uniform, polytope, class, dimensional, polytopes, schlegel, diagram, truncated, cell, with, tetrahedral, cells, visible, orthographic, projection, truncated, cell, coxeter, plane, symmetry, only, vertices, edges, drawn, geometry, uniform, polytope, uniform, po. Class of 4 dimensional polytopes Schlegel diagram for the truncated 120 cell with tetrahedral cells visible Orthographic projection of the truncated 120 cell in the H3 Coxeter plane D10 symmetry Only vertices and edges are drawn In geometry a uniform 4 polytope or uniform polychoron 91 1 93 is a 4 dimensional polytope which is vertex transitive and whose cells are uniform polyhedra and faces are regular polygons There are 47 non prismatic convex uniform 4 polytopes There are two infinite sets of convex prismatic forms along with 17 cases arising as prisms of the convex uniform polyhedra There are also an unknown number of non convex star forms Contents 1 History of discovery 2 Regular 4 polytopes 3 Convex uniform 4 polytopes 3 1 Symmetry of uniform 4 polytopes in four dimensions 3 2 Enumeration 3 3 The A4 family 3 4 The B4 family 3 4 1 Tesseract truncations 3 4 2 16 cell truncations 3 5 The F4 family 3 6 The H4 family 3 6 1 120 cell truncations 3 6 2 600 cell truncations 3 7 The D4 family 3 8 The grand antiprism 3 9 Prismatic uniform 4 polytopes 3 9 1 Convex polyhedral prisms 3 9 2 Tetrahedral prisms A3 A1 3 9 3 Octahedral prisms B3 A1 3 9 4 Icosahedral prisms H3 A1 3 9 5 Duoprisms p q 3 9 6 Polygonal prismatic prisms p 3 9 7 Polygonal antiprismatic prisms p 3 10 Nonuniform alternations 3 11 Geometric derivations for 46 nonprismatic Wythoffian uniform polychora 3 11 1 Summary of constructions by extended symmetry 4 Uniform star polychora 5 See also 6 References 7 External links History of discovery edit Convex Regular polytopes 1852 Ludwig Schlafli proved in his manuscript Theorie der vielfachen Kontinuitat that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions Regular star 4 polytopes star polyhedron cells and or vertex figures 1852 Ludwig Schlafli also found 4 of the 10 regular star 4 polytopes discounting 6 with cells or vertex figures 5 2 5 and 5 5 2 1883 Edmund Hess completed the list of 10 of the nonconvex regular 4 polytopes in his book in German Einleitung in die Lehre von der Kugelteilung mit besonderer Berucksichtigung ihrer Anwendung auf die Theorie der Gleichflachigen und der gleicheckigen Polyeder Einleitung in die Lehre von der Kugelteilung mit besonderer Berucksichtigung ihrer Anwendung auf die Theorie der Gleichflachigen und der gleicheckigen Polyeder von dr Edmund Hess Mit sechzehn lithographierten tafeln Convex semiregular polytopes Various definitions before Coxeter s uniform category 1900 Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells Platonic solids in his publication On the Regular and Semi Regular Figures in Space of n Dimensions In four dimensions this gives the rectified 5 cell the rectified 600 cell and the snub 24 cell 91 2 93 1910 Alicia Boole Stott in her publication Geometrical deduction of semiregular from regular polytopes and space fillings expanded the definition by also allowing Archimedean solid and prism cells This construction enumerated 45 semiregular 4 polytopes corresponding to the nonprismatic forms listed below The snub 24 cell and grand antiprism were missing from her list 91 3 93 1911 Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes followed Boole Stott s notations enumerating the convex uniform polytopes by symmetry based on 5 cell 8 cell 16 cell and 24 cell 1912 E L Elte independently expanded on Gosset s list with the publication The Semiregular Polytopes of the Hyperspaces polytopes with one or two types of semiregular facets 91 4 93 Convex uniform polytopes 1940 The search was expanded systematically by H S M Coxeter in his publication Regular and Semi Regular Polytopes Convex uniform 4 polytopes 1965 The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy in their publication Four Dimensional Archimedean Polytopes established by computer analysis adding only one non Wythoffian convex 4 polytope the grand antiprism 1966 Norman Johnson completes his Ph D dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter completes the basic theory of uniform polytopes for dimensions 4 and higher 1986 Coxeter published a paper Regular and Semi Regular Polytopes II which included analysis of the unique snub 24 cell structure and the symmetry of the anomalous grand antiprism 1998 91 5 93 2000 The 4 polytopes were systematically named by Norman Johnson and given by George Olshevsky s online indexed enumeration used as a basis for this listing Johnson named the 4 polytopes as polychora like polyhedra for 3 polytopes from the Greek roots poly many and choros room or space 91 6 93 The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams truncation t0 1 cantellation t0 2 runcination t0 3 with single ringed forms called rectified and bi tri prefixes added when the first ring was on the second or third nodes 91 7 93 91 8 93 2004 A proof that the Conway Guy set is complete was published by Marco Moller in his dissertation Vierdimensionale Archimedische Polytope Moller reproduced Johnson s naming system in his listing 91 9 93 2008 The Symmetries of Things 91 10 93 was published by John H Conway and contains the first print published listing of the convex uniform 4 polytopes and higher dimensional polytopes by Coxeter group family with general vertex figure diagrams for each ringed Coxeter diagram permutation snub grand antiprism and duoprisms which he called proprisms for product prisms He used his own ijk ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation and all of Johnson s names were included in the book index Nonregular uniform star 4 polytopes similar to the nonconvex uniform polyhedra 1966 Johnson describes three nonconvex uniform antiprisms in 4 space in his dissertation 91 11 93 1990 2006 In a collaborative search up to 2005 a total of 1845 uniform 4 polytopes convex and nonconvex had been identified by Jonathan Bowers and George Olshevsky 91 12 93 with an additional four discovered in 2006 for a total of 1849 The count includes the 74 prisms of the 75 non prismatic uniform polyhedra since that is a finite set the cubic prism is excluded as it duplicates the tesseract but not the infinite categories of duoprisms or prisms of antiprisms 91 13 93 2020 2023 342 new polychora were found bringing up the total number of known uniform 4 polytopes to 2191 The list has not been proven complete 91 13 93 91 14 93 Regular 4 polytopes edit Regular 4 polytopes are a subset of the uniform 4 polytopes which satisfy additional requirements Regular 4 polytopes can be expressed with Schlafli symbol p q r have cells of type p q faces of type p edge figures r and vertex figures q r The existence of a regular 4 polytope p q r is constrained by the existence of the regular polyhedra p q which becomes cells and q r which becomes the vertex figure Existence as a finite 4 polytope is dependent upon an inequality 91 15 93 sin x2061 x03C0 p sin x2061 x03C0 r gt cos x2061 x03C0 q displaystyle sin left frac pi p right sin left frac pi r right gt cos left frac pi q right The 16 regular 4 polytopes with the property that all cells faces edges and vertices are congruent 6 regular convex 4 polytopes 5 cell 3 3 3 8 cell 4 3 3 16 cell 3 3 4 24 cell 3 4 3 120 cell 5 3 3 and 600 cell 3 3 5 10 regular star 4 polytopes icosahedral 120 cell 3 5 5 2 small stellated 120 cell 5 2 5 3 great 120 cell 5 5 2 5 grand 120 cell 5 3 5 2 great stellated 120 cell 5 2 3 5 grand stellated 120 cell 5 2 5 5 2 great grand 120 cell 5 5 2 3 great icosahedral 120 cell 3 5 2 5 grand 600 cell 3 3 5 2 and great grand stellated 120 cell 5 2 3 3 Convex uniform 4 polytopes edit Symmetry of uniform 4 polytopes in four dimensions edit Main article Point groups in four dimensions Orthogonal subgroups The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups 12 mirrors 12 mirrors The 10 mirrors of B3 A1 can be decomposed into orthogonal groups 4A1 and D3 3 1 mirrors 6 mirrors There are 5 fundamental mirror symmetry point group families in 4 dimensions A4 B4 D4 F4 H4 91 7 93 There are also 3 prismatic groups A3A1 B3A1 H3A1 and duoprismatic groups I2 p I2 q Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes Each reflective uniform 4 polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction represented by rings around permutations of nodes in a Coxeter diagram Mirror hyperplanes can be grouped as seen by colored nodes separated by even branches Symmetry groups of the form a b a have an extended symmetry a b a doubling the symmetry order This includes 3 3 3 3 4 3 and p 2 p 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 4 polytope with chiral symmetry shown as empty circled nodes but the geometry is not generally adjustable to create uniform solutions Weylgroup ConwayQuaternion Abstractstructure Order Coxeterdiagram Coxeternotation Commutatorsubgroup Coxeternumber h Mirrorsm 2h Irreducible A4 1 60 I I 21 S5 120 3 3 3 3 3 3 5 10 D4 1 3 T T 2 1 2 2S4 192 31 1 1 31 1 1 6 12 B4 1 6 O O 2 2S4 S2 S4 384 4 3 3 8 4 12 F4 1 2 O O 23 3 2S4 1152 3 4 3 3 4 3 12 12 12 H4 I I 2 2 A5 A5 2 14400 5 3 3 5 3 3 30 60 Prismatic groups A3A1 1 24 O O 23 S4 D1 48 3 3 2 3 3 3 3 6 1 B3A1 1 24 O O 2 S4 D1 96 4 3 2 4 3 3 6 1 H3A1 1 60 I I 2 A5 D1 240 5 3 2 5 3 5 3 15 1 Duoprismatic groups Use 2p 2q for even integers I2 p I2 q 1 2 D2p D2q Dp Dq 4pq p 2 q p q p 2 q p q I2 2p I2 q 1 2 D4p D2q D2p Dq 8pq 2p 2 q 2p q p p q I2 2p I2 2q 1 2 D4p D4q D2p D2q 16pq 2p 2 2q 2p 2q p p q q Enumeration edit There are 64 convex uniform 4 polytopes including the 6 regular convex 4 polytopes and excluding the infinite sets of the duoprisms and the antiprismatic prisms 5 are polyhedral prisms based on the Platonic solids 1 overlap with regular since a cubic hyperprism is a tesseract 13 are polyhedral prisms based on the Archimedean solids 9 are in the self dual regular A4 3 3 3 group 5 cell family 9 are in the self dual regular F4 3 4 3 group 24 cell family Excluding snub 24 cell 15 are in the regular B4 3 3 4 group tesseract 16 cell family 3 overlap with 24 cell family 15 are in the regular H4 3 3 5 group 120 cell 600 cell family 1 special snub form in the 3 4 3 group 24 cell family 1 special non Wythoffian 4 polytope the grand antiprism TOTAL 68 8722 4 64 These 64 uniform 4 polytopes are indexed below by George Olshevsky Repeated symmetry forms are indexed in brackets In addition to the 64 above there are 2 infinite prismatic sets that generate all of the remaining convex forms Set of uniform antiprismatic prisms sr p 2 160 Polyhedral prisms of two antiprisms Set of uniform duoprisms p q A Cartesian product of two polygons The A4 family edit Further information A4 polytope The 5 cell has diploid pentachoric 3 3 3 symmetry 91 7 93 of order 120 isomorphic to the permutations of five elements because all pairs of vertices are related in the same way Facets cells are given grouped in their Coxeter diagram locations by removing specified nodes 3 3 3 uniform polytopes Name Bowers name and acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 5 Pos 2 10 Pos 1 10 Pos 0 5 Cells Faces Edges Vertices 1 5 cellPentachoron 91 7 93 pen 3 3 3 4 3 3 3 5 10 10 5 2 rectified 5 cell Rectified pentachoron rap r 3 3 3 3 3 3 3 3 2 3 3 3 10 30 30 10 3 truncated 5 cell Truncated pentachoron tip t 3 3 3 3 3 6 6 1 3 3 3 10 30 40 20 4 cantellated 5 cell Small rhombated pentachoron srip rr 3 3 3 2 3 4 3 4 2 3 4 4 1 3 3 3 3 20 80 90 30 7 cantitruncated 5 cell Great rhombated pentachoron grip tr 3 3 3 2 4 6 6 1 3 4 4 1 3 6 6 20 80 120 60 8 runcitruncated 5 cell Prismatorhombated pentachoron prip t0 1 3 3 3 3 1 3 6 6 2 4 4 6 1 3 4 4 1 3 4 3 4 30 120 150 60 3 3 3 uniform polytopes Name Bowers name and acronym Vertexfigure Coxeter diagram and Schlaflisymbols Cell counts by location Element counts Pos 3 0 10 Pos 1 2 20 Alt Cells Faces Edges Vertices 5 runcinated 5 cell Small prismatodecachoron spid t0 3 3 3 3 2 3 3 3 6 3 4 4 30 70 60 20 6 bitruncated 5 cellDecachoron deca 2t 3 3 3 4 3 6 6 10 40 60 30 9 omnitruncated 5 cell Great prismatodecachoron gippid t0 1 2 3 3 3 3 2 4 6 6 2 4 4 6 30 150 240 120 Nonuniform omnisnub 5 cellSnub decachoron snad Snub pentachoron snip 91 16 93 ht0 1 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 4 3 3 3 90 300 270 60 The three uniform 4 polytopes forms marked with an asterisk have the higher extended pentachoric symmetry of order 240 3 3 3 because the element corresponding to any element of the underlying 5 cell can be exchanged with one of those corresponding to an element of its dual There is one small index subgroup 3 3 3 order 60 or its doubling 3 3 3 order 120 defining an omnisnub 5 cell which is listed for completeness but is not uniform The B4 family edit Further information B4 polytope This family has diploid hexadecachoric symmetry 91 7 93 4 3 3 of order 24 215 16 384 4 24 permutations of the four axes 24 16 for reflection in each axis There are 3 small index subgroups with the first two generate uniform 4 polytopes which are also repeated in other families 1 4 3 3 4 3 3 and 4 3 3 all order 192 Tesseract truncations edit Name Bowers name and acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 8 Pos 2 24 Pos 1 32 Pos 0 16 Cells Faces Edges Vertices 10 tesseract or 8 cellTesseract tes 4 3 3 4 4 4 4 8 24 32 16 11 Rectified tesseract rit r 4 3 3 3 3 4 3 4 2 3 3 3 24 88 96 32 13 Truncated tesseract tat t 4 3 3 3 3 8 8 1 3 3 3 24 88 128 64 14 Cantellated tesseractSmall rhombated tesseract srit rr 4 3 3 2 3 4 4 4 2 3 4 4 1 3 3 3 3 56 248 288 96 15 Runcinated tesseract also runcinated 16 cell Small disprismatotesseractihexadecachoron sidpith t0 3 4 3 3 1 4 4 4 3 4 4 4 3 3 4 4 1 3 3 3 80 208 192 64 16 Bitruncated tesseract also bitruncated 16 cell Tesseractihexadecachoron tah 2t 4 3 3 2 4 6 6 2 3 6 6 24 120 192 96 18 Cantitruncated tesseractGreat rhombated tesseract grit tr 4 3 3 2 4 6 8 1 3 4 4 1 3 6 6 56 248 384 192 19 Runcitruncated tesseractPrismatorhombated hexadecachoron proh t0 1 3 4 3 3 1 3 8 8 2 4 4 8 1 3 4 4 1 3 4 3 4 80 368 480 192 21 Omnitruncated tesseract also omnitruncated 16 cell Great disprismatotesseractihexadecachoron gidpith t0 1 2 3 3 3 4 1 4 6 8 1 4 4 8 1 4 4 6 1 4 6 6 80 464 768 384 Related half tesseract 1 4 3 3 uniform 4 polytopes Name Bowers style acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 8 Pos 2 24 Pos 1 32 Pos 0 16 Alt Cells Faces Edges Vertices 12 Half tesseractDemitesseract 16 cell hex h 4 3 3 3 3 4 4 3 3 3 4 3 3 3 16 32 24 8 17 Cantic tesseract Truncated 16 cell thex h2 4 3 3 t 4 3 3 4 6 6 3 1 3 3 3 3 24 96 120 48 11 Runcic tesseract Rectified tesseract rit h3 4 3 3 r 4 3 3 3 3 4 3 4 2 3 3 3 24 88 96 32 16 Runcicantic tesseract Bitruncated tesseract tah h2 3 4 3 3 2t 4 3 3 2 3 4 3 4 2 3 6 6 24 120 192 96 11 Rectified tesseract rat h1 4 3 3 r 4 3 3 24 88 96 32 16 Bitruncated tesseract tah h1 2 4 3 3 2t 4 3 3 24 120 192 96 23 Rectified 24 cell rico h1 3 4 3 3 rr 3 3 4 48 240 288 96 24 Truncated 24 cell tico h1 2 3 4 3 3 tr 3 3 4 48 240 384 192 Name Bowers style acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 8 Pos 2 24 Pos 1 32 Pos 0 16 Alt Cells Faces Edges Vertices Nonuniform omnisnub tesseractSnub tesseract snet 91 17 93 Or omnisnub 16 cell ht0 1 2 3 4 3 3 1 3 3 3 3 4 1 3 3 3 4 1 3 3 3 3 1 3 3 3 3 3 4 3 3 3 272 944 864 192 16 cell truncations edit Name Bowers name and acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 8 Pos 2 24 Pos 1 32 Pos 0 16 Alt Cells Faces Edges Vertices 12 16 cellHexadecachoron 91 7 93 hex 3 3 4 8 3 3 3 16 32 24 8 22 Rectified 16 cell Same as 24 cell ico r 3 3 4 2 3 3 3 3 4 3 3 3 3 24 96 96 24 17 Truncated 16 cellTruncated hexadecachoron thex t 3 3 4 1 3 3 3 3 4 3 6 6 24 96 120 48 23 Cantellated 16 cell Same as rectified 24 cell rico rr 3 3 4 1 3 4 3 4 2 4 4 4 2 3 4 3 4 48 240 288 96 15 Runcinated 16 cell also runcinated tesseract sidpith t0 3 3 3 4 1 4 4 4 3 4 4 4 3 3 4 4 1 3 3 3 80 208 192 64 16 Bitruncated 16 cell also bitruncated tesseract tah 2t 3 3 4 2 4 6 6 2 3 6 6 24 120 192 96 24 Cantitruncated 16 cell Same as truncated 24 cell tico tr 3 3 4 1 4 6 6 1 4 4 4 2 4 6 6 48 240 384 192 20 Runcitruncated 16 cellPrismatorhombated tesseract prit t0 1 3 3 3 4 1 3 4 4 4 1 4 4 4 2 4 4 6 1 3 6 6 80 368 480 192 21 Omnitruncated 16 cell also omnitruncated tesseract gidpith t0 1 2 3 3 3 4 1 4 6 8 1 4 4 8 1 4 4 6 1 4 6 6 80 464 768 384 31 alternated cantitruncated 16 cell Same as the snub 24 cell sadi sr 3 3 4 1 3 3 3 3 3 1 3 3 3 2 3 3 3 3 3 4 3 3 3 144 480 432 96 Nonuniform Runcic snub rectified 16 cellPyritosnub tesseract pysnet sr3 3 3 4 1 3 4 4 4 2 3 4 4 1 4 4 4 1 3 3 3 3 3 2 3 4 4 176 656 672 192 Just as rectifying the tetrahedron produces the octahedron rectifying the 16 cell produces the 24 cell the regular member of the following family The snub 24 cell is repeat to this family for completeness It is an alternation of the cantitruncated 16 cell or truncated 24 cell with the half symmetry group 3 3 4 The truncated octahedral cells become icosahedra The cubes becomes tetrahedra and 96 new tetrahedra are created in the gaps from the removed vertices The F4 family edit Further information F4 polytope This family has diploid icositetrachoric symmetry 91 7 93 3 4 3 of order 24 215 48 1152 the 48 symmetries of the octahedron for each of the 24 cells There are 3 small index subgroups with the first two isomorphic pairs generating uniform 4 polytopes which are also repeated in other families 3 4 3 3 4 3 and 3 4 3 all order 576 3 4 3 uniform 4 polytopes Name Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 24 Pos 2 96 Pos 1 96 Pos 0 24 Cells Faces Edges Vertices 22 24 cell Same as rectified 16 cell Icositetrachoron 91 7 93 ico 3 4 3 6 3 3 3 3 24 96 96 24 23 rectified 24 cell Same as cantellated 16 cell Rectified icositetrachoron rico r 3 4 3 3 3 4 3 4 2 4 4 4 48 240 288 96 24 truncated 24 cell Same as cantitruncated 16 cell Truncated icositetrachoron tico t 3 4 3 3 4 6 6 1 4 4 4 48 240 384 192 25 cantellated 24 cellSmall rhombated icositetrachoron srico rr 3 4 3 2 3 4 4 4 2 3 4 4 1 3 4 3 4 144 720 864 288 28 cantitruncated 24 cellGreat rhombated icositetrachoron grico tr 3 4 3 2 4 6 8 1 3 4 4 1 3 8 8 144 720 1152 576 29 runcitruncated 24 cellPrismatorhombated icositetrachoron prico t0 1 3 3 4 3 1 4 6 6 2 4 4 6 1 3 4 4 1 3 4 4 4 240 1104 1440 576 3 4 3 uniform 4 polytopes Name Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 24 Pos 2 96 Pos 1 96 Pos 0 24 Alt Cells Faces Edges Vertices 31 snub 24 cellSnub disicositetrachoron sadi s 3 4 3 3 3 3 3 3 3 1 3 3 3 4 3 3 3 144 480 432 96 Nonuniform runcic snub 24 cellPrismatorhombisnub icositetrachoron prissi s3 3 4 3 1 3 3 3 3 3 2 3 4 4 1 3 6 6 3 Tricup 240 960 1008 288 25 cantic snub 24 cell Same as cantellated 24 cell srico s2 3 4 3 2 3 4 4 4 1 3 4 3 4 2 3 4 4 144 720 864 288 29 runcicantic snub 24 cell Same as runcitruncated 24 cell prico s2 3 3 4 3 1 4 6 6 1 3 4 4 1 3 4 4 4 2 4 4 6 240 1104 1440 576 The snub 24 cell here despite its common name is not analogous to the snub cube rather it is derived by an alternation of the truncated 24 cell Its symmetry number is only 576 the ionic diminished icositetrachoric group 3 4 3 Like the 5 cell the 24 cell is self dual and so the following three forms have twice as many symmetries bringing their total to 2304 extended icositetrachoric symmetry 3 4 3 3 4 3 uniform 4 polytopes Name Vertexfigure Coxeter diagram and Schlaflisymbols Cell counts by location Element counts Pos 3 0 48 Pos 2 1 192 Cells Faces Edges Vertices 26 runcinated 24 cellSmall prismatotetracontoctachoron spic t0 3 3 4 3 2 3 3 3 3 6 3 4 4 240 672 576 144 27 bitruncated 24 cellTetracontoctachoron cont 2t 3 4 3 4 3 8 8 48 336 576 288 30 omnitruncated 24 cellGreat prismatotetracontoctachoron gippic t0 1 2 3 3 4 3 2 4 6 8 2 4 4 6 240 1392 2304 1152 3 4 3 isogonal 4 polytope Name Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 0 48 Pos 2 1 192 Alt Cells Faces Edges Vertices Nonuniform omnisnub 24 cellSnub tetracontoctachoron snoc Snub icositetrachoron sni 91 18 93 ht0 1 2 3 3 4 3 2 3 3 3 3 4 2 3 3 3 3 4 3 3 3 816 2832 2592 576 The H4 family edit Further information H4 polytope This family has diploid hexacosichoric symmetry 91 7 93 5 3 3 of order 120 215 120 24 215 600 14400 120 for each of the 120 dodecahedra or 24 for each of the 600 tetrahedra There is one small index subgroups 5 3 3 all order 7200 120 cell truncations edit Name Bowers name and acronym Vertexfigure Coxeter diagramand Schlaflisymbols Cell counts by location Element counts Pos 3 120 Pos 2 720 Pos 1 1200 Pos 0 600 Alt Cells Faces Edges Vertices 32 120 cell hecatonicosachoron or dodecacontachoron 91 7 93 Hecatonicosachoron hi 5 3 3 4 5 5 5 120 720 1200 600 33 rectified 120 cellRectified hecatonicosachoron rahi r 5 3 3 3 3 5 3 5 2 3 3 3 720 3120 3600 1200 36 truncated 120 cellTruncated hecatonicosachoron thi t 5 3 3 3 3 10 10 1 3 3 3 720 3120 4800 2400 37 cantellated 120 cellSmall rhombated hecatonicosachoron srahi rr 5 3 3 2 3 4 5 4 2 3 4 4 1 3 3 3 3 1920 9120 10800 3600 38 runcinated 120 cell also runcinated 600 cell Small disprismatohexacosihecatonicosachoron sidpixhi t0 3 5 3 3 1 5 5 5 3 4 4 5 3 3 4 4 1 3 3 3 2640 7440 7200 2400 39 bitruncated 120 cell also bitruncated 600 cell Hexacosihecatonicosachoron xhi 2t 5 3 3 2 5 6 6 2 3 6 6 720 4320 7200 3600 42 cantitruncated 120 cellGreat rhombated hecatonicosachoron grahi tr 5 3 3 2 4 6 10 1 3 4 4 1 3 6 6 1920 9120 14400 7200 43 runcitruncated 120 cellPrismatorhombated hexacosichoron prix t0 1 3 5 3 3 1 3 10 10 2 4 4 10 1 3 4 4 1 3 4 3 4 2640 13440 18000 7200 46 omnitruncated 120 cell also omnitruncated 600 cell Great disprismatohexacosihecatonicosachoron gidpixhi t0 1 2 3 5 3 3 1 4 6 10 1 4 4 10 1 4 4 6 1 4 6 6 2640 17040 28800 14400 Nonuniform omnisnub 120 cellSnub hecatonicosachoron snahi 91 19 93 Same as the omnisnub 600 cell ht0 1 2 3 5 3 3 1 3 3 3 3 5 1 3 3 3 5 1 3 3 3 3 1 3 3 3 3 3 4 3 3 3 9840 35040 32400 7200 600 cell truncations edit Name Bowers style acronym Vertexfigure Coxeter diagramand Schlaflisymbols Symmetry Cell counts by location Element counts Pos 3 120 Pos 2 720 Pos 1 1200 Pos 0 600 Cells Faces Edges Vertices 35 600 cellHexacosichoron 91 7 93 ex 3 3 5 5 3 3 order 14400 20 3 3 3 600 1200 720 120 47 20 diminished 600 cell Grand antiprism gap Nonwythoffianconstruction 10 2 10 order 400Index 36 2 3 3 3 5 12 3 3 3 320 720 500 100 31 24 diminished 600 cell Snub 24 cell sadi Nonwythoffianconstruction 3 4 3 order 576index 25 3 3 3 3 3 3 5 3 3 3 144 480 432 96 Nonuniform bi 24 diminished 600 cellBi icositetradiminished hexacosichoron bidex Nonwythoffianconstruction order 144index 100 6 tdi 48 192 216 72 34 rectified 600 cellRectified hexacosichoron rox r 3 3 5 5 3 3 2 3 3 3 3 3 5 3 3 3 3 720 3600 3600 720 Nonuniform 120 diminished rectified 600 cellSwirlprismatodiminished rectified hexacosichoron spidrox Nonwythoffianconstruction order 1200index 12 2 3 3 3 5 2 4 4 5 5 P4 840 2640 2400 600 41 truncated 600 cellTruncated hexacosichoron tex t 3 3 5 5 3 3 1 3 3 3 3 3 5 3 6 6 720 3600 4320 1440 40 cantellated 600 cellSmall rhombated hexacosichoron srix rr 3 3 5 5 3 3 1 3 5 3 5 2 4 4 5 1 3 4 3 4 1440 8640 10800 3600 38 runcinated 600 cell also runcinated 120 cell sidpixhi t0 3 3 3 5 5 3 3 1 5 5 5 3 4 4 5 3 3 4 4 1 3 3 3 2640 7440 7200 2400 39 bitruncated 600 cell also bitruncated 120 cell xhi 2t 3 3 5 5 3 3 2 5 6 6 2 3 6 6 720 4320 7200 3600 45 cantitruncated 600 cellGreat rhombated hexacosichoron grix tr 3 3 5 5 3 3 1 5 6 6 1 4 4 5 2 4 6 6 1440 8640 14400 7200 44 runcitruncated 600 cellPrismatorhombated hecatonicosachoron prahi t0 1 3 3 3 5 5 3 3 1 3 4 5 4 1 4 4 5 2 4 4 6 1 3 6 6 2640 13440 18000 7200 46 omnitruncated 600 cell also omnitruncated 120 cell gidpixhi t0 1 2 3 3 3 5 5 3 3 1 4 6 10 1 4 4 10 1 4 4 6 1 4 6 6 2640 17040 28800 14400 The D4 family edit Further information D4 polytope This demitesseract family 31 1 1 introduces no new uniform 4 polytopes but it is worthy to repeat these alternative constructions This family has order 12 215 16 192 4 2 12 permutations of the four axes half as alternated 24 16 for reflection in each axis There is one small index subgroups that generating uniform 4 polytopes 31 1 1 order 96 31 1 1 uniform 4 polytopes Name Bowers style acronym Vertexfigure Coxeter diagram Cell counts by location Element counts Pos 0 8 Pos 2 24 Pos 1 8 Pos 3 8 Pos Alt 96 3 2 1 0 12 demitesseracthalf tesseract Same as 16 cell hex span, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.