Regular star 4-polytope
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In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the
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 ...
in three dimensions and the
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s in two dimensions. There are six
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
and ten star regular 4-polytopes, giving a total of sixteen.


History

The convex regular 4-polytopes were first described by the Swiss
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
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 ...
in the mid-19th century. He discovered that there are precisely six such figures. Schläfli also found four of the regular star 4-polytopes: the grand 120-cell,
great stellated 120-cell In geometry, the great stellated 120-cell or great stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is one of four ''regular star 4-polytopes'' discovered by Lu ...
,
grand 600-cell In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells. It is one of four ''regular star 4-polytopes'' dis ...
, and
great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ...
. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V''  2). That excludes cells and vertex figures such as the
great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...
and
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 ...
.
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 re ...
(1843–1903) published the complete list 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''.


Construction

The existence of a regular 4-polytope \ is constrained by the existence of the regular polyhedra \, \ which form its cells and a
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
constraint :\sin\fracp \sin\fracr < \cos\fracq to ensure that the cells meet to form a closed 3-surface. The six convex and ten star polytopes described are the only solutions to these constraints. There are four nonconvex Schläfli symbols that have valid cells and vertex figures , and pass the dihedral test, but fail to produce finite figures: , , , .


Regular convex 4-polytopes

The regular convex 4-polytopes are the four-dimensional analogues of 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 in three dimensions and the convex
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s in two dimensions. Five of the six are clearly analogues of the five corresponding Platonic solids. The sixth, the
24-cell In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, o ...
, has no regular analogue in three dimensions. However, there exists a pair of irregular solids, the
cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
and its dual the
rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...
, which are partial analogues to the 24-cell (in complementary ways). Together they can be seen as the three-dimensional analogue of the 24-cell. Each convex regular 4-polytope is bounded by a set of 3-dimensional '' cells'' which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion.


Properties

Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes are all
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 ...
s and given in the notation described in that article. The number following the name of the group is the order of the group.
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD). Norman Johnson advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term ''polychoron'' being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
roots ''poly'' ("many") and ''choros'' ("room" or "space"). The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula: :N_0 - N_1 + N_2 - N_3 = 0\, where ''N''''k'' denotes the number of ''k''-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.). The topology of any given 4-polytope is defined by its
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
s and
torsion coefficient A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportiona ...
s.


As configurations

A regular 4-polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices ''in'' each edge (each edge ''has'' 2 vertices), and 2 cells meet ''at'' each face (each face ''belongs to'' 2 cells), in any regular 4-polytope. Notice that the configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.


Visualization

The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The Coxeter-Dynkin diagram graphs are also given below the Schläfli symbol.


Regular star (Schläfli–Hess) 4-polytopes

The Schläfli–Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora ( four-dimensional polytopes).Coxeter, ''Star polytopes and the Schläfli function f{α,β,γ)'' p. 122 2. ''The Schläfli-Hess polytopes'' They are named in honor of their discoverers:
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 ...
and
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 re ...
. Each is represented by a Schläfli symbol {''p'',''q'',''r''} in which one of the numbers is . They are thus analogous to the regular nonconvex Kepler–Poinsot polyhedra, which are in turn analogous to the pentagram.


Names

Their names given here were given by
John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
, extending Cayley's names for the Kepler–Poinsot polyhedra: along with ''stellated'' and ''great'', he adds a ''grand'' modifier. Conway offered these operational definitions: # stellation – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
) #greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a
great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...
) #aggrandizement – replaces the cells by large ones in same 3-spaces. (Example: a
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 ...
aggrandizes into a
grand 600-cell In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells. It is one of four ''regular star 4-polytopes'' dis ...
) John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral
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 ...
), pI=polyicoshedron {3,5,} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral
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, hec ...
), with prefix modifiers: ''g'', ''a'', and ''s'' for great, (ag)grand, and stellated. The final stellation, the ''great grand stellated polydodecahedron'' contains them all as ''gaspD''.


Symmetry

All ten polychora have ,3,5( H4) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order symmetry groups: ,5,5/2 ,5/2,5 ,3,5/2 /2,5,5/2 ,5/2,3 and ,3,5/2 Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.


Properties

Note: * There are 2 unique
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, matching those of the
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, hec ...
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 ...
. * 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, which are shown as ''wireframes'' orthographic projections. * There are 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, shown as ''solids'' (face-colored) orthographic projections. The cells (polyhedra), their faces (polygons), the ''polygonal edge figures'' and ''polyhedral
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 identified by their Schläfli symbols. {, class="wikitable sortable" ! Name
Conway (abbrev.) ! Orthogonal
projection ! Schläfli
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 ...
! C
{p, q} ! F
{p} ! E
{r} ! V
{q, r} ! Dens. ! χ , - align=center BGCOLOR="#e0e0ff" , Icosahedral 120-cell
polyicosahedron (pI) , , {3,5,5/2}
, 120
{3,5}
, 1200
{3}
, 720
{5/2}
, 120
{5,5/2}
, 4 , 480 , - align=center BGCOLOR="#ffe0e0" ,
Small stellated 120-cell In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the great gran ...

stellated polydodecahedron (spD) , , {5/2,5,3}
, 120
{5/2,5}
, 720
{5/2}
, 1200
{3}
, 120
{5,3}
, 4 , −480 , - align=center BGCOLOR="#e0ffe0" , Great 120-cell
great polydodecahedron (gpD) , , {5,5/2,5}
, 120
{5,5/2}
, 720
{5}
, 720
{5}
, 120
{5/2,5}
, 6 , 0 , - align=center BGCOLOR="#e0e0ff" , Grand 120-cell
grand polydodecahedron (apD) , , {5,3,5/2}
, 120
{5,3}
, 720
{5}
, 720
{5/2}
, 120
{3,5/2}
, 20 , 0 , - align=center BGCOLOR="#ffe0e0" ,
Great stellated 120-cell In geometry, the great stellated 120-cell or great stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is one of four ''regular star 4-polytopes'' discovered by Lu ...

great stellated polydodecahedron (gspD) , , {5/2,3,5}
, 120
{5/2,3}
, 720
{5/2}
, 720
{5}
, 120
{3,5}
, 20 , 0 , - align=center BGCOLOR="#e0ffe0" ,
Grand stellated 120-cell In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is also one of two such polytopes that is self-dual. Rela ...

grand stellated polydodecahedron (aspD) , , {5/2,5,5/2}
, 120
{5/2,5}
, 720
{5/2}
, 720
{5/2}
, 120
{5,5/2}
, 66 , 0 , - align=center BGCOLOR="#e0e0ff" , Great grand 120-cell
great grand polydodecahedron (gapD) , , {5,5/2,3}
, 120
{5,5/2}
, 720
{5}
, 1200
{3}
, 120
{5/2,3}
, 76 , −480 , - align=center BGCOLOR="#ffe0e0" ,
Great icosahedral 120-cell In geometry, the great icosahedral 120-cell, great polyicosahedron or great faceted 600-cell is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arran ...

great polyicosahedron (gpI) , , {3,5/2,5}
, 120
{3,5/2}
, 1200
{3}
, 720
{5}
, 120
{5/2,5}
, 76 , 480 , - align=center BGCOLOR="#e0e0ff" ,
Grand 600-cell In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells. It is one of four ''regular star 4-polytopes'' dis ...

grand polytetrahedron (apT) , , {3,3,5/2}
, 600
{3,3}
, 1200
{3}
, 720
{5/2}
, 120
{3,5/2}
, 191 , 0 , - align=center BGCOLOR="#ffe0e0" ,
Great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ...

great grand stellated polydodecahedron (gaspD) , , {5/2,3,3}
, 120
{5/2,3}
, 720
{5/2}
, 1200
{3}
, 600
{3,3}
, 191 , 0


See also

*
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, ...
*
List of regular polytopes This article lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an ''n''-sphere, Euclidean and hyperbolic spaces. A Schläfli ...
* Infinite regular 4-polytopes: ** One regular Euclidean honeycomb: {4,3,4} ** Four compact regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5} ** Eleven paracompact regular hyperbolic honeycombs: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}. * Abstract regular 4-polytopes: **
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-icosahedra ...
{3,5,3} **
57-cell In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry or ...
{5,3,5} *
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. Th ...
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
4-polytope families constructed from these 6 regular forms. *
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 ...
* Kepler-Poinsot polyhedra — regular
star polyhedron In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave p ...
* Star polygon — regular star polygons *
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 ...
*
5-polytope In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by ( 4-polytope) facets, pairs of which share a polyhedral cell. Definition A 5-polytope is a closed five-dimensional figure with vertic ...
* 6-polytope


References


Citations


Bibliography

* * * * * * * ** (Paper 10) * *


External links

*
Jonathan Bowers, 16 regular 4-polytopesRegular 4D Polytope Foldouts
A collection of stereographic projections of 4-polytopes.
A Catalog of Uniform PolytopesDimensions
2 hour film about the fourth dimension (contains stereographic projections of all regular 4-polytopes)
Reguläre Polytope




{{DEFAULTSORT:Regular 4-polytope 4-polytopes