geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a uniform 5-polytope is a five-dimensional

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...

. By definition, a uniform 5-polytope 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 i ...

and constructed from

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 polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

s from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

History of discovery

Regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s: (convex faces) **1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

s. *Convex

semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyto ...

s: (Various definitions before Coxeter's uniform category) **1900:

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

enumerated the list of nonprismatic semiregular convex polytopes with regular facets (

convex regular 4-polytope In mathematics, a regular 4-polytope or regular polychoron 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 co ...

s) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions''. *Convex uniform polytopes: **1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication ''Regular and Semi-Regular Polytopes I, II, and III''. **1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, ''The Theory of Uniform Polytopes and Honeycombs'', University of Toronto * Non-convex uniform polytopes: **1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation. **2000-2024: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes, with a current count of 1348 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.

Regular 5-polytopes

Regular 5-polytopes can be represented by the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

, with s 4-polytope facets around each

face The face is the front of the 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 affect th ...

. There are exactly three such regular polytopes, all convex: * -

5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°. The ...

* - 5-cube * -

5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5- cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regula ...

There are no nonconvex regular polytopes in 5 dimensions or above.

Convex uniform 5-polytopes

There are 104 known convex uniform 5-polytopes, plus a number of infinite families of

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

prisms, and polygon-polyhedron duoprisms. All except the ''grand antiprism prism'' are based on

s, reflection symmetry generated with

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 ...

Symmetry of uniform 5-polytopes in four dimensions

The

is the regular form in the A₅ family. The 5-cube and

are the regular forms in the B₅ family. The bifurcating graph of the D₅ family contains the

, as well as a

5-demicube In Five-dimensional space, five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with Alternation (geometry), alternated vertices removed. It was discovered by Thorold ...

which is an alternated 5-cube. Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a

, represented by rings around permutations of nodes in a

. Mirror

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

s can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form ,b,b,a have an extended symmetry, a,b,b,a, like ,3,3,3 doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry. If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions. Coxeter diagram finite rank5 correspondence

Coxeter diagram finite rank5 correspondence

;Fundamental families ;Uniform prisms There are 5 finite categorical

uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...

prism PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

atic families of polytopes based on the nonprismatic

s. There is one infinite family of 5-polytopes based on prisms of the uniform

s ××. ;Uniform duoprisms There are 3 categorical

atic families of polytopes based on

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

s of the

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

and

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 ...

s: ×.

Enumerating the convex uniform 5-polytopes

Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

family: A₅ ⁴** 19 uniform 5-polytopes *

Hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

Orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular polytope, regular, convex polytope that exists in ''n''-dimensions, dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensi ...

family: B₅ ,3³** 31 uniform 5-polytopes * Demihypercube D₅/E₅ family: ^2,1,1** 23 uniform 5-polytopes (8 unique) * Polychoral prisms: ** 56 uniform 5-polytope (45 unique) constructions based on prismatic families: ,3,3� nbsp; ,3,3� nbsp; ,3,3� nbsp; ^1,1,1� nbsp; ** One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two

grand antiprism In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polyto ...

s connected by polyhedral prisms. That brings the tally to: 19+31+8+45+1=104 In addition there are: * Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: 'p''� 'q''� nbsp; * Infinitely many uniform 5-polytope constructions based on duoprismatic families: ,3� 'p'' ,3� 'p'' ,3� 'p''

The A₅ family

There are 19 forms based on all permutations of the

s with one or more rings. (16+4-1 cases) They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron). The A₅ family has symmetry of order 720 (6

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440. The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).

The B₅ family

The B₅ family has symmetry of order 3840 (5!×2⁵). This family has 2⁵−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the

. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D₅ family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.) For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both. The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2. {, class="wikitable" !rowspan=2, # !rowspan=2, Base point !rowspan=2, Name

!colspan=5, Element counts !rowspan=2, Vertex
figure !colspan=6 , Facet counts by location: ,3,3,3, - BGCOLOR="#e0e0f0" !4, , 3, , 2, , 1, , 0 !
,3,3BR>(10) !
,3,2BR>(40) !
,2,3BR>(80) !
,3,3BR>(80) !
,3,3BR>(32) ! Alt , - BGCOLOR="#f0e0e0" !20 , , (0,0,0,0,1)√2, ,

triacontaditeron (tac)
, , 32, , 80, , 80, , 40, , 10 , ,
{3,3,4}, , - , , - , , - , , - , , Schlegel wireframe 5-cell

{3,3,3}, , , - BGCOLOR="#f0e0e0" !21 , , (0,0,0,1,1)√2, , Rectified 5-orthoplex
rectified triacontaditeron (rat)
, , 42, , 240, , 400, , 240, , 40 , , Rectified pentacross verf

{ }×{3,4}, , Schlegel wireframe 16-cell

{3,3,4} , , - , , - , , - , , Schlegel half-solid rectified 5-cell

r{3,3,3} , , , - BGCOLOR="#f0e0e0" !22 , , (0,0,0,1,2)√2, , Truncated 5-orthoplex
truncated triacontaditeron (tot)
, , 42, , 240, , 400, , 280, , 80 , , Truncated pentacross

(Octah.pyr), , Schlegel wireframe 16-cell

{3,3,4} , , - , , - , , - , , Schlegel half-solid truncated pentachoron

t{3,3,3}, , , - BGCOLOR="#e0f0e0" !23 , , (0,0,1,1,1)√2, , Birectified 5-cube
penteractitriacontaditeron (nit)
(Birectified 5-orthoplex)
, , 42, , 280, , 640, , 480, , 80 , , Birectified penteract verf

{4}×{3}, , Schlegel half-solid rectified 16-cell

r{3,3,4} , , - , , - , , - , , Schlegel half-solid rectified 5-cell

r{3,3,3} , , , -BGCOLOR="#f0e0e0" !24 , , (0,0,1,1,2)√2, , Cantellated 5-orthoplex
small rhombated triacontaditeron (sart)
, , 82, , 640, , 1520, , 1200, , 240 , , Cantellated pentacross verf

Prism-wedge, , Schlegel half-solid rectified 16-cell

r{3,3,4}, , Octahedral prism

{ }×{3,4} , , - , , - , , Schlegel half-solid cantellated 5-cell

rr{3,3,3} , , , - BGCOLOR="#f0e0e0" !25 , , (0,0,1,2,2)√2, , Bitruncated 5-orthoplex
bitruncated triacontaditeron (bittit)
, , 42, , 280, , 720, , 720, , 240 , , Bitruncated pentacross verf

, ,

t{3,3,4} , , - , , - , , - , , Schlegel half-solid bitruncated 5-cell

2t{3,3,3} , , , - BGCOLOR="#f0e0e0" !26 , , (0,0,1,2,3)√2, , Cantitruncated 5-orthoplex
great rhombated triacontaditeron (gart)
, , 82, , 640, , 1520, , 1440, , 480 , , Canitruncated 5-orthoplex verf

, ,

t{3,3,4}, , Octahedral prism

{ }×{3,4} , , -, , - , , Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !27 , , (0,1,1,1,1)√2, , Rectified 5-cube
rectified penteract (rin)
, , 42, , 200, , 400, , 320, , 80 , , Rectified 5-cube verf

{3,3}×{ }, , Schlegel half-solid rectified 8-cell

r{4,3,3}, , - , , - , , - , , Schlegel wireframe 5-cell

{3,3,3} , , , -BGCOLOR="#f0e0e0" !28 , , (0,1,1,1,2)√2, , Runcinated 5-orthoplex
small prismated triacontaditeron (spat)
, , 162, , 1200, , 2160, , 1440, , 320 , , Runcinated pentacross verf

, ,

r{4,3,3} , , Cuboctahedral prism

{ }×r{3,4} , , 3-4 duoprism

{3}×{4}, , , , Schlegel half-solid runcinated 5-cell

t_0,3{3,3,3} , , , - BGCOLOR="#e0f0e0" !29 , , (0,1,1,2,2)√2, , Bicantellated 5-cube
small birhombated penteractitriacontaditeron (sibrant)
(Bicantellated 5-orthoplex)
, , 122, , 840, , 2160, , 1920, , 480 , , Bicantellated penteract verf

, ,

rr{3,3,4}, , - , , 3-4 duoprism

{4}×{3}, , - , , Schlegel half-solid cantellated 5-cell

rr{3,3,3} , , , - BGCOLOR="#f0e0e0" !30 , , (0,1,1,2,3)√2, , Runcitruncated 5-orthoplex
prismatotruncated triacontaditeron (pattit)
, , 162, , 1440, , 3680, , 3360, , 960 , , Runcitruncated 5-orthoplex verf

, ,

rr{3,3,4} , , Cuboctahedral prism

{ }×r{3,4} , , 6-4 duoprism

{6}×{4}, , - , , Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !31 , , (0,1,2,2,2)√2, , Bitruncated 5-cube
bitruncated penteract (bittin)
, , 42, , 280, , 720, , 800, , 320 , , Bitruncated penteract verf

, ,

2t{4,3,3}, , - , , - , , - , , Schlegel half-solid truncated pentachoron

t{3,3,3} , , , - BGCOLOR="#f0e0e0" !32 , , (0,1,2,2,3)√2, , Runcicantellated 5-orthoplex
prismatorhombated triacontaditeron (pirt)
, , 162, , 1200, , 2960, , 2880, , 960 , , Runcicantellated 5-orthoplex verf

, ,

2t{4,3,3}, , Truncated octahedral prism

{ }×t{3,4}, , 3-4 duoprism

{3}×{4} , , - , , Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , , , - BGCOLOR="#e0f0e0" !33 , , (0,1,2,3,3)√2, , Bicantitruncated 5-cube
great birhombated triacontaditeron (gibrant)
(Bicantitruncated 5-orthoplex)
, , 122, , 840, , 2160, , 2400, , 960 , , Bicantellated penteract verf

, ,

Schlegel half-solid cantitruncated 16-cell

tr{3,3,4}, , - , , 3-4 duoprism

{4}×{3}, , - , , Schlegel half-solid cantellated 5-cell

rr{3,3,3} , , , - BGCOLOR="#f0e0e0" !34 , , (0,1,2,3,4)√2, , Runcicantitruncated 5-orthoplex
great prismated triacontaditeron (gippit)
, , 162, , 1440, , 4160, , 4800, , 1920 , ,

, ,

tr{3,3,4} , , Truncated octahedral prism

{ }×t{3,4} , , 6-4 duoprism

{6}×{4}, , - , , Schlegel half-solid omnitruncated 5-cell

t_0,1,2,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !35 , , (1,1,1,1,1), , 5-cube
penteract (pent)
, , 10, , 40, , 80, , 80, , 32 , , 5-cube verf

{3,3,3}, , Schlegel wireframe 8-cell

{4,3,3}, , - , , - , , - , , - , , , - BGCOLOR="#e0f0e0" !36 , , (1,1,1,1,1)
+ (0,0,0,0,1)√2, ,

Stericated 5-cube In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations ( sterication) of the regular 5-cube. There are eight degrees of sterication for the 5-cube, including permutations of runcination, ...

small cellated penteractitriacontaditeron (scant)
(Stericated 5-orthoplex)
, , 242, , 800, , 1040, , 640, , 160 , , Stericated penteract verf

Tetr.antiprm, , Schlegel wireframe 8-cell

{4,3,3}, , Schlegel wireframe 8-cell

{4,3}×{ }, , 3-4 duoprism

{4}×{3}, , Tetrahedral prism

{ }×{3,3}, , Schlegel wireframe 5-cell

{3,3,3} , , , - BGCOLOR="#e0e0f0" !37 , , (1,1,1,1,1)
+ (0,0,0,1,1)√2, , Runcinated 5-cube
small prismated penteract (span)
, , 202, , 1240, , 2160, , 1440, , 320 , , Runcinated penteract verf

, ,

t_0,3{4,3,3}, , - , , 3-4 duoprism

{4}×{3}, , Octahedral prism

{ }×r{3,3}, , Schlegel half-solid rectified 5-cell

r{3,3,3} , , , - BGCOLOR="#f0e0e0" !38 , , (1,1,1,1,1)
+ (0,0,0,1,2)√2, , Steritruncated 5-orthoplex
celliprismated triacontaditeron (cappin)
, , 242, , 1520, , 2880, , 2240, , 640 , , Steritruncated 5-orthoplex verf

, ,

t_0,3{4,3,3} , , Schlegel wireframe 8-cell

{4,3}×{ } , , 6-4 duoprism

{6}×{4} , , Truncated tetrahedral prism

{ }×t{3,3} , , Schlegel half-solid truncated pentachoron

t{3,3,3} , , , - BGCOLOR="#e0e0f0" !39 , , (1,1,1,1,1)
+ (0,0,1,1,1)√2, ,

Cantellated 5-cube In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the ...

small rhombated penteract (sirn)
, , 122, , 680, , 1520, , 1280, , 320 , , Cantellated 5-cube vertf

Prism-wedge, , Schlegel half-solid cantellated 8-cell

rr{4,3,3}, , - , , - , , Tetrahedral prism

{ }×{3,3}, , Schlegel half-solid rectified 5-cell

r{3,3,3} , , , - BGCOLOR="#e0f0e0" !40 , , (1,1,1,1,1)
+ (0,0,1,1,2)√2, , Stericantellated 5-cube
cellirhombated penteractitriacontaditeron (carnit)
(Stericantellated 5-orthoplex)
, , 242, , 2080, , 4720, , 3840, , 960 , , Stericantellated 5-orthoplex verf

, ,

rr{4,3,3}, , Rhombicuboctahedral prism

rr{4,3}×{ }, , 3-4 duoprism

{4}×{3}, , Cuboctahedral prism

{ }×rr{3,3}, , Schlegel half-solid cantellated 5-cell

rr{3,3,3} , , , - BGCOLOR="#e0e0f0" !41 , , (1,1,1,1,1)
+ (0,0,1,2,2)√2, , Runcicantellated 5-cube
prismatorhombated penteract (prin)
, , 202, , 1240, , 2960, , 2880, , 960 , , Runcicantellated 5-cube verf

, ,

t_0,2,3{4,3,3}, , - , , 3-4 duoprism

{4}×{3}, , Truncated tetrahedral prism

{ }×t{3,3}, , Schlegel half-solid bitruncated 5-cell

2t{3,3,3} , , , - BGCOLOR="#f0e0e0" !42 , , (1,1,1,1,1)
+ (0,0,1,2,3)√2, , Stericantitruncated 5-orthoplex
celligreatorhombated triacontaditeron (cogart)
, , 242, , 2320, , 5920, , 5760, , 1920 , , Stericanitruncated 5-orthoplex verf

, ,

t_0,2,3{4,3,3}, , Rhombicuboctahedral prism

rr{4,3}×{ }, , 6-4 duoprism

{6}×{4}, , Truncated octahedral prism

{ }×tr{3,3}, , Schlegel half-solid cantitruncated 5-cell

tr{3,3,3} , , , - BGCOLOR="#e0e0f0" !43 , , (1,1,1,1,1)
+ (0,1,1,1,1)√2, , Truncated 5-cube
truncated penteract (tan)
, , 42, , 200, , 400, , 400, , 160 , , Truncated 5-cube verf

Tetrah.pyr, , Schlegel half-solid truncated tesseract

t{4,3,3}, , - , , - , , - , , Schlegel wireframe 5-cell

{3,3,3} , , , - BGCOLOR="#e0e0f0" !44 , , (1,1,1,1,1)
+ (0,1,1,1,2)√2, , Steritruncated 5-cube
celliprismated triacontaditeron (capt)
, , 242, , 1600, , 2960, , 2240, , 640 , , Steritruncated 5-cube verf

, ,

t{4,3,3}, , Truncated cubic prism

t{4,3}×{ }, , 8-3 duoprism

{8}×{3}, , Tetrahedral prism

{ }×{3,3}, , Schlegel half-solid runcinated 5-cell

t_0,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !45 , , (1,1,1,1,1)
+ (0,1,1,2,2)√2, , Runcitruncated 5-cube
prismatotruncated penteract (pattin)
, , 202, , 1560, , 3760, , 3360, , 960 , , Runcitruncated 5-cube verf

, ,

Schlegel half-solid runcitruncated 8-cell

t_0,1,3{4,3,3} , , - , , 8-3 duoprism

{8}×{3}, , Octahedral prism

{ }×r{3,3} , , Schlegel half-solid cantellated 5-cell

rr{3,3,3} , , , - BGCOLOR="#e0f0e0" !46 , , (1,1,1,1,1)
+ (0,1,1,2,3)√2, , Steriruncitruncated 5-cube
celliprismatotruncated penteractitriacontaditeron (captint)
(Steriruncitruncated 5-orthoplex)
, , 242, , 2160, , 5760, , 5760, , 1920 , , Steriruncitruncated 5-cube verf

, ,

t_0,1,3{4,3,3}, , Truncated cubic prism

t{4,3}×{ }, , 8-6 duoprism

{8}×{6}, , Truncated tetrahedral prism

{ }×t{3,3}, , Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !47 , , (1,1,1,1,1)
+ (0,1,2,2,2)√2, , Cantitruncated 5-cube
great rhombated penteract (girn)
, , 122, , 680, , 1520, , 1600, , 640 , , Canitruncated 5-cube verf

, ,

Schlegel half-solid cantitruncated 8-cell

tr{4,3,3}, , - , , - , , Tetrahedral prism

{ }×{3,3}, , Schlegel half-solid truncated pentachoron

t{3,3,3} , , , - BGCOLOR="#e0e0f0" !48 , , (1,1,1,1,1)
+ (0,1,2,2,3)√2, , Stericantitruncated 5-cube
celligreatorhombated penteract (cogrin)
, , 242, , 2400, , 6000, , 5760, , 1920 , , Stericanitruncated 5-cube verf

, ,

tr{4,3,3}, , Truncated cuboctahedral prism

tr{4,3}×{ }, , 8-3 duoprism

{8}×{3}, , Cuboctahedral prism

{ }×rr{3,3}, , Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , , , - BGCOLOR="#e0e0f0" !49 , , (1,1,1,1,1)
+ (0,1,2,3,3)√2, , Runcicantitruncated 5-cube
great prismated penteract (gippin)
, , 202, , 1560, , 4240, , 4800, , 1920 , , Runcicantitruncated 5-cube verf

, ,

Schlegel half-solid omnitruncated 8-cell

t_0,1,2,3{4,3,3}, , - , , 8-3 duoprism

{8}×{3}, , Truncated tetrahedral prism

{ }×t{3,3}, , Schlegel half-solid cantitruncated 5-cell

tr{3,3,3} , , , - BGCOLOR="#e0f0e0" !50 , , (1,1,1,1,1)
+ (0,1,2,3,4)√2, , Omnitruncated 5-cube
great cellated penteractitriacontaditeron (gacnet)
(omnitruncated 5-orthoplex)
, , 242, , 2640, , 8160, , 9600, , 3840 , , Omnitruncated 5-cube verf

Irr. {3,3,3}, , Schlegel half-solid omnitruncated 8-cell

tr{4,3}×{ }, , Truncated cuboctahedral prism

tr{4,3}×{ }, , 8-6 duoprism

{8}×{6}, , Truncated octahedral prism

{ }×tr{3,3}, , Schlegel half-solid omnitruncated 5-cell

t_0,1,2,3{3,3,3} , , , - BGCOLOR="#d0f0f0" !51 , ,

hemipenteract (hin)
= , 26 , 120 , 160 , 80 , 16 , Demipenteract verf

r{3,3,3} , Schlegel wireframe 16-cell

h{4,3,3} , - , - , - , - , (16)
Schlegel wireframe 5-cell

{3,3,3} , - BGCOLOR="#d0f0f0" !52 , , Cantic 5-cube
Truncated hemipenteract (thin)
= , 42 , 280 , 640 , 560 , 160 , Truncated 5-demicube verf

h₂{4,3,3} , - , - , - , (16)
Schlegel half-solid rectified 5-cell

r{3,3,3} , (16)
Schlegel half-solid truncated pentachoron

t{3,3,3} , - BGCOLOR="#d0f0f0" !53 , , Runcic 5-cube
Small rhombated hemipenteract (sirhin)
= , 42 , 360 , 880 , 720 , 160 , , Schlegel half-solid rectified 8-cell

h₃{4,3,3} , - , - , - , (16)
Schlegel half-solid rectified 5-cell

r{3,3,3} , (16)
Schlegel half-solid cantellated 5-cell

rr{3,3,3} , - BGCOLOR="#d0f0f0" !54 , , Steric 5-cube
Small prismated hemipenteract (siphin)
= , 82 , 480 , 720 , 400 , 80 , , Schlegel wireframe 16-cell

h{4,3,3} , Tetrahedral prism

h{4,3}×{} , - , - , (16)
Schlegel wireframe 5-cell

{3,3,3} , (16)
Schlegel half-solid runcinated 5-cell

t_0,3{3,3,3} , - BGCOLOR="#d0f0f0" !55 , , Runcicantic 5-cube
Great rhombated hemipenteract (girhin)
= , 42 , 360 , 1040 , 1200 , 480 , , Schlegel half-solid bitruncated 8-cell

h_2,3{4,3,3} , - , - , - , (16)
Schlegel half-solid bitruncated 5-cell

2t{3,3,3} , (16)
Schlegel half-solid cantitruncated 5-cell

tr{3,3,3} , - BGCOLOR="#d0f0f0" !56 , , Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
= , 82 , 720 , 1840 , 1680 , 480 , , Schlegel half-solid truncated 16-cell

h₂{4,3,3} , Truncated tetrahedral prism

h₂{4,3}×{} , - , - , (16)
Schlegel half-solid cantellated 5-cell

rr{3,3,3} , (16)
Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , - BGCOLOR="#d0f0f0" !57 , , Steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
= , 82 , 560 , 1280 , 1120 , 320 , , Schlegel half-solid rectified 8-cell

h₃{4,3,3} , Tetrahedral prism

h{4,3}×{} , - , - , (16)
Schlegel half-solid truncated pentachoron

t{3,3,3} , (16)
Schlegel half-solid runcitruncated 5-cell

t_0,1,3{3,3,3} , - BGCOLOR="#d0f0f0" !58 , , Steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
= , 82 , 720 , 2080 , 2400 , 960 , , Schlegel half-solid bitruncated 8-cell

h_2,3{4,3,3} , Truncated tetrahedral prism

h₂{4,3}×{} , - , - , (16)
Schlegel half-solid cantitruncated 5-cell

tr{3,3,3} , (16)
Schlegel half-solid omnitruncated 5-cell

t_0,1,2,3{3,3,3} , - BGCOLOR="#d0f0f0" !Nonuniform , , Alternated runcicantitruncated 5-orthoplex
Snub prismatotriacontaditeron (snippit)
Snub hemipenteract (snahin)
= , 1122 , 6240 , 10880 , 6720 , 960 , , Schlegel half-solid alternated cantitruncated 16-cell

Schlegel half-solid alternated cantitruncated 16-cell

sr{3,3,4} , sr{2,3,4} , sr{3,2,4} , - , ht_0,1,2,3{3,3,3} , (960)
Schlegel wireframe 5-cell

Irr. {3,3,3} , - BGCOLOR="#d0f0f0" !Nonuniform , , Edge-snub 5-orthoplex
Pyritosnub penteract (pysnan)
, 1202 , 7920 , 15360 , 10560 , 1920 , , sr₃{3,3,4} , sr₃{2,3,4} , sr₃{3,2,4} , Icosahedral prism

s{3,3}×{ } , ht_0,1,2,3{3,3,3} , (960)
Tetrahedral prism

Irr. {3,3}×{ } , - BGCOLOR="#d0f0f0" !Nonuniform , , Snub 5-cube
Snub penteract (snan)
, 2162 , 12240 , 21600 , 13440 , 960 , , ht_0,1,2,3{3,3,4} , ht_0,1,2,3{2,3,4} , ht_0,1,2,3{3,2,4} , ht_0,1,2,3{3,3,2} , ht_0,1,2,3{3,3,3} , (1920)
Schlegel wireframe 5-cell

Irr. {3,3,3}

The D₅ family

The D₅ family has symmetry of order 1920 (5! x 2⁴). This family has 23 Wythoffian uniform polytopes, from ''3×8-1'' permutations of the D₅

with one or more rings. 15 (2×8-1) are repeated from the B₅ family and 8 are unique to this family, though even those 8 duplicate the alternations from the B₅ family. In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5-polytopes 51 through 58 above. {, class="wikitable" !rowspan=2, # !rowspan=2,

symbols
Johnson and Bowers names !colspan=5, Element counts !rowspan=2, Vertex
figure !colspan=6 , Facets by location: ^1,2,1 , - !4 !3 !2 !1 !0 !
,3,3BR>(16) !
^1,1,1BR>(10) !
,3� nbsp;BR>(40) !
nbsp;� � nbsp;BR>(80) !
,3,3BR>(16) ! Alt , - ! 1, =
h{4,3,3,3},

Hemipenteract (hin) , 26 , 120 , 160 , 80 , 16 ,
r{3,3,3} ,
{3,3,3} ,
h{4,3,3} , - , - , - , , - ! 2, =
h₂{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin) , 42 , 280 , 640 , 560 , 160 , ,
t{3,3,3} ,
h₂{4,3,3} , - , - ,
r{3,3,3} , , - ! 3, =
h₃{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin) , 42 , 360 , 880 , 720 , 160 , ,
rr{3,3,3} ,
h₃{4,3,3} , - , - ,
r{3,3,3} , , - ! 4, =
h₄{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin) , 82 , 480 , 720 , 400 , 80 , ,
t_0,3{3,3,3} ,
h{4,3,3} ,
h{4,3}×{} , - ,
{3,3,3} , , - ! 5, =
h_2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin) , 42 , 360 , 1040 , 1200 , 480 , ,
2t{3,3,3} ,
h_2,3{4,3,3} , - , - ,
tr{3,3,3} , , - ! 6, =
h_2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin) , 82 , 720 , 1840 , 1680 , 480 , ,
t_0,1,3{3,3,3} ,
h₂{4,3,3} ,
h₂{4,3}×{} , - ,
rr{3,3,3} , , - ! 7, =
h_3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin) , 82 , 560 , 1280 , 1120 , 320 , ,
t_0,1,3{3,3,3} ,
h₃{4,3,3} ,
h{4,3}×{} , - ,
t{3,3,3} , , - ! 8, =
h_2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin) , 82 , 720 , 2080 , 2400 , 960 , ,
t_0,1,2,3{3,3,3} ,
h_2,3{4,3,3} ,
h₂{4,3}×{} , - ,
tr{3,3,3} , , - bgcolor="#D0F0F0" ! Nonuniform , =
ht_0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin) , 1122 , 6240 , 10880 , 6720 , 960 , , ht_0,1,2,3{3,3,3} ,
sr{3,3,4} , sr{2,3,4} , sr{3,2,4} , ht_0,1,2,3{3,3,3} , (960)

Irr. {3,3,3}

Uniform prismatic forms

There are 5 finite categorical

atic families of polytopes based on the nonprismatic uniform

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: Vertex (geometry), vertices, Edge (geo ...

s. For simplicity, most alternations are not shown.

A₄ × A₁

This prismatic family has 9 forms: The A₁ x A₄ family has symmetry of order 240 (2*5!). {, class="wikitable" !rowspan=2, # !rowspan=2,

and Schläfli
symbols
Name !colspan=5 rowspan=1, Element counts , - ! Facets, , Cells, , Faces, , Edges, , Vertices , - , 59 , = {3,3,3}×{ }
5-cell prism (penp) , 7, , 20, , 30, , 25, , 10 , - , 60 , = r{3,3,3}×{ }
Rectified 5-cell prism (rappip) , 12, , 50, , 90, , 70, , 20 , - , 61 , = t{3,3,3}×{ }
Truncated 5-cell prism (tippip) , 12, , 50, , 100, , 100, , 40 , - , 62 , = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip) , 22, , 120, , 250, , 210, , 60 , - BGCOLOR="#e0f0e0" , 63 , = t_0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip) , 32, , 130, , 200, , 140, , 40 , - BGCOLOR="#e0f0e0" , 64 , = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap) , 12, , 60, , 140, , 150, , 60 , - , 65 , = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip) , 22, , 120, , 280, , 300, , 120 , - , 66 , = t_0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip) , 32, , 180, , 390, , 360, , 120 , - BGCOLOR="#e0f0e0" , 67 , = t_0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism (gippiddip) , 32, , 210, , 540, , 600, , 240

B₄ × A₁

This prismatic family has 16 forms. (Three are shared with ,4,3� nbsp;family) The A₁×B₄ family has symmetry of order 768 (2⁵4!). The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes. {, class="wikitable" !rowspan=2, # !rowspan=2,

and Schläfli
symbols
Name !colspan=5 rowspan=1, Element counts , - ! Facets, , Cells, , Faces, , Edges, , Vertices , - BGCOLOR="#f0e0e0" , 6'', , = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube) , 10, , 40, , 80, , 80, , 32 , - BGCOLOR="#f0e0e0" , 68, , = r{4,3,3}×{ }
Rectified tesseractic prism (rittip) , 26, , 136, , 272, , 224, , 64 , - BGCOLOR="#f0e0e0" , 69, , = t{4,3,3}×{ }
Truncated tesseractic prism (tattip) , 26, , 136, , 304, , 320, , 128 , - BGCOLOR="#f0e0e0" , 70, , = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip) , 58, , 360, , 784, , 672, , 192 , - BGCOLOR="#e0f0e0" , 71, , = t_0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip) , 82, , 368, , 608, , 448, , 128 , - BGCOLOR="#e0f0e0" , 72, , = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp) , 26, , 168, , 432, , 480, , 192 , - BGCOLOR="#f0e0e0" , 73, , = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip) , 58, , 360, , 880, , 960, , 384 , - BGCOLOR="#f0e0e0" , 74, , = t_0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp) , 82, , 528, , 1216, , 1152, , 384 , - BGCOLOR="#e0f0e0" , 75, , = t_0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip) , 82, , 624, , 1696, , 1920, , 768 , - BGCOLOR="#e0e0f0" , 76, , = {3,3,4}×{ }
16-cell prism (hexip) , 18, , 64, , 88, , 56, , 16 , - BGCOLOR="#e0e0f0" , 77, , = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism) , 26, , 144, , 288, , 216, , 48 , - BGCOLOR="#e0e0f0" , 78, , = t{3,3,4}×{ }
Truncated 16-cell prism (thexip) , 26, , 144, , 312, , 288, , 96 , - BGCOLOR="#e0e0f0" , 79, , = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism) , 50, , 336, , 768, , 672, , 192 , - BGCOLOR="#e0e0f0" , 80, , = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism) , 50, , 336, , 864, , 960, , 384 , - BGCOLOR="#e0e0f0" , 81, , = t_0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip) , 82, , 528, , 1216, , 1152, , 384 , - BGCOLOR="#a0e0f0" , 82, , = sr{3,3,4}×{ }
snub 24-cell prism (sadip) , 146, , 768, , 1392, , 960, , 192 , - BGCOLOR="#a0e0f0" , Nonuniform, ,
rectified tesseractic alterprism (rita) , 50, , 288, , 464, , 288, , 64 , - BGCOLOR="#a0e0f0" , Nonuniform, ,
truncated 16-cell alterprism (thexa) , 26, , 168, , 384, , 336, , 96 , - BGCOLOR="#a0e0f0" , Nonuniform, ,
bitruncated tesseractic alterprism (taha) , 50, , 288, , 624, , 576, , 192

F₄ × A₁

This prismatic family has 10 forms. The A₁ x F₄ family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry 3,4,32], order 4608. The last one, snub 24-cell prism, (blue background) has ⁺,4,3,2symmetry, order 1152. {, class="wikitable" !rowspan=2, # !rowspan=2,

and Schläfli
symbols
Name !colspan=5 rowspan=1, Element counts , - ! Facets, , Cells, , Faces, , Edges, , Vertices , - , 7, = {3,4,3}×{ }
24-cell prism (icope) , 26, , 144, , 288, , 216, , 48 , - , 9, = r{3,4,3}×{ }
rectified 24-cell prism (ricope) , 50, , 336, , 768, , 672, , 192 , - , 0, = t{3,4,3}×{ }
truncated 24-cell prism (ticope) , 50, , 336, , 864, , 960, , 384 , - , 83, , = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope) , 146, , 1008, , 2304, , 2016, , 576 , - BGCOLOR="#b0f0b0" , 84, , = t_0,3{3,4,3}×{ }
runcinated 24-cell prism (spiccup) , 242, , 1152, , 1920, , 1296, , 288 , - BGCOLOR="#b0f0b0" , 85, , = 2t{3,4,3}×{ }
bitruncated 24-cell prism (contip) , 50, , 432, , 1248, , 1440, , 576 , - , 86, , = tr{3,4,3}×{ }
cantitruncated 24-cell prism (gricope) , 146, , 1008, , 2592, , 2880, , 1152 , - , 87, , = t_0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism (pricope) , 242, , 1584, , 3648, , 3456, , 1152 , - BGCOLOR="#b0f0b0" , 88, , = t_0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism (gippiccup) , 242, , 1872, , 5088, , 5760, , 2304 , - BGCOLOR="#b0e0f0" , 2, = s{3,4,3}×{ }
snub 24-cell prism (sadip) , 146, , 768, , 1392, , 960, , 192

H₄ × A₁

This prismatic family has 15 forms: The A₁ x H₄ family has symmetry of order 28800 (2*14400). {, class="wikitable" !rowspan=2, # !rowspan=2,

and Schläfli
symbols
Name !colspan=5 rowspan=1, Element counts , - ! Facets, , Cells, , Faces, , Edges, , Vertices , - BGCOLOR="#f0e0e0" , 89, , = {5,3,3}×{ }
120-cell prism (hipe) , 122, , 960, , 2640, , 3000, , 1200 , - BGCOLOR="#f0e0e0" , 90, , = r{5,3,3}×{ }
Rectified 120-cell prism (rahipe) , 722, , 4560, , 9840, , 8400, , 2400 , - BGCOLOR="#f0e0e0" , 91, , = t{5,3,3}×{ }
Truncated 120-cell prism (thipe) , 722, , 4560, , 11040, , 12000, , 4800 , - BGCOLOR="#f0e0e0" , 92, , = rr{5,3,3}×{ }
Cantellated 120-cell prism (srahip) , 1922, , 12960, , 29040, , 25200, , 7200 , - BGCOLOR="#e0f0e0" , 93, , = t_0,3{5,3,3}×{ }
Runcinated 120-cell prism (sidpixhip) , 2642, , 12720, , 22080, , 16800, , 4800 , - BGCOLOR="#e0f0e0" , 94, , = 2t{5,3,3}×{ }
Bitruncated 120-cell prism (xhip) , 722, , 5760, , 15840, , 18000, , 7200 , - BGCOLOR="#f0e0e0" , 95, , = tr{5,3,3}×{ }
Cantitruncated 120-cell prism (grahip) , 1922, , 12960, , 32640, , 36000, , 14400 , - BGCOLOR="#f0e0e0" , 96, , = t_0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism (prixip) , 2642, , 18720, , 44880, , 43200, , 14400 , - BGCOLOR="#e0f0e0" , 97, , = t_0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism (gidpixhip) , 2642, , 22320, , 62880, , 72000, , 28800 , - BGCOLOR="#e0e0f0" , 98, , = {3,3,5}×{ }
600-cell prism (exip) , 602, , 2400, , 3120, , 1560, , 240 , - BGCOLOR="#e0e0f0" , 99, , = r{3,3,5}×{ }
Rectified 600-cell prism (roxip) , 722, , 5040, , 10800, , 7920, , 1440 , - BGCOLOR="#e0e0f0" , 100, , = t{3,3,5}×{ }
Truncated 600-cell prism (texip) , 722, , 5040, , 11520, , 10080, , 2880 , - BGCOLOR="#e0e0f0" , 101, , = rr{3,3,5}×{ }
Cantellated 600-cell prism (srixip) , 1442, , 11520, , 28080, , 25200, , 7200 , - BGCOLOR="#e0e0f0" , 102, , = tr{3,3,5}×{ }
Cantitruncated 600-cell prism (grixip) , 1442, , 11520, , 31680, , 36000, , 14400 , - BGCOLOR="#e0e0f0" , 103, , = t_0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism (prahip) , 2642, , 18720, , 44880, , 43200, , 14400

Duoprism prisms

Uniform duoprism prisms, {''p''}×{''q''}×{ }, form an infinite class for all integers ''p'',''q''>2. {4}×{4}×{ } makes a lower symmetry form of the 5-cube. The extended f-vector of {''p''}×{''q''}×{ } is computed as (''p'',''p'',1)*(''q'',''q'',1)*(2,1) = (2''pq'',5''pq'',4''pq''+2''p''+2''q'',3''pq''+3''p''+3''q'',''p''+''q''+2,1). {, class="wikitable" , - !rowspan=2,

!rowspan=2, Names !colspan=6, Element counts , - ! 4-faces ! Cells ! Faces ! Edges ! Vertices , - align=center , , , {''p''}×{''q''}×{ }, , ''p''+''q''+2, , 3''pq''+3''p''+3''q'', , 4''pq''+2''p''+2''q'', , 5''pq'', , 2''pq'' , - align=center , , , {''p''}²×{ }, , 2(''p''+1), , 3''p''(''p''+1), , 4''p''(''p''+1), , 5''p''², , 2''p''² , - align=center , , , {3}²×{ }, , 8, , 36, , 48, , 45, , 18 , - align=center , , , {4}²×{ } = 5-cube, , 10, , 40, , 80, , 80, , 32

Grand antiprism prism

The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

, 40

pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of ten triangles fo ...

s, 700

triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

s, 20

pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

s), and 322 hypercells (2

s , 20

prisms , and 300 tetrahedral prisms ). {, class="wikitable" !rowspan=2, # !rowspan=2, Name !colspan=5, Element counts , - ! Facets, , Cells, , Faces, , Edges, , Vertices , - , 104, , grand antiprism prism (gappip), , 322, , 1360, , 1940, , 1100, , 200

Notes on the Wythoff construction for the uniform 5-polytopes

Construction of the reflective 5-dimensional

s are done through a

process, and represented through a

, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 5-polytopes are named in relation to the

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s in each family. Some families have two regular constructors and thus may have two ways of naming them. Here are the primary operators available for constructing and naming the uniform 5-polytopes. The last operation, the snub, and more generally the alternation, are the operations that can create nonreflective forms. These are drawn with "hollow rings" at the nodes. The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity. {, class="wikitable" !Operation !width=200 colspan=2, Extended

!width=80,

!Description , - align=center ! Parent , t₀{p,q,r,s} , {p,q,r,s} , , Any regular 5-polytope , - align=center ! Rectified , t₁{p,q,r,s}, , r{p,q,r,s} , , align=left, The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual. , - align=center ! Birectified , t₂{p,q,r,s}, , 2r{p,q,r,s} , , align=left, Birectification reduces faces to points, cells to their

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, ...

. , - align=center ! Trirectified , t₃{p,q,r,s}, , 3r{p,q,r,s} , , align=left, Trirectification reduces cells to points. (Dual rectification) , - align=center ! Quadrirectified , t₄{p,q,r,s}, , 4r{p,q,r,s} , , align=left, Quadrirectification reduces 4-faces to points. (Dual) , - align=center ! Truncated , t_0,1{p,q,r,s}, , t{p,q,r,s} , , align=left, Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.
, - align=center ! Cantellated , t_0,2{p,q,r,s}, , rr{p,q,r,s} , , align=left, In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place.
, - align=center ! Runcinated , colspan=2, t_0,3{p,q,r,s} , , align=left, Runcination reduces cells and creates new cells at the vertices and edges. , - align=center ! Stericated , t_0,4{p,q,r,s}, , 2r2r{p,q,r,s} , , align=left, Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.) , - align=center !

Omnitruncated In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each Flag (geometry), flag of the original polytope and a Facet (geometry), facet for each face of any dimension of the original pol ...

, colspan=2, t_0,1,2,3,4{p,q,r,s} , , align=left, All four operators, truncation, cantellation, runcination, and sterication are applied. , - align=center !Half , colspan=2, h{2p,3,q,r} , , align=left, Alternation, same as , - align=center !Cantic , colspan=2, h₂{2p,3,q,r} , , align=left, Same as , - align=center !Runcic , colspan=2, h₃{2p,3,q,r} , , align=left, Same as , - align=center !Runcicantic , colspan=2, h_2,3{2p,3,q,r} , , align=left, Same as , - align=center !Steric , colspan=2, h₄{2p,3,q,r} , , align=left, Same as , - align=center !Steriruncic , colspan=2, h_3,4{2p,3,q,r} , , align=left, Same as , - align=center !Stericantic , colspan=2, h_2,4{2p,3,q,r} , , align=left, Same as , - align=center !Steriruncicantic , colspan=2, h_2,3,4{2p,3,q,r} , , align=left, Same as , - align=center !Snub , colspan=2, s{p,2q,r,s} , , align=left, Alternated truncation , - align=center !Snub rectified , colspan=2, sr{p,q,2r,s} , , align=left, Alternated truncated rectification , - align=center ! , colspan=2, ht_0,1,2,3{p,q,r,s} , , align=left, Alternated runcicantitruncation , - align=center !Full snub , colspan=2, ht_0,1,2,3,4{p,q,r,s} , , align=left, Alternated omnitruncation

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space. {, class=wikitable , + Fundamental groups , - !# !colspan=3,

!Forms , - align=center , 1, ,

{\tilde{A_4

, , ^[5/sup>">.html" ;"title="^{[5">^{[5/sup>">[(3,3,3,3,3)">">, 7
, - align=center
, 2, , ${\tilde{C_4$ , , , , , , 19
, - align=center
, 3, , ${\tilde{B_4$ , , [4,3,3,4,1⁺, = , , 23 (8 new)
, - align=center
, 4, , ${\tilde{D_4$ , , = , , 9 (0 new)
, - align=center
, 5, , ${\tilde{F_4$ , , , , , , 31 (21 new)

There are three regular honeycombs of Euclidean 4-space:
* tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
* 24-cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
** Truncated 24-cell honeycomb with symbols t{3,4,3,3},
** Snub 24-cell honeycomb, with symbols s{3,4,3,3}, and constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
* 16-cell honeycomb, with symbols {3,3,4,3},

Other families that generate uniform honeycombs:
* There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h{4,3²,4} it is geometrically identical to the 16-cell honeycomb, =
* There are 7 uniquely ringed forms from the ${\tilde{A_4$ , family, all new, including:
** 4-simplex honeycomb
** Truncated 4-simplex honeycomb
** Omnitruncated 4-simplex honeycomb
* There are 9 uniquely ringed forms in the ${\tilde{D_4$ : ^1,1,1,1 family, two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = .

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

{, class=wikitable
, + Prismatic groups
, -
!#
!colspan=2, 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 ...

!Coxeter diagram

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.

Coxeter was born in England and educated ...

, -
, 1, , ${\tilde{C_3$ × ${\tilde{I_1$ , , ,3,4,2,∞,
, -
, 2, , ${\tilde{B_3$ × ${\tilde{I_1$ , , ,3^1,1,2,∞,
, -
, 3, , ${\tilde{A_3$ × ${\tilde{I_1$ , , ^[4/sup>,2,∞">.html" ;"title="^{[4">^{[4/sup>,2,∞">
, -
, 4, , ${\tilde{C_2$ × ${\tilde{I_1$ x ${\tilde{I_1$ , , [4,4,2,∞,2,∞,
, -
, 5, , ${\tilde{H_2$ × ${\tilde{I_1$ x ${\tilde{I_1$ , , [6,3,2,∞,2,∞], ,
, -
, 6, , ${\tilde{A_2$ × ${\tilde{I_1$ x ${\tilde{I_1$ , , ^[3/sup>,2,∞,2,∞], ,
, -
, 7, , ${\tilde{I_1$ × ${\tilde{I_1$ x ${\tilde{I_1$ x ${\tilde{I_1$ , , ��,2,∞,2,∞,2,∞,
, -
, 8, , ${\tilde{A_2$ x ${\tilde{A_2$ , , ^[3/sup>,2,3^{[3">.html" ;"title="^{[3">^{[3/sup>,2,3^{[3/sup>">"><sup>[3<_a>_sup>,2,3<sup>.html" ;"title=".html" ;"title="^{[3">^{[3/sup>,2,3^{[3">.html" ;"title="^{[3">^{[3/sup>,2,3^{[3/sup>">
, -
, 9, , ${\tilde{A_2$ × ${\tilde{B_2$ , , ^[3/sup>,2,4,4">
, -
, 10, , ${\tilde{A_2$ × ${\tilde{G_2$ , , ^[3/sup>,2,6,3">
">-
, 11, , ${\tilde{B_2$ × ${\tilde{B_2$ , ,
, -
, 12, , ${\tilde{B_2$ × ${\tilde{G_2$ , , [4,4,2,6,3,
, -
, 13, , ${\tilde{G_2$ × ${\tilde{G_2$ , ,

Regular and uniform hyperbolic honeycombs

;Hyperbolic compact groups

There are 5 Coxeter-Dynkin diagram#Compact, compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.

{, class="wikitable"
, valign=top align=
${\widehat{AF_4$ = [(3,3,3,3,4)]:
, valign=top align=
${\bar{DH_4$ = [5,3,3^1,1]:
, valign=top align= ${\bar{H_4$ = [3,3,3,5]:

${\bar{BH_4$ = [4,3,3,5]:

${\bar{K_4$ = ,3,3,5

There are 5 regular compact convex hyperbolic honeycombs in H⁴ space:Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213
{, class="wikitable"
, + Compact regular convex hyperbolic honeycombs
, -
!Honeycomb name
! Schläfli
Symbol
{p,q,r,s}
!Coxeter diagram

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.

Coxeter was born in England and educated ...

!Facet
type
{p,q,r}
!Cell
type
{p,q}
!Face
type
{p}
!Face
figure
{s}
!Edge
figure
{r,s}
! Vertex
figure
{q,r,s}
! Dual
, - BGCOLOR="#ffe0e0" align=center
, Order-5 5-cell (pente), , {3,3,3,5}, , , , {3,3,3}, , {3,3}, , {3}, , {5}, , {3,5}, , {3,3,5}, , {5,3,3,3}
, - BGCOLOR="#e0e0ff" align=center
, Order-3 120-cell (hitte), , {5,3,3,3}, , , , {5,3,3}, , {5,3}, , {5}, , {3}, , {3,3}, , {3,3,3}, , {3,3,3,5}
, - BGCOLOR="#ffe0e0" align=center
, Order-5 tesseractic (pitest), , {4,3,3,5}, , , , {4,3,3}, , {4,3}, , {4}, , {5}, , {3,5}, , {3,3,5}, , {5,3,3,4}
, - BGCOLOR="#e0e0ff" align=center
, Order-4 120-cell (shitte), , {5,3,3,4}, , , , {5,3,3}, , {5,3}, , {5}, , {4}, , {3,4}, , {3,3,4}, , {4,3,3,5}
, - BGCOLOR="#e0ffe0" align=center
, Order-5 120-cell (phitte), , {5,3,3,5}, , , , {5,3,3}, , {5,3}, , {5}, , {5}, , {3,5}, , {3,3,5}, , Self-dual

There are also 4 regular compact hyperbolic star-honeycombs in H⁴ space:
{, class="wikitable"
, + Compact regular hyperbolic star-honeycombs
, -
!Honeycomb name
! Schläfli
Symbol
{p,q,r,s}
!Coxeter diagram

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.

Coxeter was born in England and educated ...

!Facet
type
{p,q,r}
!Cell
type
{p,q}
!Face
type
{p}
!Face
figure
{s}
!Edge
figure
{r,s}
! Vertex
figure
{q,r,s}
! Dual
, - BGCOLOR="#ffe0e0" align=center
, Order-3 small stellated 120-cell, , {5/2,5,3,3}, , , , {5/2,5,3}, , {5/2,5}, , {5}, , {5}, , {3,3}, , {5,3,3}, , {3,3,5,5/2}
, - BGCOLOR="#e0e0ff" align=center
, Order-5/2 600-cell, , {3,3,5,5/2}, , , , {3,3,5}, , {3,3}, , {3}, , {5/2}, , {5,5/2}, , {3,5,5/2}, , {5/2,5,3,3}
, - BGCOLOR="#ffe0e0" align=center
, Order-5 icosahedral 120-cell, , {3,5,5/2,5}, , , , {3,5,5/2}, , {3,5}, , {3}, , {5}, , {5/2,5}, , {5,5/2,5}, , {5,5/2,5,3}
, - BGCOLOR="#e0e0ff" align=center
, Order-3 great 120-cell, , {5,5/2,5,3}, , , , {5,5/2,5}, , {5,5/2}, , {5}, , {3}, , {5,3}, , {5/2,5,3}, , {3,5,5/2,5}

;Hyperbolic paracompact groups

There are 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite facets or vertex figure

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions

Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s.

{, class=wikitable
, align=
${\bar{P_4$ = ,3^[4/sup>">.html" ;"title=",3^{[4">,3^{[4/sup>

${\bar{BP_4$ = [4,3^{[4">">,3<sup>[4<_a>_sup>.html" ;"title=".html" ;"title=",3^{[4">,3^{[4/sup>">.html" ;"title=",3^{[4">,3^{[4/sup>

${\bar{BP_4$ = [4,3^{[4/sup>]:

${\bar{FR_4$ = [(3,3,4,3,4)]:

${\bar{DP_4$ = [3^�[]]:

, align=
${\bar{N_4$ = ,/3\,3,4

${\bar{O_4$ = ,4,3^1,1

${\bar{S_4$ = ,3^2,1

${\bar{M_4$ = ,3^1,1,1

, align=
${\bar{R_4$ = ,4,3,4

Notes

References

* T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics

The ''Messenger of Mathematics'' is a defunct British mathematics journal. The founding editor-in-chief was William Allen Whitworth with Charles Taylor and volumes 1–58 were published between 1872 and 1929. James Whitbread Lee Glaisher was t ...
, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
* A. Boole Stott: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
* H.S.M. Coxeter:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)
** H.S.M. Coxeter, ''The Beauty of Geometry: Twelve Essays'' (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591(p. 287 5D Euclidean groups, p. 298 Four-dimensionsal honeycombs)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2

External links

* – includes nonconvex forms as well as the duplicate constructions from the B₅ and D₅ families

{{Honeycombs

5-polytopes

eo:5-hiperpluredro}}}}}}}}}}}}}}}}}}}}}}

History of discovery

Regular 5-polytopes

Convex uniform 5-polytopes

Symmetry of uniform 5-polytopes in four dimensions

Enumerating the convex uniform 5-polytopes

The A5 family

The B5 family

The D5 family

Uniform prismatic forms

A4 × A1

B4 × A1

F4 × A1

H4 × A1