This article lists the
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s and regular
polytope compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connecte ...
s in
Euclidean,
spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
and
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
spaces.
The
Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an ''n''-polytope equivalently describes a tessellation of an (''n'' − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
, which
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called
Coxeter notation. Another related symbol is the
Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the
cube has Schläfli symbol , and with its
octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
,
,3or , it is represented by Coxeter diagram .
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
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
. Infinite forms
tessellate
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
a one-lower-dimensional Euclidean space.
Infinite forms can be extended to tessellate a
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
. 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, like making a vertex with seven
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
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.
A more general definition of regular polytopes which do not have simple Schläfli symbols includes
regular skew polytopes and
regular skew apeirotopes with nonplanar
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
or
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s.
Overview
This table shows a summary of regular polytope counts by dimension.
Note that the Euclidean and hyperbolic tilings are given one dimension more than what would be expected. This is because of an analogy with finite polytopes: a convex regular ''n''-polytope can be seen as a tessellation of (''n''−1)-dimensional spherical space. Thus the three regular tilings of the Euclidean plane (by triangles, squares, and hexagons) are listed under dimension ''three'' rather than two.
There are no Euclidean regular star tessellations in any number of dimensions.
One dimension
A one-dimensional polytope or 1-polytope is a closed
line segment, bounded by its two endpoints. A 1-polytope is regular by definition and is represented by
Schläfli symbol , or a
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
with a single ringed node, .
Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol .
Although trivial as a polytope, it appears as the
edges of polygons and other higher dimensional polytopes. It is used in the definition of
uniform prisms like Schläfli symbol ×, or Coxeter diagram as a
Cartesian product of a line segment and a regular polygon.
Two dimensions (polygons)
The two-dimensional polytopes are called
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 to ...
s. Regular polygons are
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
and
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
. A p-gonal regular polygon is represented by
Schläfli symbol .
Usually only
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
s are considered regular, but
star polygons, like the
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 ...
, 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 be completed.
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.
Convex
The Schläfli symbol represents a
regular ''p''-gon.
Spherical
The regular
digon
In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
can be considered to be a
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
or
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
. For example, digon can be realised non-degenerately as a
spherical lune
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from ''luna'', the ...
. A
monogon
In geometry, a monogon, also known as a henagon, is a polygon with one edge and one vertex. It has Schläfli symbol .Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388
In Euclidean geometry
In Eucli ...
could also be realised on the sphere as a single point with a great circle through it. However, a monogon is not a valid
abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
because its single edge is incident to only one vertex rather than two.
Stars
There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers . They are called
star polygons and share the same
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
s of the convex regular polygons.
In general, for any natural number ''n'', there are n-pointed star regular polygonal stars with Schläfli symbols for all m such that ''m'' < ''n''/2 (strictly speaking =) and ''m'' and ''n'' are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
(as such, all stellations of a polygon with a prime number of sides will be regular stars). Cases where ''m'' and ''n'' are not coprime are called
compound polygons.
Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: , , , , ), however these do not appear to have been studied in detail.
There also exist ''failed'' star polygons, such as the ''piangle'', which do not cover the surface of a circle finitely many times.
Skew polygons
In 3-dimensional space, a
regular skew polygon
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrume ...
is called an ''antiprismatic polygon'', with the
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
of an
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 ...
, and a subset of edges, zig-zagging between top and bottom polygons.
In 4-dimensions a regular skew polygon can have vertices on a
Clifford torus and related by a
Clifford displacement. Unlike antiprismatic skew polygons, skew polygons on double rotations can include an odd-number of sides.
They can be seen in the
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
s of the
convex regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star reg ...
s, seen as regular plane polygons in the perimeter of Coxeter plane projection:
Three dimensions (polyhedra)
In three dimensions, polytopes are called
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 ...
:
A regular polyhedron with
Schläfli symbol , Coxeter diagrams , has a regular face type , and regular
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
.
A
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
(of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. 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 ...
, 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 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 ...
:
:
By enumerating the
permutations, we find five convex forms, four star forms and three plane tilings, all with polygons and limited to: , , , , and .
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
Convex
The five convex regular
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 ...
are called the
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s. 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is given with each vertex count. All these polyhedra have an
Euler characteristic (χ) of 2.
Spherical
In
spherical geometry, regular
spherical polyhedra
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most ...
(
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 ...
s of the
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
) exist that would otherwise be degenerate as polytopes. These are the
hosohedra
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...
and their dual
dihedra
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
.
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
calls these cases "improper" tessellations.
The first few cases (n from 2 to 6) are listed below.
Star-dihedra and hosohedra and also exist for any star polygon .
Stars
The regular
star polyhedra are called the
Kepler–Poinsot polyhedra and there are four of them, based on the
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
s of the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
and
icosahedron :
As
spherical tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...
s, these star forms overlap the sphere multiple times, called its ''density'', being 3 or 7 for these forms. The tiling images show a single
spherical polygon
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
face in yellow.
There are infinitely many ''failed'' star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are , , , , , , and .
Skew polyhedra
Regular skew polyhedra are generalizations to the set of
regular polyhedron
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 ...
which include the possibility of nonplanar
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s.
For 4-dimensional skew polyhedra, Coxeter offered a modified
Schläfli symbol for these figures, with implying 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
, ''m'' l-gons around a vertex, and ''n''-gonal holes. Their vertex figures are
skew polygon
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not ...
s, zig-zagging between two planes.
The regular skew polyhedra, represented by , follow this equation:
: 2 sin(π/l) sin(π/m) = cos(π/n)
Four of them can be seen in 4-dimensions as a subset of faces of four
regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star reg ...
s, sharing the same
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
and
edge arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
:
Four dimensions
Regular
4-polytopes
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
with
Schläfli symbol have cells of type
, faces of type
, edge figures
, and vertex figures
.
* A
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
(of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
* An
edge 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.
The existence of a regular 4-polytope
is constrained by the existence of the regular polyhedra
. A suggested name for 4-polytopes is "polychoron".
Each will exist in a space dependent upon this expression:
:
::
: Hyperspherical 3-space honeycomb or 4-polytope
::
: 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 for convex 4-polytopes is zero:
Convex
The 6 convex
regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star reg ...
s are shown in the table below. All these 4-polytopes have an
Euler characteristic (χ) of 0.
Spherical
Di-4-topes and
hoso-4-topes exist as regular tessellations of the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
.
Regular di-4-topes (2 facets) include: , , , , , , and their hoso-4-tope
duals
''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers.
Track listing
:* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...
(2 vertices): , , , , , . 4-polytopes of the form are the same as . There are also the cases which have dihedral cells and hosohedral vertex figures.
Stars
There are ten
regular star 4-polytopes, which are called the
Schläfli–Hess 4-polytopes. Their vertices are based on the convex
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
' and
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from " ...
'.
Ludwig Schläfli
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
found four of them and skipped the last six because he would not allow forms that failed the
Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2).
Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes.
See also
* Schläfli–Hess polychoron
* Hess polytope
References
* ''Regular Polytopes
In mathematics, a regu ...
(1843–1903) completed the full list of ten in his German book ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' (188
There are 4 unique
edge arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
s and 7 unique
face arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...
s from these 10 regular star 4-polytopes, shown as
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s:
There are 4 ''failed'' potential regular star 4-polytopes permutations: , , , . Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.
Five and more dimensions
In
five dimensions, a regular polytope can be named as
where
is the 4-face type,
is the cell type,
is the face type, and
is the face figure,
is the edge figure, and
is the vertex figure.
: A
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
(of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex.
: An
edge 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
(of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
: A
face 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
(of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.
A regular 5-polytope
exists only if
and
are regular 4-polytopes.
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. There are no non-convex regular polytopes in five dimensions or higher.
Convex
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.
There are also improper cases where some numbers in the Schläfli symbol are 2. For example, is an improper regular spherical polytope whenever is a regular spherical polytope, and is an improper regular spherical polytope whenever is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as .
5 dimensions
6 dimensions
7 dimensions
8 dimensions
9 dimensions
10 dimensions
...
Non-convex
There are no non-convex regular polytopes in five dimensions or higher, excluding
hosotopes formed from lower-dimensional non-convex regular polytopes.
Regular projective polytopes
A projective regular (''n''+1)-polytope exists when an original regular ''n''-spherical tessellation, , is
centrally symmetric
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
. Such a polytope is named hemi-
, and contain half as many elements. Coxeter gives a symbol /2, while McMullen writes
h/2 with ''h'' as the
coxeter number
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
.
Even-sided
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s have hemi-''2n''-gon projective polygons, /2.
There are 4 regular
projective polyhedra
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
Projec ...
related to 4 of 5
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s.
The hemi-cube and hemi-octahedron generalize as hemi-
''n''-cubes and hemi-''n''-
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
es in any dimensions.
Regular projective polyhedra
Regular projective 4-polytopes
In 4-dimensions 5 of 6 convex regular 4-polytopes generate projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.
Regular projective 5-polytopes
There are only 2 convex regular projective hemi-polytopes in dimensions 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below in dimension 5, for example:
Apeirotopes
An
apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.
Definition Abstract apeirotope
An abstract ''n''-polytope is a partially ordered set ''P'' (whose elements are called ''faces'') such tha ...
or infinite polytope is a
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
which has infinitely many
facets
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
. An ''n''-apeirotope is an infinite ''n''-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc.
There are two main geometric classes of apeirotope:
*Regular
honeycombs in ''n'' dimensions, which completely fill an ''n''-dimensional space.
*Regular
skew apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.
Definition Abstract apeirotope
An abstract ''n''-polytope is a partially ordered set ''P'' (whose elements are called ''faces'') such tha ...
s, comprising an ''n''-dimensional manifold in a higher space.
One dimension (apeirogons)
The straight
apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its
Schläfli symbol is , and Coxeter diagram .
...
![Regular apeirogon](https://upload.wikimedia.org/wikipedia/commons/3/30/Regular_apeirogon.svg)
...
It exists as the limit of the ''p''-gon as ''p'' tends to infinity, as follows:
Apeirogons in the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
, most notably the ''regular apeirogon'', , can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by
horocycle
In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...
s or
hypercycles rather than
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s.
Regular apeirogons that are scaled to converge at infinity have the symbol and exist on horocycles, while more generally they can exist on hypercycles.
Above are two regular hyperbolic apeirogons in the
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk th ...
, the right one shows perpendicular reflection lines of divergent
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s, separated by length λ.
Skew apeirogons
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.
Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular
skew apeirogon
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lin ...
traces out a helical spiral and may be either left- or right-handed.
Two dimensions (apeirohedra)
Euclidean tilings
There are three regular tessellations of the plane. All three have an
Euler characteristic (χ) of 0.
There are two improper regular tilings: , an apeirogonal
dihedron
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
, made from two
apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
s, each filling half the plane; and secondly, its dual, , an apeirogonal
hosohedron
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune havin ...
, seen as an infinite set of parallel lines.
Euclidean star-tilings
There are no regular plane tilings of
star polygons. 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 are ''
hyperbolic tilings''. There are infinitely many regular tilings in H
2. As stated above, every positive integer pair such that 1/''p'' + 1/''q'' < 1/2 gives a hyperbolic tiling. In fact, for the general
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
(''p'', ''q'', ''r'') the same holds true for 1/''p'' + 1/''q'' + 1/''r'' < 1.
There are a number of different ways to display the hyperbolic plane, including the
Poincaré disc model
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luci ...
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
A fisheye lens is an ultra wide-angle lens that produces strong visual distortion intended to create a wide panoramic or hemispherical image. Fisheye lenses achieve extremely wide angles of view, well beyond any rectilinear lens. Instead of pr ...
.
There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form , with p+q
listed above as tessellations)
* , , ...
* , , ...
* , , ...
* , , ...
* , , ...
* , , ...
* , , ...
* ...
* , , ...
A sampling:
The tilings have ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
vertices, on the ''edge'' of the Poincaré disc model. Their duals have ideal apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
al faces, meaning that they are inscribed in horocycle
In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...
s. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in hypercycles; in what is symbolised above, infinitely many tiles still fit around each ultra-ideal vertex.[Roice Nelson and Henry Segerman]
Visualizing Hyperbolic Honeycombs
/ref> (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
lines meet at an ultra-ideal point.)
Hyperbolic star-tilings
There are 2 infinite forms of hyperbolic tilings whose 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 ...
or 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s are star polygons: and their duals with ''m'' = 7, 9, 11, .... The tilings are stellation
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...
s of the tilings while the dual tilings are faceting
Stella octangula as a faceting of the cube
In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices.
New edges of a faceted pol ...
s of the tilings and greatenings of the tilings.
The patterns and continue for odd ''m'' < 7 as 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 ...
: when ''m'' = 5, we obtain 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 ...
and 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 ...
, and when ''m'' = 3, the case degenerates to a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
. The other two Kepler–Poinsot polyhedra (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 ...
and 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 ...
) do not have regular hyperbolic tiling analogues. If ''m'' is even, depending on how we choose to define , we can either obtain degenerate double covers of other tilings or compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive struct ...
tilings.
Skew apeirohedra in Euclidean 3-space
There are three regular skew apeirohedra in Euclidean 3-space, with regular skew polygon
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrume ...
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s. They share the same vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
and edge arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
of 3 convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells.
Twenty-eight such honey ...
s.
* 6 squares around each vertex:
* 4 hexagons around each vertex:
* 6 hexagons around each vertex:
There are thirty regular apeirohedra in Euclidean 3-space. These include those listed above, as well as 8 other "pure" apeirohedra, all related to the cubic honeycomb, , with others having skew polygon faces: 4, 4, 6, a, b, .*3, 6,4, 4,4, and 6,3.
Skew apeirohedra in hyperbolic 3-space
There are 31 regular skew apeirohedra in hyperbolic 3-space:
* 14 are compact: , , , , , , , , , , ,, , and .
* 17 are paracompact: , , , , , , , , , , , , , , , , and .
Three dimensions (4-apeirotopes)
Tessellations of Euclidean 3-space
There is only one non-degenerate regular tessellation of 3-space ('' honeycombs''), :
Improper tessellations of Euclidean 3-space
There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra
In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...
, dihedra
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
, , and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higher-dimensional analogues of the order-2 apeirogonal tiling
In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tiling of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schl ...
and apeirogonal hosohedron
In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol
...
.
Tessellations of hyperbolic 3-space
There are ten flat regular honeycombs of hyperbolic 3-space: (previously listed above as tessellations)
*4 are compact: , , , and
*while 6 are paracompact: , , , , , , , , , , and .
Tessellations of hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
can be called '' hyperbolic honeycombs''. There are 15 hyperbolic honeycombs in H3, 4 compact and 11 paracompact.
There are also 11 paracompact H3 honeycombs (those with infinite (Euclidean) cells and/or vertex figures): , , , , , , , , , , and .
Noncompact solutions exist as Lorentzian Coxeter group Lorentzian may refer to
* Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution
* Lorentz transformation
* Lorentzian manifold
See also
*Lorentz (disambiguation)
*Lorenz (disambiguati ...
s, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.
There are no regular hyperbolic star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.
Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.[
]
Four dimensions (5-apeirotopes)
Tessellations of Euclidean 4-space
There are three kinds of infinite regular tessellations ( honeycombs) that can tessellate Euclidean four-dimensional space:
There are also the two improper cases and .
There are three flat regular honeycombs of Euclidean 4-space:
* , , and .
There are seven flat regular convex honeycombs of hyperbolic 4-space:
*5 are compact: , , , ,
*2 are paracompact: , and .
There are four flat regular star honeycombs of hyperbolic 4-space:
* , , , and .
Tessellations of hyperbolic 4-space
There are seven convex regular honeycombs and four star-honeycombs in H4 space. Five convex ones are compact, and two are paracompact.
Five compact regular honeycombs in H4:
The two paracompact regular H4 honeycombs are: , .
Noncompact solutions exist as Lorentzian Coxeter group Lorentzian may refer to
* Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution
* Lorentz transformation
* Lorentzian manifold
See also
*Lorentz (disambiguation)
*Lorenz (disambiguati ...
s, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.
Star tessellations of hyperbolic 4-space
There are four regular star-honeycombs in H4 space, all compact:
Five dimensions (6-apeirotopes)
There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations)
*
There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations)
* , , , , and
Tessellations of Euclidean 5-space
The hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 -hypercube ...
is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
facets, four around every ridge
A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...
.
In E5, there are also the improper cases , , , , , and . In E''n'', and are always improper Euclidean tessellations.
Tessellations of hyperbolic 5-space
There are 5 regular honeycombs in H5, all paracompact, which include infinite (Euclidean) facets or vertex figures: , , , , and .
There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.
Since there are no regular star ''n''-polytopes for ''n'' ≥ 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in H''n'' for ''n'' ≥ 5.
6 dimensions and higher (7-apeirotopes+)
Tessellations of hyperbolic 6-space and higher
There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form not covered above (p,q,r,s,... natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
above 2, or infinity) will form a noncompact tessellation of hyperbolic ''n''-space.[
]
Compound polytopes
Two dimensional compounds
For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols for all m such that m < n/2 (strictly speaking =) and m and n are coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
. When m and n are not coprime, the star polygon obtained will be a regular polygon with ''n''/''m'' sides. A new figure is obtained by rotating these regular ''n''/''m''-gons one vertex to the left on the original polygon until the number of vertices rotated equals ''n''/''m'' minus one, and combining these figures. An extreme case of this is where ''n''/''m'' is 2, producing a figure consisting of ''n''/2 straight line segments; this is called a degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
star polygon.
In other cases where ''n'' and ''m'' have a common factor, a star polygon for a lower ''n'' is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form ''k'' as being more correct, where usually ''k'' = ''m''.
A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form ''k'', as 2, rather than the commonly used .
Coxeter's extended notation for compounds is of the form ''c'' 'd'''e'', indicating that ''d'' distinct 's together cover the vertices of ''c'' times and the facets of ''e'' times. If no regular exists, the first part of the notation is removed, leaving 'd'''e''; the opposite holds if no regular exists. The dual of ''c'' 'd'''e'' is ''e'' 'd'''c''. If ''c'' or ''e'' are 1, they may be omitted. For compound polygons, this notation reduces to 'k'' for example, the hexagram
, can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green).
A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
may be written thus as
Regular skew polygons also create compounds, seen in the edges of prismatic compound of antiprisms, for instance:
Three dimensional compounds
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is 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 ...
, edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...
, and 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 congruent ...
. With this definition there are 5 regular compounds.
Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, 'd'' denotes the components of the compound: ''d'' separate 's. The material ''before'' the square brackets denotes the vertex arrangement of the compound: ''c'' 'd''is a compound of ''d'' 's sharing the vertices of an counted ''c'' times. The material ''after'' the square brackets denotes the facet arrangement of the compound: 'd'''e'' is a compound of ''d'' 's sharing the faces of counted ''e'' times. These may be combined: thus ''c'' 'd'''e'' is a compound of ''d'' 's sharing the vertices of counted ''c'' times ''and'' the faces of counted ''e'' times. This notation can be generalised to compounds in any number of dimensions.
Euclidean and hyperbolic plane compounds
There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.
The Euclidean and hyperbolic compound families 2 (4 ≤ ''p'' ≤ ∞, ''p'' an integer) are analogous to the spherical stella octangula
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...
, 2 .
Four dimensional compounds
Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
lists 32 regular compounds of regular 4-polytopes in his book ''Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
''.[. Table VII, p. 305] McMullen adds six in his paper ''New Regular Compounds of 4-Polytopes'', in which he also proves that the list is now complete. In the following tables, the superscript (var) indicates that the labeled compounds are distinct from the other compounds with the same symbols.
There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different.
There are also fourteen ''partially regular'' compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds.
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...
and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive.
Euclidean 3-space compounds
The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a re ...
s, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is 'd''
Five dimensions and higher compounds
There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds (16, 240, or 480 7-simplices), and six known eight-dimensional ones (16, 240, or 480 8-cube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by Schl ...
s or 8-orthoplex
In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells ''4-faces'', 1792 ''5-faces'', 1024 ''6-faces'', and 256 ''7-faces''.
It has two constr ...
es). There is also one compound of ''n''-simplices in ''n''-dimensional space provided that ''n'' is one less than a power of two, and also two compounds (one of ''n''-cubes and a dual one of ''n''-orthoplexes) in ''n''-dimensional space if ''n'' is a power of two.
The Coxeter notation for these compounds are (using α''n'' = , β''n'' = , γ''n'' = ):
* 7-simplexes: ''c''γ7 7">6''c''α7'c''β7, where ''c'' = 1, 15, or 30
* 8-orthoplexes: ''c''γ8 8">6''c''β8* 8-cubes: 8">6''c''γ8'c''β8
The general cases (where ''n'' = 2''k'' and ''d'' = 22''k'' − ''k'' − 1, ''k'' = 2, 3, 4, ...):
* Simplexes: γ''n''−1 ''n''−1">'d''α''n''−1''n''−1
* Orthoplexes: γ''n'' ''n''">'d''β''n''* Hypercubes: ''n''">'d''γ''n''''n''
Euclidean honeycomb compounds
A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 -hypercube ...
s, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δ''n'' ''n''">'d''δ''n''''n'' where δ''n'' = when ''n'' = 2 and when ''n'' ≥ 3.
Abstract polytopes
The abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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. Se
this atlas
for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11-cell
In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedr ...
, , and the 57-cell, , which have regular projective polyhedra as cells and vertex figures.
The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the ''null polytope'' or empty set. These abstract elements can be mapped into ordinary space or ''realised'' as geometrical figures. Some abstract polyhedra have well-formed or ''faithful'' realisations, others do not. A ''flag'' is a connected set of elements of each dimension - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be ''regular'' if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
Five such regular abstract polyhedra, which can not be realised faithfully, were identified by H. S. M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
in his book ''Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'' (1977) and again by J. M. Wills
''J. The Jewish News of Northern California'', formerly known as ''Jweekly'', is a weekly print newspaper in Northern California, with its online edition updated daily. It is owned and operated by San Francisco Jewish Community Publications In ...
in his paper "The combinatorially regular polyhedra of index 2" (1987). They are all topologically equivalent to toroid
In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
s. Their construction, by arranging ''n'' faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.
:
These occur as dual pairs as follows:
*The medial rhombic triacontahedron
__NOTOC__
In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual ...
and dodecadodecahedron
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and .
The e ...
are dual to each other.
*The medial triambic icosahedron and ditrigonal dodecadodecahedron are dual to each other.
*The excavated dodecahedron
In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its exterior surface represents the Ef1g1 stellation of the icosahedron. It appears in Magnus We ...
is self-dual.
See also
* 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 ...
** Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
** 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 ...
* 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 ...
** Regular polyhedron
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 ...
(5 regular Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s and 4 Kepler–Poinsot solids)
*** Uniform polyhedron
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 ...
* 4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
** Regular 4-polytope
In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regu ...
(16 regular 4-polytopes, 4 convex and 10 star (Schläfli–Hess))
** Uniform 4-polytope
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
* Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
** Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619).
Notation of Eucli ...
** Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells.
Twenty-eight such honey ...
* Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
** Uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
* Regular map (graph theory)
In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such tha ...
Notes
References
*. See in particular Summary Tables II,III,IV,V, pp. 212–213.
**Originally published in .
* See in particular Tables I and II: Regular polytopes and honeycombs, pp. 294–296.
*
*
*.
hyperbolichoneycombs.org/
*. Reprint of 1930 ed., published by E. P. Dutton. See in particular Chapter X: The Regular Polytopes.
External links
Regular 4d Polytope Foldouts
* ttp://www.stat.berkeley.edu/~evans/shapiro/TesseractApplet.html Polytope Viewerbr>Polytopes and optimal packing of p points in n dimensional spheres
An atlas of small regular polytopes
Regular polyhedra through time
I. Hubard, ''Polytopes, Maps and their Symmetries''
Regular Star Polytopes
Nan Ma
{{DEFAULTSORT:Polytopes, Regular
Regular
Multi-dimensional geometry
Lists of shapes