that has eleven Point . There are 4 regular forms: , , , ....
s , ,
dodecagram
...n-agrams
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....

Three dimensions

The regular star polyhedra
Star polyhedron

In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvex polygon giving it a star-like visual quality.There are two general kinds of star polyhedron:...
are called the Kepler-Poinsot solid
Kepler-Poinsot solid

The Kepler-Poinsot polyhedra are the four Regular polyhedron Star polyhedron. They may be obtained by stellation the regular convex or Platonic solids, and differ from these in having regular star polygons for their faces or vertex figures....
s and there are four of them, based on the vertices of the dodecahedron
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....
and icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
:

Name Schläfli
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....

Faces
Edges Vertices
χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Symmetry
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
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....
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....
12
30 12
I_h Great dodecahedron
Great dodecahedron
Great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
12
30 12
I_h Small stellated dodecahedron
Great stellated dodecahedron
Great stellated dodecahedron

In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
12
30 20
2 I_h Great icosahedron
Great icosahedron
Great icosahedron

In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
20
30 12
2 I_h Great stellated dodecahedron

Four dimensions

There are ten regular star polychora, which can be called Schläfli-Hess polychora
Schläfli-Hess polychoron

In four dimensional geometry, Schl?fli-Hess polychora are the complete set of 10 Regular polytope self-intersecting Star polytope . They are named in honor of their discoverers: Ludwig Schl?fli and Edmund Hess....
and their vertices are based on the convex 120-cell
120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
and 600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
:

Ludwig Schläfli
Ludwig Schläfli

Ludwig Schl?fli was a Switzerland geometry and complex analysis who was one of the key figures in developing the notion of higher dimensional spaces....
found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
on cells or vertex figures (For zero-hole toruses: F+V-E=2). Edmund Hess
Edmund Hess

Edmund Hess was a Germany mathematician who discovered several regular polytopes.See also* Schl?fli-Hess polychoron* Hess polytope...
(1843-1903) completed the full list of ten in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.

There are 4 failed potential nonconvex regular polychora permutations: , , , . Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

Name
Schläfli
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....

Cells
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....

Faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....

Edges
Edge (geometry)

In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....

Vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
and
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....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Dual
Great grand stellated 120-cell
Great grand stellated 120-cell

In geometry, the great grand stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron....
120
720
1200
600
0 Grand 600-cell
Grand 600-cell
Grand 600-cell

In geometry, the Grand 600-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron. It is the only one with 600 cells....
600
1200
720
120
0 Great grand stellated 120-cell
Great stellated 120-cell
Great stellated 120-cell

In geometry, the great stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is one of four regular star polychora discovered by Ludwig Schl?fli....
120
720
720
120
0 Grand 120-cell
Grand 120-cell
Grand 120-cell

In geometry, the grand 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is one of four regular star polychora discovered by Ludwig Schl?fli....
120
720
720
120
0 Great stellated 120-cell
Grand stellated 120-cell
Grand stellated 120-cell

In geometry, the grand stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the grand 600-cell, icosahedral 120-cell, and the same face arrangement as the great stellated 120-cell....
120
720
720
120
0 Self-dual
Small stellated 120-cell
Small stellated 120-cell

In geometry, the small stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the great grand 120-cell....
120
720
1200
120
-480 Icosahedral 120-cell
Icosahedral 120-cell
Icosahedral 120-cell

In geometry, the icosahedral 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is constructed by 5 icosahedron around each edge in a pentagram figure....
120
1200
720
120
480 Small stellated 120-cell
Great icosahedral 120-cell
Great icosahedral 120-cell

In geometry, the great icosahedral 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the great stellated 120-cell, and grand stellated 120-cell, and face arrangement of the grand 600-cell....
120
1200
720
120
480 Great grand 120-cell
Great grand 120-cell
Great grand 120-cell

In geometry, the great grand 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the small stellated 120-cell....
120
720
1200
120
-480 Great icosahedral 120-cell
Great 120-cell
Great 120-cell

In geometry, the great 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the 600-cell, icosahedral 120-cell as well as the same face arrangement as the grand 120-cell....
120
720
720
120
0 Self-dual

There are 7 unique face arrangements from these 10 nonconvex polychora, shown as orthogonal projections:

Icosahedral 120-cell

In geometry, the icosahedral 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is constructed by 5 icosahedron around each edge in a pentagram figure....

Great 120-cell

In geometry, the great 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the 600-cell, icosahedral 120-cell as well as the same face arrangement as the grand 120-cell....
and
Grand 120-cell

In geometry, the grand 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is one of four regular star polychora discovered by Ludwig Schl?fli....

Small stellated 120-cell

In geometry, the small stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the great grand 120-cell....

Great grand 120-cell

In geometry, the great grand 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the small stellated 120-cell....

Great stellated 120-cell

In geometry, the great stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It is one of four regular star polychora discovered by Ludwig Schl?fli....
and
Grand stellated 120-cell

In geometry, the grand stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the grand 600-cell, icosahedral 120-cell, and the same face arrangement as the great stellated 120-cell....

Great icosahedral 120-cell

In geometry, the great icosahedral 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron.It has the same edge arrangement as the great stellated 120-cell, and grand stellated 120-cell, and face arrangement of the grand 600-cell....
and
Grand 600-cell

In geometry, the Grand 600-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron. It is the only one with 600 cells....

Great grand stellated 120-cell

In geometry, the great grand stellated 120-cell is a star polychoron with Schl?fli symbol . It is one of 10 regular Schl?fli-Hess polychoron....

Higher dimensions
There are no non-convex regular polytopes in five dimensions or higher.

Tessellations

The classical convex polytopes may be considered tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
s, or tilings of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

Two dimensions

There is one tessellation of the line, giving one polytope, the (two-dimensional) apeirogon
Apeirogon

An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides....
. This has infinitely many vertices and edges. Its 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....
is .

......

Three dimensions

Euclidean (plane) tilings

There are three regular tessellations of the plane.
Name Schläfli
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....

Symbol
Face
type
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....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Symmetry
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
Square tiling
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....
0 p4m Self-dual
Triangular tiling
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....
0 p6m Hexagonal tiling
Hexagonal tiling
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....
0 p6m Triangular tiling

There is one degenerate regular tiling, , made from two apeirogon
Apeirogon

An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides....
s, each filling half the plane. This tiling is related to a 2-faced dihedron
Dihedron

A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. It is Mathematical degeneracy if its faces are flat....
, , on the sphere.

Euclidean star-tilings

There are no regular plane tilings of 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. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like , , , , etc, but none repeat periodically.

Hyperbolic tilings

Tessellations of hyperbolic 2-space
Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
can be called hyperbolic tilings.

There are infinitely many regular tilings in H². As stated above, every positive integer pairs such that 1/p + 1/q < 1/2 is a hyperbolic tiling.

A sampling:
Name Schläfli
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....

Symbol
Face
type
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....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Symmetry
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
Order-5 square tiling
Order-5 square tiling

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
0 *542
Order-4 pentagonal tiling
Order-4 pentagonal tiling

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
0 *542
Order-7 triangular tiling
Order-7 triangular tiling

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
0 *732
Order-3 heptagonal tiling
Order-3 heptagonal tiling

In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....
0 *732
Order-6 square tiling 0 *642
Order-4 hexagonal tiling 0 *642
Order-5 pentagonal tiling 0 *552 Self-dual
Order-8 triangular tiling 0 *832
Order-3 octagonal tiling 0 *832
Order-7 square tiling 0 *742
Order-4 heptagonal tiling 0 *742
Order-6 pentagonal tiling 0 *652
Order-5 hexagonal tiling 0 *652
Order-9 triangular tiling 0 *932
Order-3 enneagonal tiling 0 *932
Order-8 square tiling 0 *842
Order-4 octagonal tiling 0 *842
Order-7 pentagonal tiling 0 *752
Order-5 heptagonal tiling 0 *752
Order-6 hexagonal tiling 0 *662 Self-dual

There are 2 infinite forms of hyperbolic tilings whose faces
Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
or vertex figures
Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....
are star polygons: and their duals with m=7,9,11,...

There are a number of different ways to display the hyperbolic plane
Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
, including the Poincaré disc model below which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens
Fisheye lens

In photography, a fisheye lens is a wide-angle lens that takes in an extremely wide, Sphere image. Originally developed for use in meteorology to study cloud formation and called "whole-sky lenses", fisheye lenses quickly became popular in general photography for their unique, distorted appearance....
.

Four dimensions

Tessellations of Euclidean 3-space

There is only one regular tessellation of 3-space (honeycombs

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
. There are 4 regular honeycombs in H³:

Name Schläfli
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....

Symbol
Cell
type
Face
type
Edge
figure
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....

χ
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
Dual
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....
Order-3 icosahedral honeycomb 0 Self-dual
Order-5 cubic honeycomb 0
Order-4 dodecahedral honeycomb 0
Order-5 dodecahedral honeycomb 0 Self-dual

Here are some projected images: The first shows the perspective from the center of the disc in a Beltrami-Klein model, and the second and third from the outside with a Poincaré disk model
Poincaré disk model

In geometry, the Poincar? disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or ball , and the straight lines of the hyperbolic geometry are segments of circles contained in the disk orthogonal to the boundary of the disk...
.

(8 dodecahedra at a vertex)

(20 cubes at a vertex)

(12 icosahedra at a vertex)

There are also 11 H3 honeycombs which have infinite (Euclidean) cells and/or vertex figures: , , , , , , , , , , .

Five dimensions

Tessellations of Euclidean 4-space

There are three kinds of infinite regular tessellations (honeycombs
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
) that can tessellate four dimensional space:

Name Schläfli
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....

Symbol
Facet
type
Cell
type
Face
type
Face
figure
Edge
figure
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....

Dual
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....
Tesseractic honeycomb Self-dual
Hexadecachoric honeycomb
Icositetrachoric honeycomb

Projected portion of
(Tesseractic honeycomb)
Projected portion of
(Hexadecachoronic honeycomb)
Projected portion of
(Icositetrachoronic honeycomb)

Tessellations of hyperbolic 4-space

There are five kinds of convex regular honeycombs
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
and four kinds of star-honeycombs in H⁴ space. [Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213]

Five convex regular honeycombs in H⁴:
Name Schläfli
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....

Symbol
Facet
type
Cell
type
Face
type
Face
figure
Edge
figure
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....

Dual
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....
Order-5 pentachoric honeycomb
Order-3 hecatonicosachoric honeycomb
Order-5 tesseractic honeycomb
Order-4 hecatonicosachoric honeycomb
Order-5 hecatonicosachoric honeycomb Self-dual

Four regular star-honeycombs in H⁴ space:

Name Schläfli
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....

Symbol
Facet
type
Cell
type
Face
type
Face
figure
Edge
figure
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....

Dual
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....
Order-3 small stellated hecatonicosachoric honeycomb
Pentagrammic-order hexacosichoric honeycomb
Order-5 icosahedral hecatonicosachoric honeycomb
Order-3 great hecatonicosachoric honeycomb

There are also 2 H4 honeycombs with infinite (Euclidean) facets or vertex figures: ,

Higher dimensions

Tessellations of Euclidean Space

The hypercube honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
facets, four around every ridge
Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a Facet ....
.

Name Schläfli
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....

Facet
type 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....
Dual
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....
Square tiling
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....
Self-dual
Cubic honeycomb
Cubic honeycomb

The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
Self-dual
Tesseractic honeycomb Self-dual
Penteractic honeycomb Self-dual
Hexeractic honeycomb Self-dual
Hepteractic honeycomb Self-dual
Octeractic honeycomb Self-dual
n-hypercube honeycomb Self-dual

Tessellations of hyperbolic space

There are no finite-faceted regular tessellations of hyperbolic space of dimension 5 or higher.

There are 5 regular honeycombs in H5 with infinite (Euclidean) facets or vertex figures: .

Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular tessellations of hyperbolic space of dimension 6 or higher.

Apeirotopes

An apeirotope is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope does not curl back.

Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different things.

Two dimensions

A regular apeirogon
Apeirogon

An apeirogon is a Degeneracy polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides....
is a regular division of an infinitely long line into equal segments, joined by vertices. It has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it can form a straight line or a zig-zag. In three dimensions, it traces out a helical spiral. The zig-zag and spiral forms are said to be skew.

Three dimensions

An apeirohedron
Apeirohedron

An apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded....
is an infinite polyhedral surface. Like an apeirogon, it can be flat or skew. A flat apeirohedron is just a tiling of the plane. A skew apeirohedron is an intricate honeycomb-like structure which divides space into two regions.

There are thirty regular apeirohedra in Euclidean space. See section 7E of Abstract Regular Polytopes, by McMullen and Schulte. These include the tessellations of type and above, as well as (in the plane) polytopes of type: , and , and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)

Four and higher dimensions

The apeirochora have not been completely classified as of 2006.

Abstract polytopes

The abstract polytope
Abstract polytope

In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorics properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths etc....
s arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, euclidean and hyperbolic space, tessellations of other manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See for a sample. Some notable examples of abstract polytopes that do not appear elsewhere in this list are the 11-cell
11-cell

In mathematics, the 11-cell is a duality abstract polytope . Its 11 cells are hemi-icosahedron. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has...
and the 57-cell
57-cell

In mathematics, the 57-cell is a duality abstract polytope . Its 57 Cell s are hemi-dodecahedron. It also has 57 vertices, 171 edges and 171 faces....
.

See also

Polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
Regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
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....
Polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
Regular polyhedron
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
(5 regular 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....
s and 4 Kepler-Poinsot solid
Kepler-Poinsot solid

The Kepler-Poinsot polyhedra are the four Regular polyhedron Star polyhedron. They may be obtained by stellation the regular convex or Platonic solids, and differ from these in having regular star polygons for their faces or vertex figures....
s)
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....
Polychoron
Polychoron

In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
Convex regular 4-polytope
Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is 4-dimensional polytope which is both regular polytope and convex set. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
(6 regular polychora)
Uniform polychoron
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
Schläfli-Hess polychoron
Schläfli-Hess polychoron

In four dimensional geometry, Schl?fli-Hess polychora are the complete set of 10 Regular polytope self-intersecting Star polytope . They are named in honor of their discoverers: Ludwig Schl?fli and Edmund Hess....
(10 regular star polychora)
Tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
Tilings of regular polygons
Convex uniform honeycomb
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
Regular polytope
Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
Uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....

External links

(Look up Hexacosichoron and Hecatonicosachoron)

List of regular polytopes

Regular polytope summary count by dimension

One-dimensional regular polytopes

Two-dimensional regular polytopes

Three-dimensional regular polytopes

Four-dimensional regular polytopes

Five-dimensional regular polytopes

Classical convex polytopes

Two dimensions

Three dimensions

Four dimensions

Five dimensions

Higher dimensions

Finite non-convex polytopes - star-polytopes

Two dimensions

Three dimensions

Four dimensions

Higher dimensions

Tessellations

Two dimensions

Three dimensions

Euclidean (plane) tilings

Euclidean star-tilings

Hyperbolic tilings

Four dimensions

Tessellations of Euclidean 3-space

Tessellations of hyperbolic 3-space

Five dimensions

Tessellations of Euclidean 4-space

Tessellations of hyperbolic 4-space

Higher dimensions

Tessellations of Euclidean Space

Tessellations of hyperbolic space

Apeirotopes

Two dimensions

Three dimensions

Four and higher dimensions

Abstract polytopes

See also

External links