en>Anthony Appleyard: Anthony Appleyard moved page Reciprocal Gamma function to Reciprocal gamma function: Requested at WP:RM as uncontroversial (permalink)

2013-10-20T15:24:36Z

Anthony Appleyard moved page Reciprocal Gamma function to Reciprocal gamma function: Requested at WP:RM as uncontroversial (permalink)

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{| class=wikitable align=right width=400
|- align=center valign=top
|[[File:Schlegel wireframe 8-cell.png|100px]] [[Tesseract]] {{CDD|node_1|4|node|3|node|3|node}}
|[[File:Schlegel half-solid truncated tesseract.png|100px]] '''Truncated tesseract''' {{CDD|node_1|4|node_1|3|node|3|node}}
|[[File:Schlegel half-solid rectified 8-cell.png|100px]] [[Rectified tesseract]] {{CDD|node|4|node_1|3|node|3|node}}
|[[File:Schlegel half-solid bitruncated 8-cell.png|100px]] '''Bitruncated tesseract''' {{CDD|node|4|node_1|3|node_1|3|node}}
|- align=center
|colspan=4|[[Schlegel diagram]]s centered on [4,3] (cells visible at [3,3])
|- align=center valign=top
|[[File:Schlegel wireframe 16-cell.png|100px]] [[16-cell]] {{CDD|node|4|node|3|node|3|node_1}}
|[[File:Schlegel half-solid truncated 16-cell.png|100px]] '''Truncated 16-cell''' {{CDD|node|4|node|3|node_1|3|node_1}}
|[[File:Schlegel half-solid rectified 16-cell.png|100px]] Rectified 16-cell ([[24-cell]]) {{CDD|node|4|node|3|node_1|3|node}}
|[[File:Schlegel half-solid bitruncated 16-cell.png|100px]] '''Bitruncated tesseract''' {{CDD|node|4|node_1|3|node_1|3|node}}
|- align=center
|colspan=4|Schlegel diagrams centered on [3,3] (cells visible at [4,3])
|}
In [[geometry]], a '''truncated tesseract''' is a [[uniform polychoron]] (4-dimensional uniform [[polytope]]) formed as the [[Truncation (geometry)|truncation]] of the regular [[tesseract]].

There are three trunctions, including a [[bitruncation]], and a tritruncation, which creates the ''truncated 16-cell''.

== Truncated tesseract==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|Truncated tesseract
|-
|align=center colspan=3|[[Image:Schlegel half-solid truncated tesseract.png|220px]] [[Schlegel diagram]] ([[tetrahedron]] cells visible)
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t{4,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node_1|4|node_1|3|node|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|24
|8 [[truncated cube|''3.8.8'']] [[Image:Truncated hexahedron.png|20px]] 16 [[Tetrahedron|''3.3.3'']] [[Image:Tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|88
|64 [[triangle|{3}]] 24 [[octagon|{8}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|128
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|64
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Truncated 8-cell verf.png|80px]] Isosceles triangular pyramid
|-
|bgcolor=#e7dcc3|Dual
|colspan=2|[[Tetrakis 16-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|BC4, [4,3,3], order 384
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[16-cell|12]]'' 13 ''[[Cantellated tesseract|14]]''
|}
The '''truncated tesseract''' is bounded by 24 [[cell (mathematics)|cells]]: 8 [[truncated cube]]s, and 16 [[tetrahedron|tetrahedra]].

=== Alternate names===
* Truncated tesseract ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Truncated tesseract (Acronym tat) ([[George Olshevsky]], and Jonathan Bowers)<ref>Klitizing, (o3o3o4o - tat)</ref>

===Construction===

The truncated tesseract may be constructed by [[Truncation (geometry)|truncating]] the vertices of the [[tesseract]] at <math>1/(\sqrt{2}+2)</math> of the edge length. A regular tetrahedron is formed at each truncated vertex.

The [[Cartesian coordinate]]s of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:

:<math>\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)</math>

===Projections===

[[Image:3D stereoscopic projection truncated tesseract.PNG|left|thumb|A [[stereogram|stereoscopic]] 3D projection of a truncated tesseract.]]
In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:

* The projection envelope is a [[cube]].
* Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope.
* The other 6 truncated cubes project onto the square faces of the envelope.
* The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.

=== Images ===
{{4-cube Coxeter plane graphs|t01|150}}

{| class="wikitable"
|[[Image:Truncated tesseract net.png|200px]] A polyhedral [[Net (polytope)|net]]
|[[Image:Truncated tesseract stereographic (tC).png|200px]] Truncated tesseract projected onto the [[3-sphere]] with a [[stereographic projection]] into 3-space.
|}

=== Related polytopes ===
The ''[[Truncation (geometry)|truncated]] [[tesseract]]'', is third in a sequence of truncated [[hypercube]]s:
{{Truncated hypercube polytopes}}

==Bitruncated tesseract==
{| class="wikitable" align="right" width="360"
|-
!bgcolor=#e7dcc3 colspan=3|Bitruncated tesseract
|-
|colspan=3|[[Image:Schlegel half-solid bitruncated 16-cell.png|175px]][[Image:Schlegel half-solid bitruncated 8-cell.png|175px]] Two [[Schlegel diagram]]s, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden.
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2| 2t{4,3,3} 2t{3,31,1} h2,3{4,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node|4|node_1|3|node_1|3|node}} {{CDD|nodes_11|split2|node_1|3|node}} {{CDD|nodes_10ru|split2|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|Cells
|24
|8 [[Truncated octahedron|''4.6.6'']] [[Image:Truncated octahedron.png|20px]] 16 [[Truncated tetrahedron|''3.6.6'']] [[Image:Truncated tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|120
|32 [[triangle|{3}]] 24 [[square (geometry)|{4}]] 64 [[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|192
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|96
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Bitruncated 8-cell verf.png|60px]][[File:Cantitruncated demitesseract verf.png|60px]] Digonal [[disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
|colspan=2|BC4, [3,3,4], order 384 D4, [31,1,1], order 192
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[vertex-transitive]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Runcinated tesseract|15]]'' 16 ''[[Truncated 16-cell|17]]''
|}
The '''bitruncated tesseract''' or '''bitruncated 16-cell''' is constructed by a [[bitruncation]] operation applied to the [[tesseract]]. It can also be called a '''runcicantic tesseract''' with half the vertices of a [[runcicantellated tesseract]] with a {{CDD|node_h|4|node|3|node_1|3|node_1}} construction.

=== Alternate names===
* Bitruncated tesseract/Runcicantic tesseract ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Bitruncated tesseract (Acronym tah) ([[George Olshevsky]], and Jonathan Bowers)<ref>Klitizing, (o3x3x4o - tah)</ref>

===Construction===

A tesseract is bitruncated by [[Truncation (geometry)|truncating]] its [[Cell (mathematics)|cells]] beyond their mid-points, turning the eight [[cube]]s into eight [[truncated octahedron|truncated octahedra]]. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other.

The [[Cartesian coordinate]]s of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of:

:<math>\left(0,\ \pm\sqrt{2},\ \pm2\sqrt{2},\ \pm2\sqrt{2}\right)</math>

===Structure===

The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.

===Projections===
{{4-cube Coxeter plane graphs|t12|150}}

=== Stereographic projections ===

The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a [[truncated cube|truncated cubical]] envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells.

{| class=wikitable width=600
|+ [[Stereographic projection]]s
|- valign=top
|[[File:Bitruncated tesseract stereographic.png|300px]]
|[[Image:Bitrunc tessa schlegel.png|300px]] Colored transparently with pink triangles, blue squares, and gray hexagons
|}

=== Related polytopes ===
The ''[[bitruncation|bitruncated]] [[tesseract]]'' is second in a sequence of bitruncated [[hypercube]]s:
{{Bitruncated hypercube polytopes}}

== Truncated 16-cell==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|-
!bgcolor=#e7dcc3 colspan=3|'''Truncated 16-cell''' '''Cantic tesseract'''
|-
|colspan=3 align=center|[[Image:Schlegel half-solid truncated 16-cell.png|250px]] [[Schlegel diagram]] ([[octahedron]] cells visible)
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform polychoron]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t{4,3,3} t{3,31,1} h2{4,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
|colspan=2|{{CDD|node_1|3|node_1|3|node|4|node}} {{CDD|node_1|3|node_1|split1|nodes}} {{CDD|nodes_10ru|split2|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|Cells
|24
|8 [[Octahedron|''3.3.3.3'']] [[Image:Octahedron.png|20px]] 16 [[Truncated tetrahedron|''3.6.6'']] [[Image:Truncated tetrahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Faces
|96
|64 [[triangle|{3}]] 32 [[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|120
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|48
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[Image:Truncated 16-cell verf.png|50px]][[File:Truncated demitesseract verf.png|50px]] [[square pyramid]]
|-
|bgcolor=#e7dcc3|Dual
|colspan=2|[[Hexakis tesseract]]
|-
|bgcolor=#e7dcc3|[[Coxeter groups]]
|colspan=2|BC4 [3,3,4], order 384 D4 [31,1,1], order 192
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]]
|-
|bgcolor=#e7dcc3|Uniform index
|colspan=2|''[[Bitruncated tesseract|16]]'' 17 ''[[Cantitruncated tesseract|18]]''
|}

The '''truncated 16-cell''' or '''cantic tesseract''' which is bounded by 24 [[cell (mathematics)|cells]]: 8 regular [[octahedron|octahedra]], and 16 [[truncated tetrahedron|truncated tetrahedra]]. It has half the vertices of a [[cantellated tesseract]] with construction {{CDD|node_h|4|node|3|node_1|3|node}}.

It is related to, but not to be confused with, the [[24-cell]], which is a [[List_of_regular_polytopes#Four_dimensional_regular_polytopes|regular polychoron]] bounded by 24 regular octahedra.

=== Alternate names===
* Truncated 16-cell/Cantic tesseract ([[Norman Johnson (mathematician)|Norman W. Johnson]])
* Truncated hexadecachoron (Acronym thex) ([[George Olshevsky]], and Jonathan Bowers)<ref>Klitizing, (x3x3o4o - thex)</ref>

===Construction===

The truncated 16-cell may be constructed from the [[16-cell]] by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures).

(Truncating a 16-cell at 1/2 of the edge length results in the [[24-cell]], which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.)

The [[Cartesian coordinate]]s of the vertices of a truncated 16-cell having edge length 2√2 are given by all permutations, and sign combinations:
: (0,0,1,2)

An alternate construction begins with a [[demihypercube|demitesseract]] with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of
: (1,1,3,3), with an even number of each sign.

===Structure===

The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.

===Projections===

====Centered on octahedron====
[[Image:Truncated16cell-trunc-tetrahedron-small.gif|thumb|left|Octahedron-first parallel projection into 3 dimensions, with octahedral cells highlighted]]

The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

* The projection envelope is a [[truncated octahedron]].
* The 6 square faces of the envelope are the images of 6 of the octahedral cells.
* An octahedron lies at the center of the envelope, joined to the center of the 6 square faces by 6 edges. This is the image of the other 2 octahedral cells.
* The remaining space between the envelope and the central octahedron is filled by 8 truncated tetrahedra (distorted by projection). These are the images of the 16 truncated tetrahedral cells, a pair of cells to each image.

This layout of cells in projection is analogous to the layout of faces in the projection of the [[truncated octahedron]] into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.

{{clear}}

====Centered on truncated tetrahedron====
[[Image:Truncated 16cell-trunc-tetrahedron-first.gif|thumb|left|Projection of truncated 16-cell into 3 dimensions, centered on truncated tetrahedral cell, with hidden cells culled]]

The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

* The projection envelope is a [[truncated cube]].
* The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to 4 of the triangular faces of the envelope.
* The remaining space in the envelope is filled by 4 other truncated tetrahedra.
* These volumes are the images of the cells lying on the near side of the truncated 16-cell; the other cells project onto the same layout except in the dual configuration.
* The six octagonal faces of the projection envelope are the images of the remaining 6 truncated tetrahedral cells.

{{clear}}

=== Images ===

{{4-cube Coxeter plane graphs|t23|150}}

{| class="wikitable"
|[[Image:Truncated 16-cell net.png|160px]] [[Net (polytope)|Net]]
|[[Image:Truncated cross stereographic close-up.png|200px]] [[Stereographic projection]] (centered on [[truncated tetrahedron]])
|}

== Related uniform polytopes ==
{{Demitesseract family}}

{{Tesseract family}}

== Notes ==
{{reflist}}

== References ==
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Models 13, 16, 17}}
* {{KlitzingPolytopes|polychora.htm|4D|uniform polytopes (polychora)}} o3o3o4o - tat, o3x3x4o - tah, x3x3o4o - thex

== External links ==
* [http://www.software3d.com/Tat.php Paper model of truncated tesseract] created using nets generated by [[Stella (software)|Stella4D]] software

{{Polytopes}}

[[Category:Four-dimensional geometry]]
[[Category:Polychora]]

Reciprocal gamma function - Revision history

en>Anthony Appleyard: Anthony Appleyard moved page Reciprocal Gamma function to Reciprocal gamma function: Requested at WP:RM as uncontroversial (permalink)