Regular 4-polytope

{{Short description|Four-dimensional analogues of the regular polyhedra in three dimensions}} [[File:Hypercube.svg|thumb|The tesseract is one of 6 convex regular 4-polytopes]] 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 convex and ten star regular 4-polytopes, giving a total of sixteen.

== History == The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.{{Sfn|Coxeter|1973|p=141|loc=§7-x. Historical remarks}} He discovered that there are precisely six such figures.

Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V'' {{=}} 2). That excludes cells and vertex figures such as the great dodecahedron {5,{{sfrac|5|2}}} and small stellated dodecahedron {{(}}{{sfrac|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 4-polytope <math>\{p,q,r\}</math> is constrained by the existence of the regular polyhedra <math>\{p,q\}, \{q,r\}</math> which form its cells and a dihedral angle constraint :<math>\sin\frac{\pi}p \sin\frac{\pi}r > \cos\frac{\pi}q</math>

to ensure that the cells meet to form a closed 3-surface.

The six convex and ten star polytopes described are the only solutions to these constraints.

There are four nonconvex Schläfli symbols {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,{{sfrac|5|2}},3}, {4,3,{{sfrac|5|2}}}, {{(}}{{sfrac|5|2}},3,4}, {{(}}{{sfrac|5|2}},3,{{sfrac|5|2}}}.

==Regular convex 4-polytopes== The regular convex 4-polytopes are the four-dimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.

Each convex regular 4-polytope is bounded by a set of 3-dimensional ''cells'' which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the ''surface'' of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).

=== Properties === Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content within the same radius.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions}} The 4-simplex (5-cell) has the smallest content, and the 120-cell has the largest.

The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes 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.

{| class="wikitable sortable" style="text-align:center" ! Names ||Image|| Family || Schläfli Coxeter || V || E || F || C || Vert. fig. || Dual !colspan=2 | Symmetry group |- BGCOLOR="#e8ffe8" | 5-cell pentachoron pentatope 4-simplex || 125px || ''n''-simplex (An family) || {3,3,3} {{dark mode invert|{{CDD|node_1|3|node|3|node|3|node}}}} || 5 || 10 || 10 {3} || 5 {3,3} || {3,3} || ''self-dual'' || ''A''4 [3,3,3]|| 120 |- style="background: linear-gradient(0deg, rgb(240,224,240) 0%, rgb(224,224,255) 10%, rgb(224,224,255) 100%);" | 16-cell hexadecachoron 4-orthoplex || 125px|| ''n''-orthoplex (Bn family) || {3,3,4} {{dark mode invert|{{CDD|node_1|3|node|3|node|4|node}}}} || 8 || 24 || 32 {3} || 16 {3,3} || {3,4} || 8-cell || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | ''B''4 [4,3,3] || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | 384 |- style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(255,224,224) 90%, rgb(240,224,240) 100%);" | 8-cell octachoron tesseract 4-cube || 125px || hypercube ''n''-cube (Bn family) || {4,3,3} {{dark mode invert|{{CDD|node_1|4|node|3|node|3|node}}}} || 16 || 32 || 24 {4} || 8 {4,3} || {3,3} || 16-cell |- BGCOLOR="#e8ffe8" | 24-cell icositetrachoron octaplex polyoctahedron (pO) || 125px || Fn family || {3,4,3} {{dark mode invert|{{CDD|node_1|3|node|4|node|3|node}}}} || 24 || 96 || 96 {3} || 24 {3,4} || {4,3} || ''self-dual'' || ''F''4 [3,4,3] || 1152 |- style="background: linear-gradient(0deg, rgb(240,224,240) 0%, rgb(224,224,255) 10%, rgb(224,224,255) 100%);" | 600-cell hexacosichoron tetraplex polytetrahedron (pT) || 125px || n-pentagonal polytope (Hn family) || {3,3,5} {{dark mode invert|{{CDD|node_1|3|node|3|node|5|node}}}} || 120 || 720 || 1200 {3} || 600 {3,3} || {3,5} || 120-cell || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | ''H''4 [5,3,3] || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | 14400 |- style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(255,224,224) 90%, rgb(240,224,240) 100%);" | 120-cell hecatonicosachoron dodecacontachoron dodecaplex polydodecahedron (pD) || 125px|| n-pentagonal polytope (Hn family) || {5,3,3} {{dark mode invert|{{CDD|node_1|5|node|3|node|3|node}}}} || 600 || 1200 || 720 {5} || 120 {5,3} || {3,3} || 600-cell |}

John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).<ref>{{harvnb|Conway|Burgiel|Goodman-Strauss|2008|loc=Ch. 26. Higher Still}}</ref>

Norman Johnson advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term ''polychoron'' being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots ''poly'' ("many") and ''choros'' ("room" or "space").<ref>[https://www.mit.edu/~hlb/Associahedron/program.pdf "Convex and abstract polytopes", Programme and abstracts, MIT, 2005]</ref><ref>{{cite book |first=Norman W. |last=Johnson |title=Geometries and Transformations |chapter-url=https://books.google.com/books?id=adBVDwAAQBAJ&pg=PA246 |date=2018 |publisher=Cambridge University Press |isbn=978-1-107-10340-5 |pages=246– |chapter=§ 11.5 Spherical Coxeter groups}}</ref>

The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula: :<math>N_0 - N_1 + N_2 - N_3 = 0\,</math> where ''N''''k'' denotes the number of ''k''-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.<ref name="richeson">{{cite book |first=David S. |last=Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology |url=https://books.google.com/books?id=zyIRIcRSNwsC |date=2012 |publisher=Princeton University Press |isbn=978-0-691-15457-2 |chapter-url=https://books.google.com/books?id=zyIRIcRSNwsC&pg=PA253 |chapter=23. Henri Poincaré and the Ascendancy of Topology |pages=256–}}</ref>

=== As configurations === A regular 4-polytope can be completely described as a configuration matrix containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices ''in'' each edge (each edge ''has'' 2 vertices), and 2 cells meet ''at'' each face (each face ''belongs to'' 2 cells), in any regular 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.<ref>{{harvnb|Coxeter|1973|loc=§ 1.8 Configurations}}</ref><ref>Coxeter, Complex Regular Polytopes, p. 117</ref> {| class=wikitable !5-cell {3,3,3} !16-cell {3,3,4} !8-cell {4,3,3} !24-cell {3,4,3} !600-cell {3,3,5} !120-cell {5,3,3} |- style="font-size:85%;" |bgcolor="#e8ffe8"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}5 & 4 & 6 & 4 \\ 2 & 10 & 3 & 3 \\ 3 & 3 & 10 & 2 \\ 4 & 6 & 4 & 5 \end{matrix}\end{bmatrix}</math>}} |bgcolor="#FFe0e0"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}8 & 6 & 12 & 8 \\ 2 & 24 & 4 & 4 \\ 3 & 3 & 32 & 2 \\ 4 & 6 & 4 & 16 \end{matrix}\end{bmatrix}</math>}} |bgcolor="#e0e0ff"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}16 & 4 & 6 & 4 \\ 2 & 32 & 3 & 3 \\ 4 & 4 & 24 & 2 \\ 8 & 12 & 6 & 8 \end{matrix}\end{bmatrix}</math>}} |bgcolor="#e8ffe8"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math>}} |bgcolor="#ffe0e0"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math>}} |bgcolor="#e0e0ff"|{{dark mode invert|<math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math>}} |}

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

{| class="wikitable" style="text-align:center;" |- ! A4 = [3,3,3] ||colspan=2| B4 = [4,3,3] || F4 = [3,4,3] ||colspan=2| H4 = [5,3,3] |- ! 5-cell || 16-cell || 8-cell || 24-cell || 600-cell || 120-cell |- ! {3,3,3} || {3,3,4} || {4,3,3} || {3,4,3} || {3,3,5} || {5,3,3} |- !{{CDD|node_1|3|node|3|node|3|node}} !{{CDD|node_1|3|node|3|node|4|node}} !{{CDD|node_1|4|node|3|node|3|node}} !{{CDD|node_1|3|node|4|node|3|node}} !{{CDD|node_1|3|node|3|node|5|node}} !{{CDD|node_1|5|node|3|node|3|node}} |- |colspan=6|Solid 3D orthographic projections |- valign=top style="line-height:120%" | 125px Tetrahedral envelope (cell/vertex-centered) | 125px Cubic envelope (cell-centered) | 125px Cubic envelope (cell-centered) | 125px Cuboctahedral envelope (cell-centered) | 125px Pentakis icosidodecahedral envelope (vertex-centered) | 125px Truncated rhombic triacontahedron envelope (cell-centered) |- |colspan=6|Wireframe Schlegel diagrams (Perspective projection) |- valign=bottom | 125px Cell-centered | 125px Cell-centered | 125px Cell-centered | 125px Cell-centered | 125px Vertex-centered | 125px Cell-centered |- |colspan=6|Wireframe stereographic projections (3-sphere) |- valign=top | 125px | 125px | 125px | 125px | 125px|class=skin-invert | 125px |}

== Regular star (Schläfli–Hess) 4-polytopes == [[File:Relationship among regular star polychora.png|320px|thumb|This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.<ref>{{harvnb|Conway|Burgiel|Goodman-Strauss|2008|p=406, Fig 26.2}}</ref>]] [[File:Relationship among regular star polychora-8.png|320px|thumb|A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations {a,g,s} are commutable, defining a cubic framework. There are 7 densities seen in vertical positioning, with 2 dual forms having the same density.]] The '''Schläfli–Hess 4-polytopes''' are the complete set of 10 regular self-intersecting '''star polychora''' (four-dimensional polytopes).<ref>Coxeter, ''Star polytopes and the Schläfli function f{α,β,γ)'' p. 122 2. ''The Schläfli-Hess polytopes''</ref> 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 {{sfrac|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 with longer edges in same lines. (Example: a pentagon stellates into a pentagram) #'''greatening''' – replaces the faces with large ones in same planes. (Example: an icosahedron greatens into a great icosahedron) #'''aggrandizement''' – replaces the cells with large ones in same 3-spaces. (Example: a 600-cell aggrandizes into a grand 600-cell)

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicosahedron {3,5,{{sfrac|5|2}}} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), 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] (H4) hexacosichoric symmetry. They are generated from 6 related Goursat tetrahedra rational-order 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 star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

=== Properties === Note: * There are 2 unique vertex arrangements, matching those of the 120-cell and 600-cell. * There are 4 unique edge arrangements, which are shown as ''wireframes'' orthographic projections. * There are 7 unique face arrangements, shown as ''solids'' (face-colored) orthographic projections.

The cells (polyhedra), their faces (polygons), the ''polygonal edge figures'' and ''polyhedral vertex figures'' are identified by their Schläfli symbols.

{| class="wikitable sortable" ! Name Conway (abbrev.) ! Orthogonal projection ! Schläfli Coxeter ! C {p, q} ! F {p} ! E {r} ! V {q, r} !Dens. ! χ |- align=center BGCOLOR="#e0e0ff" | Icosahedral 120-cell polyicosahedron (pI) | 75px | {3,5,5/2} {{dark mode invert|{{CDD|node_1|3|node|5|node|5|rat|d2|node}}}} | 120 {3,5} 25px | 1200 {3} 25px | 720 {5/2} 25px | 120 {5,5/2} 25px | 4 | 480 |- align=center BGCOLOR="#ffe0e0" | Small stellated 120-cell stellated polydodecahedron (spD) | 75px | {5/2,5,3} {{dark mode invert|{{CDD|node|3|node|5|node|5|rat|d2|node_1}}}} | 120 {5/2,5} 25px | 720 {5/2} 25px | 1200 {3} 25px | 120 {5,3} 25px | 4 | −480 |- align=center BGCOLOR="#e0ffe0" | Great 120-cell great polydodecahedron (gpD) | 75px | {5,5/2,5} {{dark mode invert|{{CDD|node_1|5|node|5|rat|d2|node|5|node}}}} | 120 {5,5/2} 25px | 720 {5} 25px | 720 {5} 25px | 120 {5/2,5} 25px | 6 | 0 |- align=center BGCOLOR="#e0e0ff" | Grand 120-cell grand polydodecahedron (apD) | 75px | {5,3,5/2} {{dark mode invert|{{CDD|node_1|5|node|3|node|5|rat|d2|node}}}} | 120 {5,3} 25px | 720 {5} 25px | 720 {5/2} 25px | 120 {3,5/2} 25px | 20 | 0 |- align=center BGCOLOR="#ffe0e0" | Great stellated 120-cell great stellated polydodecahedron (gspD) | 75px | {5/2,3,5} {{dark mode invert|{{CDD|node|5|node|3|node|5|rat|d2|node_1}}}} | 120 {5/2,3} 25px | 720 {5/2} 25px | 720 {5} 25px | 120 {3,5} 25px | 20 | 0 |- align=center BGCOLOR="#e0ffe0" | Grand stellated 120-cell grand stellated polydodecahedron (aspD) | 75px | {5/2,5,5/2} {{dark mode invert|{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}}} | 120 {5/2,5} 25px | 720 {5/2} 25px | 720 {5/2} 25px | 120 {5,5/2} 25px | 66 | 0 |- align=center BGCOLOR="#e0e0ff" | Great grand 120-cell great grand polydodecahedron (gapD) | 75px | {5,5/2,3} {{dark mode invert|{{CDD|node_1|5|node|5|rat|d2|node|3|node}}}} | 120 {5,5/2} 25px | 720 {5} 25px | 1200 {3} 25px | 120 {5/2,3} 25px | 76 | −480 |- align=center BGCOLOR="#ffe0e0" | Great icosahedral 120-cell great polyicosahedron (gpI) | 75px | {3,5/2,5} {{dark mode invert|{{CDD|node|5|node|5|rat|d2|node|3|node_1}}}} | 120 {3,5/2} 25px | 1200 {3} 25px | 720 {5} 25px | 120 {5/2,5} 25px | 76 | 480 |- align=center BGCOLOR="#e0e0ff" | Grand 600-cell grand polytetrahedron (apT) | 75px | {3,3,5/2} {{dark mode invert|{{CDD|node_1|3|node|3|node|5|rat|d2|node}}}} | 600 {3,3} 25px | 1200 {3} 25px | 720 {5/2} 25px | 120 {3,5/2} 25px | 191 | 0 |- align=center BGCOLOR="#ffe0e0" | Great grand stellated 120-cell great grand stellated polydodecahedron (gaspD) | 75px | {5/2,3,3} {{dark mode invert|{{CDD|node|3|node|3|node|5|rat|d2|node_1}}}} | 120 {5/2,3} 25px | 720 {5/2} 25px | 1200 {3} 25px | 600 {3,3} 25px | 191 | 0 |}

== See also == * Regular polytope * List of regular polytopes * Infinite regular 4-polytopes: ** One regular Euclidean honeycomb: {4,3,4} ** Four compact regular hyperbolic honeycombs: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5} ** Eleven paracompact regular hyperbolic honeycombs: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}. * Abstract regular 4-polytopes: ** 11-cell {3,5,3} ** 57-cell {5,3,5} * Uniform 4-polytope uniform 4-polytope families constructed from these 6 regular forms. * Platonic solid * Kepler-Poinsot polyhedra — regular star polyhedron * Star polygon — regular star polygons * 4-polytope * 5-polytope * 6-polytope

== Notes == {{Notelist}}

== References ==

=== Citations === {{Reflist}}

=== Bibliography === {{refbegin}} * {{cite book | last=Coxeter | first=H.S.M. | author-link=H. S. M. Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | title-link=Regular Polytopes (book) | publisher=Dover | place=New York | edition=3rd }}

* {{cite book |first=H.S.M. |last=Coxeter |title=Introduction to Geometry |publisher=Wiley |year=1969 |edition=2nd |isbn=0-471-50458-0 }}

* {{cite book |author=D.M.Y. Sommerville |author-link=Duncan Sommerville |title=Introduction to the Geometry of '''n''' Dimensions |chapter=X. The Regular Polytopes |chapter-url=https://books.google.com/books?id=4vXDDwAAQBAJ&pg=PA161 |pages=159–192 |date=2020 |orig-year=1930 |publisher=Courier Dover |isbn=978-0-486-84248-6 }}

* {{cite book |first1=John H. |last1=Conway |author-link=John Horton Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |title=The Symmetries of Things |year=2008 |chapter=26. Regular Star-polytopes |pages=404–8 |isbn=978-1-56881-220-5 }}

* {{cite web |first=Edmund |last=Hess |author-link=Edmund Hess |title=Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder |date=1883 |url=http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001 }}

* {{cite journal |first=Edmund |last=Hess |title=Uber die regulären Polytope höherer Art |journal=Sitzungsber Gesells Beförderung Gesammten Naturwiss Marburg |pages=31–57 |year=1885 }}

* {{cite book |editor1-first=F. Arthur |editor1-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivić |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |url=https://archive.org/details/kaleidoscopessel0000coxe |url-access=registration |isbn=978-0-471-01003-6 }} ** (Paper 10) {{cite journal |first=H.S.M. |last=Coxeter |title=Star Polytopes and the Schläfli Function f(α,β,γ) |journal=Elemente der Mathematik |volume=44 |issue=2 |pages=25–36 |year=1989 |url=https://eudml.org/doc/141447 }}

* {{cite book |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |publisher=Cambridge University Press |edition=2nd |year=1991 |isbn=978-0-521-39490-1 }}

* {{cite book |first1=Peter |last1=McMullen |first2=Egon |last2=Schulte |year=2002 |title=Abstract Regular Polytopes |series=Encyclopedia of Mathematics and its Applications |volume=92 |publisher=Cambridge University Press |url=https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA1 |isbn=978-0-521-81496-6 }} [http://assets.cambridge.org/052181/4960/sample/0521814960ws.pdf Chapter I Classical Regular Polytopes] (Sample text) {{refend}}

== External links == *{{Mathworld | urlname=RegularPolychoron | title=Regular polychoron }} *[http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, 16 regular 4-polytopes] *[http://www.weimholt.com/andrew/polytope.shtml Regular 4D Polytope Foldouts] {{Webarchive|url=https://web.archive.org/web/20110717184828/http://weimholt.com/andrew/polytope.shtml |date=2011-07-17 }} *[http://www.math.cmu.edu/~fho/jenn/polytopes/index.html Catalog of Polytope Images] A collection of stereographic projections of 4-polytopes. *[http://www.math.cmu.edu/~fho/jenn/polytopes A Catalog of Uniform Polytopes] *[http://www.dimensions-math.org/ Dimensions] 2 hour film about the fourth dimension (contains stereographic projections of all regular 4-polytopes) *[https://web.archive.org/web/20061107052613/http://www.mathematik.uni-regensburg.de/Goette/sterne/ Reguläre Polytope] * [https://web.archive.org/web/20070704012333/http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora] *[https://pages.uoregon.edu/koch/hypersolids/hypersolids.html Hypersolids]

{{DEFAULTSORT:Regular 4-polytope}} Category:Regular 4-polytopes