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In mathematics, a regular 4-polytope is a
regular 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 instrum ...
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 equiv ...
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 A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...
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, mathematical structure, structure, space, Mathematica ...
Ludwig Schläfli 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, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
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 penta ...
and small stellated dodecahedron . Edmund Hess (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 symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
s 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, oc ...
, 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 ref ...
s and given in the notation described in that article. The number following the name of the group is the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of the group. John Conway 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
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 numbers and torsion coefficients.


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


Regular star (Schläfli–Hess) 4-polytopes

The Schläfli–Hess 4-polytopes are the complete set of 10
regular 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 instrum ...
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 and Edmund Hess. Each is represented by a
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
{''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, 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 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 ...
– replaces edges by longer edges in same lines. (Example: a
pentagon In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...
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 aro ...
) #greatening – replaces the faces by large ones in same planes. (Example: an
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...
greatens into a great icosahedron) #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) 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, heca ...
), 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 arrangements, 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, 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 ...
. * There are 4 unique edge arrangements, which are shown as ''wireframes''
orthographic projections Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal t ...
. * There are 7 unique face arrangements, 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 lines ...
s'' are identified by their
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
s. {, class="wikitable sortable" ! Name
Conway (abbrev.) ! Orthogonal
projection ! Schläfli
Coxeter ! 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
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
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
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 In geometry, the great grand 120-cell or great grand 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 small stella ...

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
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
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
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 *
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 Schläfl ...
* 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 Abstract may refer to: * ''Abstract'' (album), 1962 album by Joe Harriott * Abstract of title a summary of the documents affecting title to parcel of land * Abstract (law), a summary of a legal document * Abstract (summary), in academic publishi ...
regular 4-polytopes: ** 11-cell {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
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, se ...
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 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 operation ...
— 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), ...
*
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 vertices ...
* 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