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| [[File:Hypercube.svg|thumb|The [[tesseract]] is one of 6 convex regular polychora]]
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| In [[mathematics]], a '''convex regular polychoron''' is a [[polychoron]] (4-polytope) that is both [[regular polytope|regular]] and [[Convex polytope|convex]]. These are the four-dimensional analogs of the [[Platonic solid]]s (in three dimensions) and the [[regular polygon]]s (in two dimensions).
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| These polychora were first described by the Swiss [[mathematician]] [[Ludwig Schläfli]] in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the [[24-cell]]) which has no exact three-dimensional equivalent.
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| Each convex regular polychoron is bounded by a set of 3-dimensional ''[[cell (mathematics)|cells]]'' which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.
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| == Properties ==
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| The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all [[Coxeter group]]s and given in the notation described in that article. The number following the name of the group is the [[order (group theory)|order]] of the group.
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| {| class="wikitable" style="margin: auto; text-align: center;"
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| ! Names || Family || [[Schläfli symbol|Schläfli<br>symbol]] || Vertices || Edges || Faces || Cells || [[Vertex figure]]s || [[Dual polytope]]
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| !colspan=2 | [[Coxeter group|Symmetry group]]
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| | [[5-cell]]<BR>pentachoron<BR>pentatope<BR>hyperpyramid<BR>hypertetrahedron<BR>4-simplex || [[simplex]]<BR>(n-simplex) || {3,3,3} || 5 || 10 || 10<br>[[triangle]]s || 5<br>[[tetrahedra]] || [[tetrahedra]] || (self-dual) || ''A''<sub>4</sub> || 120
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| | [[8-cell]]<BR>octachoron<BR>tesseract<BR>hypercube<BR>4-cube || [[hypercube]]<BR>(n-cube) || {4,3,3} || 16 || 32 || 24<br>[[Square (geometry)|square]]s || 8<br>[[cube]]s || [[tetrahedra]] || 16-cell || ''B''<sub>4</sub> || 384
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| | [[16-cell]]<BR>hexadecachoron<BR>hyperoctahedron<BR>4-orthoplex || [[cross-polytope]]<BR>(n-orthoplex) || {3,3,4} || 8 || 24 || 32<br>[[triangle]]s || 16<br>[[tetrahedra]] || [[octahedra]] || 8-cell || ''B''<sub>4</sub> || 384
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| | [[24-cell]]<BR>icositetrachoron<BR>octaplex<BR>polyoctahedron || || {3,4,3} || 24 || 96 || 96<br>[[triangle]]s || 24<br>[[octahedra]] || [[cube]]s || (self-dual) || ''F''<sub>4</sub> || 1152
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| | [[120-cell]]<BR>hecatonicosachoron<BR>dodecaplex<BR>hyperdodecahedron<BR>polydodecahedron || dodecahedral [[pentagonal polytope]]<BR>(n-pentagonal polytope) || {5,3,3} || 600 || 1200 || 720<br>[[pentagon]]s || 120<br>[[dodecahedra]] || [[tetrahedra]] || 600-cell || ''H''<sub>4</sub> || 14400
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| | [[600-cell]]<BR>hexacosichoron<BR>tetraplex<BR>hypericosahedron<BR>polytetrahedron || icosahedral [[pentagonal polytope]]<BR>(n-pentagonal polytope) || {3,3,5} || 120 || 720 || 1200<br>[[triangle]]s || 600<br>[[tetrahedra]] || [[icosahedra]] || 120-cell || ''H''<sub>4</sub> || 14400
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| |}
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| Since the boundaries of each of these figures is [[homeomorphism|topologically equivalent]] to a [[3-sphere]], whose [[Euler characteristic]] is zero, we have the 4-dimensional analog of Euler's polyhedral formula:
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| :<math>N_0 - N_1 + N_2 - N_3 = 0\,</math>
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| where ''N''<sub>''k''</sub> denotes the number of ''k''-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).
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| ==Visualizations==
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| The following table shows some 2 dimensional projections of these polychora. Various other visualizations can be found in the external links below. The [[Coxeter-Dynkin diagram]] graphs are also given below the [[Schläfli symbol]].
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| {| class="wikitable" style="margin: auto; text-align: center;" width=630
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| ! [[5-cell]] || [[8-cell|Tesseract]] || [[16-cell]] || [[24-cell]] || [[120-cell]] || [[600-cell]]
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| ! {3,3,3} || {4,3,3} || {3,3,4} || {3,4,3} || {5,3,3} || {3,3,5}
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| |-
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| !{{CDD|node_1|3|node|3|node|3|node}}
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| !{{CDD|node_1|4|node|3|node|3|node}}
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| !{{CDD|node_1|3|node|3|node|4|node}}
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| !{{CDD|node_1|3|node|4|node|3|node}}
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| !{{CDD|node_1|5|node|3|node|3|node}}
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| !{{CDD|node_1|3|node|3|node|5|node}}
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| |-
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| |colspan=6|Wireframe [[orthographic projection]]s inside [[Petrie polygon]]s.
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| |- valign=top
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| | [[Image:4-simplex_t0.svg|105px]]
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| | [[Image:4-cube_t0.svg|105px]]
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| | [[Image:4-cube_t3.svg|105px]]
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| | [[Image:24-cell_t0_F4.svg|105px]]
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| | [[Image:120-cell_graph_H4.svg|105px]]
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| | [[Image:600-cell_graph_H4.svg|105px]]
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| |colspan=6|Solid [[orthographic projection]]s
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| |- valign=top
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| | [[Image:Tetrahedron.png|105px]]<BR>[[Tetrahedron|tetrahedral<BR>envelope]]<br />(cell/vertex-centered)
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| | [[Image:Hexahedron.png|105px]]<BR>[[Cube|cubic envelope]]<br />(cell-centered)
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| | [[File:16-cell ortho cell-centered.png|105px]]<BR>[[Cube|Cubic envelope]]<br />(cell-centered)
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| | [[Image:Ortho solid 24-cell.png|105px]]<BR>[[Cuboctahedron|cuboctahedral<BR>envelope]]<br />(cell-centered)
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| | [[Image:Ortho solid 120-cell.png|105px]]<BR>[[Truncated rhombic triacontahedron|truncated rhombic<BR>triacontahedron<BR>envelope]]<br />(cell-centered)
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| | [[Image:Ortho solid 600-cell.png|105px]]<BR>[[Pentakis icosidodecahedron|Pentakis icosidodecahedral<BR>envelope]]<br />(vertex-centered)
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| |colspan=6|Wireframe [[Schlegel diagram]]s ([[Perspective projection]])
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| | [[Image:Schlegel wireframe 5-cell.png|105px]]<BR>(Vertex-centered)
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| | [[Image:Schlegel wireframe 8-cell.png|105px]]<BR>(Cell-centered)
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| | [[Image:Schlegel wireframe 16-cell.png|105px]]<BR>(Cell-centered)
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| | [[Image:Schlegel wireframe 24-cell.png|105px]]<BR>(Cell-centered)
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| | [[Image:Schlegel wireframe 120-cell.png|105px]]<BR>(Cell-centered)
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| | [[Image:Schlegel_wireframe_600-cell_vertex-centered.png|105px]]<BR>(Vertex-centered)
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| |colspan=6|Wireframe [[stereographic projection]]s (Hyperspherical)
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| | [[Image:Stereographic polytope 5cell.png|105px]]
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| | [[Image:Stereographic_polytope_8cell.png|105px]]
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| | [[Image:Stereographic_polytope_16cell.png|105px]]
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| | [[Image:Stereographic_polytope_24cell.png|105px]]
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| | [[Image:Stereographic_polytope_120cell.png|105px]]
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| | [[Image:Stereographic_polytope_600cell.png|105px]]
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| |}
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| ==See also==
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| * Infinite regular polychora:
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| ** [[List_of_regular_polytopes#Tessellations_of_Euclidean_3-space|One regular Euclidean honeycomb:]] {4,3,4}
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| ** [[List_of_regular_polytopes#Tessellations_of_hyperbolic_3-space|Four regular hyperbolic honeycombs:]] {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
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| * Nonconvex regular polychora:
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| **[[Schläfli-Hess polychoron]] - Ten nonconvex regular polychora
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| *[[Abstract polytope|Abstract regular polychora]]:
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| ** [[11-cell]] {3,5,3}
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| ** [[57-cell]] {5,3,5}
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| *[[Uniform polychoron]] Polychoron families constructed from the from these 6 regular forms.
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| *[[Regular polytope]]
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| *[[Platonic solid]]
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| ==References==
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| *[[H. S. M. Coxeter]], ''Introduction to Geometry, 2nd ed.'', John Wiley & Sons Inc., 1969. ISBN 0-471-50458-0.
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| *H. S. M. Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
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| * [[Duncan MacLaren Young Sommerville|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
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| ==External links==
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| *{{Mathworld | urlname=RegularPolychoron | title=Regular polychoron }}
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| *[http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, 16 regular polychora]
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| *[http://www.weimholt.com/andrew/polytope.shtml Regular 4D Polytope Foldouts]
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| *[http://www.math.cmu.edu/~fho/jenn/polytopes/index.html Catalog of Polytope Images] A collection of stereographic projections of 4-polytopes.
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| *[http://www.math.cmu.edu/~fho/jenn/polytopes A Catalog of Uniform Polytopes]
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| *[http://www.dimensions-math.org/ Dimensions] 2 hour film about the fourth dimension (contains stereographic projections of all regular polychorons)
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| {{4D regular polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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