Regular 4polytope
In mathematics, a regular 4polytope is a regular fourdimensional polytope. They are the fourdimensional analogs of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
Regular 4polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid19th century, although the full set were not discovered until later.
There are six convex and ten star regular 4polytopes, giving a total of sixteen.
Contents
History
The convex regular 4polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid19th century. He discovered that there are precisely six such figures.
Schläfli also found four of the regular star 4polytopes: the grand 120cell, great stellated 120cell, grand 600cell, and great grand stellated 120cell). He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zerohole tori: F − E + V = 2). That excludes cells and vertex figures as {5,5/2} and {5/2,5}.
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 4polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint
to ensure that the cells meet to form a closed 3surface.
The six convex and ten star polytopes described are the only solutions to these constraints.
There are four nonconvex Schläfli symbols {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.
Regular convex 4polytopes
The regular convex 4polytopes are the fourdimensional analogs of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.
Five of them may be thought of as close analogs of the Platonic solids. One additional figure, the 24cell, has no close threedimensional equivalent.
Each convex regular 4polytope is bounded by a set of 3dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.
Properties
The following tables lists some properties of the six convex regular 4polytopes. The symmetry groups of these 4polytopes are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.
Names  Image  Family 
Schläfli Coxeter 
V  E  F  C  Vert. fig. 
Dual  Symmetry group  

5cell pentachoron pentatope 4simplex 
nsimplex (A_{n} family) 
{3,3,3} 
5  10  10 {3} 
5 {3,3} 
{3,3}  (selfdual) 
A_{4} [3,3,3] 
120  
8cell octachoron tesseract 4cube 
ncube (B_{n} family) 
{4,3,3} 
16  32  24 {4} 
8 {4,3} 
{3,3}  16cell 
B_{4} [4,3,3] 
384  
16cell hexadecachoron 4orthoplex 
northoplex (B_{n} family) 
{3,3,4} 
8  24  32 {3} 
16 {3,3} 
{3,4}  8cell 
B_{4} [4,3,3] 
384  
24cell icositetrachoron octaplex polyoctahedron (pO) 
F_{n} family  {3,4,3} 
24  96  96 {3} 
24 {3,4} 
{4,3}  (selfdual) 
F_{4} [3,4,3] 
1152  
120cell hecatonicosachoron dodecacontachoron dodecaplex polydodecahedron (pD) 
npentagonal polytope (H_{n} family) 
{5,3,3} 
600  1200  720 {5} 
120 {5,3} 
{3,3}  600cell 
H_{4} [5,3,3] 
14400  
600cell hexacosichoron tetraplex polytetrahedron (pT) 
npentagonal polytope (H_{n} family) 
{3,3,5} 
120  720  1200 {3} 
600 {3,3} 
{3,5}  120cell 
H_{4} [5,3,3] 
14400 
John Conway advocates the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), dodecaplex or polydodecahedron (pD), and tetraplex or polytetrahedron (pT).^{[1]}
Norman Johnson advocates the names ncell, or pentachoron, tesseract or octachoron, hexadecachoron, icositetrachoron, hecatonicosachoron (or dodecacontachoron), and hexacosichoron, coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").^{[2]}^{[3]}
The Euler characteristic for all 4polytopes is zero, we have the 4dimensional analog of Euler's polyhedral formula:
where N_{k} denotes the number of kfaces in the polytope (a vertex is a 0face, an edge is a 1face, etc.).
The topology of any given 4polytope is defined by its Betti numbers and torsion coefficients.^{[4]}
As configurations
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[5]}^{[6]}
5cell {3,3,3} 
16cell {3,3,4} 
tesseract {4,3,3} 
24cell {3,4,3} 
600cell {3,3,5} 
120cell {5,3,3} 

Visualization
The following table shows some 2dimensional projections of these 4polytopes. Various other visualizations can be found in the external links below. The CoxeterDynkin diagram graphs are also given below the Schläfli symbol.
A_{4} = [3,3,3]  BC_{4} = [4,3,3]  F_{4} = [3,4,3]  H_{4} = [5,3,3]  

5cell  8cell  16cell  24cell  120cell  600cell 
{3,3,3}  {4,3,3}  {3,3,4}  {3,4,3}  {5,3,3}  {3,3,5} 
Solid 3D orthographic projections  
Tetrahedral envelope (cell/vertexcentered) 
Cubic envelope (cellcentered) 
cubic envelope (cellcentered) 
Cuboctahedral envelope (cellcentered) 
Truncated rhombic triacontahedron envelope (cellcentered) 
pentakis icosidodecahedral envelope (vertexcentered) 
Wireframe Schlegel diagrams (Perspective projection)  
Cellcentered 
Cellcentered 
Cellcentered 
Cellcentered 
Cellcentered 
Vertexcentered 
Wireframe stereographic projections (3sphere)  
Regular star (Schläfli–Hess) 4polytopes
The Schläfli–Hess 4polytopes are the complete set of 10 regular selfintersecting star polychora (fourdimensional polytopes).^{[8]} They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2. 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 – replaces edges by longer edges in same lines. (Example: a pentagon stellates into a pentagram)
 greatening – replaces the faces by large ones in same planes. (Example: an icosahedron greatens into a great icosahedron)
 aggrandizement – replaces the cells by large ones in same 3spaces. (Example: a 600cell aggrandizes into a grand 600cell)
John Conway names the 10 forms from 3 regular celled 4polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600cell), pI=polyicoshedron {3,5,5/2} (an icosahedral 120cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120cell), 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 [3,3,5] (H_{4}) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rationalorder symmetry groups: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].
Each group has 2 regular starpolychora, except for two groups which are selfdual, having only one. So there are 4 dualpairs and 2 selfdual forms among the ten regular star polychora.
Properties
Note:
 There are 2 unique vertex arrangements, matching those of the 120cell and 600cell.
 There are 4 unique edge arrangements, which are shown as wireframes orthographic projections.
 There are 7 unique face arrangements, shown as solids (facecolored) orthographic projections.
The cells (polyhedra), their faces (polygons), the polygonal edge figures and polyhedral vertex figures are identified by their Schläfli symbols.
Name Conway (abbrev.) 
Orthogonal projection 
Schläfli Coxeter 
C {p, q} 
F {p} 
E {r} 
V {q, r} 
Dens.  χ 

Icosahedral 120cell polyicosahedron (pI) 
{3,5,5/2} 
120 {3,5} 
1200 {3} 
720 {5/2} 
120 {5,5/2} 
4  480  
Small stellated 120cell stellated polydodecahedron (spD) 
{5/2,5,3} 
120 {5/2,5} 
720 {5/2} 
1200 {3} 
120 {5,3} 
4  −480  
Great 120cell great polydodecahedron (gpD) 
{5,5/2,5} 
120 {5,5/2} 
720 {5} 
720 {5} 
120 {5/2,5} 
6  0  
Grand 120cell grand polydodecahedron (apD) 
{5,3,5/2} 
120 {5,3} 
720 {5} 
720 {5/2} 
120 {3,5/2} 
20  0  
Great stellated 120cell great stellated polydodecahedron (gspD) 
{5/2,3,5} 
120 {5/2,3} 
720 {5/2} 
720 {5} 
120 {3,5} 
20  0  
Grand stellated 120cell 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  
Great grand 120cell great grand polydodecahedron (gapD) 
{5,5/2,3} 
120 {5,5/2} 
720 {5} 
1200 {3} 
120 {5/2,3} 
76  −480  
Great icosahedral 120cell great polyicosahedron (gpI) 
{3,5/2,5} 
120 {3,5/2} 
1200 {3} 
720 {5} 
120 {5/2,5} 
76  480  
Grand 600cell grand polytetrahedron (apT) 
{3,3,5/2} 
600 {3,3} 
1200 {3} 
720 {5/2} 
120 {3,5/2} 
191  0  
Great grand stellated 120cell 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
 Infinite regular 4polytopes:
 One regular Euclidean honeycomb: {4,3,4}
 Four compact regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
 Eleven paracompact regular hyperbolic honeycombs: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
 Abstract regular 4polytopes:
 Uniform 4polytope uniform 4polytope families constructed from these 6 regular forms.
 Platonic solid
 KeplerPoinsot polyhedra – regular star polyhedron
 Star polygon – regular star polygons
References
Citations
 ^ Conway, 2008, Chapter 26, Higher Still
 ^ "Convex and abstract polytopes", Programme and abstracts, MIT, 2005
 ^ Johnson (2015), Chapter 11, Section 11.5 Spherical Coxeter groups
 ^ Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ^ Coxeter, Complex Regular Polytopes, p.117
 ^ The Symmetries of Things, John Conway, (2008), p. 406, Fig 26.2
 ^ Coxeter, Star polytopes and the Schläfli function f{α,β,γ) p. 122 2. The SchläfliHess polytopes
Bibliography
 H. S. M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons Inc., 1969. ISBN 0471504580.
 H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808.
 D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26, Regular Starpolytopes, pp. 404–408)
 Edmund Hess, (1883) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [1].
 Edmund Hess Uber die regulären Polytope höherer Art, Sitzungsber Gesells Beförderung gesammten Naturwiss Marburg, 1885, 3157

Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [2]
 (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36]
 H. S. M. Coxeter, Regular Complex Polytopes, 2nd. ed., Cambridge University Press 1991. ISBN 9780521394901. [3]
 Peter McMullen and Egon Schulte, Abstract Regular Polytopes, 2002, PDF
External links
 Weisstein, Eric W. "Regular polychoron". MathWorld.
 Jonathan Bowers, 16 regular 4polytopes
 Regular 4D Polytope Foldouts
 Catalog of Polytope Images A collection of stereographic projections of 4polytopes.
 A Catalog of Uniform Polytopes
 Dimensions 2 hour film about the fourth dimension (contains stereographic projections of all regular 4polytopes)

Olshevsky, George. "Hecatonicosachoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Olshevsky, George. "Hexacosichoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Olshevsky, George. "Stellation". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Olshevsky, George. "Greatening". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Olshevsky, George. "Aggrandizement". Glossary for Hyperspace. Archived from the original on 4 February 2007.
 Reguläre Polytope
 The Regular Star Polychora