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Added link

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2009-05-22T22:06:06Z

←Created page with 'In mathematics — specifically, in geometric measure theory — a '''uniformly distributed measure''' on a metric space is one for which the me…'

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{| class=wikitable align=right width=520
|- align=center valign=top
|[[File:5-simplex t0.svg|120px]] [[File:5-simplex t0 A4.svg|120px]] [[5-simplex]] {{CDD|node_1|3|node|3|node|3|node|3|node}}
|[[File:5-simplex t04.svg|120px]] [[File:5-simplex t04 A4.svg|120px]] '''Stericated 5-simplex''' {{CDD|node_1|3|node|3|node|3|node|3|node_1}}
|- align=center valign=top
|[[File:5-simplex t014.svg|120px]] [[File:5-simplex t014 A4.svg|120px]] '''Steritruncated 5-simplex''' {{CDD|node_1|3|node_1|3|node|3|node|3|node_1}}
|[[File:5-simplex t024.svg|120px]] [[File:5-simplex t024 A4.svg|120px]] '''Stericantellated 5-simplex''' {{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}
|- align=center valign=top
|[[File:5-simplex t0124.svg|120px]] [[File:5-simplex t0124 A4.svg|120px]] '''Stericantitruncated 5-simplex''' {{CDD|node_1|3|node|3|node_1|3|node|3|node_1}}
|[[File:5-simplex t0134.svg|120px]] [[File:5-simplex t0134 A4.svg|120px]] '''Steriruncitruncated 5-simplex''' {{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1}}
|- align=center valign=top
|colspan=2|[[File:5-simplex t01234.svg|180px]] [[File:5-simplex t01234 A4.svg|180px]] '''Steriruncicantitruncated 5-simplex''' (Omnitruncated 5-simplex) {{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
|-
!colspan=4|[[Orthogonal projection]]s in A5 and A4 [[Coxeter plane]]s
|}

In five-dimensional [[geometry]], a '''stericated 5-simplex''' is a convex [[uniform 5-polytope]] with fourth-order [[Truncation (geometry)|truncations]] ([[sterication]]) of the regular [[5-simplex]].

There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an '''expanded 5-simplex''', with the first and last nodes ringed, for being constructible by an [[Expansion (geometry)|expansion]] operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an [[#Omnitruncated 5-simplex|omnitruncated 5-simplex]] with all of the nodes ringed.

== Stericated 5-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|2r2r{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD||node_1|3|node||3|node||3|node||3|node_1}} or {{CDD|node|split1|nodes|3ab|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces
|62
|6+6 [[Pentachoron|{3,3,3}]][[Image:Schlegel wireframe 5-cell.png|25px]] 15+15 [[Tetrahedral prism|{}×{3,3}]][[Image:Tetrahedral prism.png|25px]] 20 [[Duoprism|{3}×{3}]][[Image:3-3 duoprism.png|25px]]
|-
|bgcolor=#e7dcc3|Cells
|180
|60 [[Tetrahedron|{3,3}]][[Image:Tetrahedron.png|25px]] 120 [[Triangular prism|{}×{3}]][[Image:Triangular prism.png|25px]]
|-
|bgcolor=#e7dcc3|Faces
|210
|120 [[Triangle|{3}]] 90 [[Square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|120
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|30
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericated hexateron verf.png|80px]] [[16-cell|Tetrahedral antiprism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A5×2, [[3,3,3,3]], order 1440
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]]
|}

A '''stericated 5-simplex''' can be constructed by an [[Expansion (geometry)|expansion]] operation applied to the regular [[5-simplex]], and thus is also sometimes called an '''expanded 5-simplex'''. It has 30 [[Vertex (geometry)|vertices]], 120 [[Edge (geometry)|edges]], 210 [[Face (geometry)|faces]] (120 [[triangle]]s and 90 [[Square (geometry)|squares]]), 180 [[Cell (geometry)|cells]] (60 [[Tetrahedron|tetrahedra]] and 120 [[triangular prism]]s) and 62 [[4-face]]s (12 [[5-cell]]s, 30 [[tetrahedral prism]]s and 20 [[Duoprism|3-3 duoprisms]]).

=== Alternate names ===
* Expanded 5-simplex
* Stericated hexateron
* Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)<ref>Klitizing, (x3o3o3o3x - scad)</ref>

===Cross-sections===

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a [[runcinated pentachoron]]. This cross-section divides the stericated hexateron into two [[Cupola (geometry)#Hypercupolas|pentachoral hypercupolas]] consisting of 6 [[Pentachoron|pentachora]], 15 [[tetrahedral prism]]s and 10 [[Duoprism|3-3 duoprisms]] each.

=== Coordinates ===

The vertices of the ''stericated 5-simplex'' can be constructed on a [[hyperplane]] in 6-space as permutations of (0,1,1,1,1,2). This represents the positive [[orthant]] [[Facet (geometry)|facet]] of the [[stericated 6-orthoplex]].

A second construction in 6-space, from the center of a [[rectified 6-orthoplex]] is given by coordinate permutations of:
: (1,-1,0,0,0,0)

The [[Cartesian coordinates]] in 5-space for the normalized vertices of an origin-centered '''stericated hexateron''' are:

:<math>\left(\pm1,\ 0,\ 0,\ 0,\ 0\right)</math>
:<math>\left(0,\ \pm1,\ 0,\ 0,\ 0\right)</math>
:<math>\left(0,\ 0,\ \pm1,\ 0,\ 0\right)</math>
:<math>\left(\pm1/2,\ 0,\ \pm1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right)</math>
:<math>\left(\pm1/2,\ 0,\ \pm1/2,\ \sqrt{1/8},\ \sqrt{3/8}\right)</math>
:<math>\left( 0,\ \pm1/2,\ \pm1/2,\ -\sqrt{1/8},\ \sqrt{3/8}\right)</math>
:<math>\left( 0,\ \pm1/2,\ \pm1/2,\ \sqrt{1/8},\ -\sqrt{3/8}\right)</math>
:<math>\left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt{1/2},\ 0\right)</math>

=== Root system ===
Its 30 vertices represent the root vectors of the [[simple Lie group]] A5. It is also the [[vertex figure]] of the [[5-simplex honeycomb]].

=== Images ===

{{5-simplex2 Coxeter plane graphs|t04|100}}

{| class=wikitable
|- align=center
|[[File:Stericated hexateron ortho.svg|160px]] orthogonal projection with [6] symmetry
|}

==Steritruncated 5-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Steritruncated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t0,2,3{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|3|node_1||3|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces
|62
|6 [[Truncated 5-cell|t{3,3,3}]] 15 {}×[[Truncated tetrahedron|t{3,3}]] 20 [[Duoprism|{3}×{6}]] 15 {}×[[Tetrahedron|{3,3}]] 6 [[Runcinated 5-cell|rr{3,3,3}]]
|-
|bgcolor=#e7dcc3|Cells
|330
|
|-
|bgcolor=#e7dcc3|Faces
|570
|
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|420
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|120
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Steritruncated 5-simplex verf.png|100px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A5 [3,3,3,3], order 720
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Steritruncated hexateron
* Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)<ref>Klitizing, (x3x3o3o3x - cappix)</ref>

=== Coordinates ===
The coordinates can be made in 6-space, as 180 permutations of:
: (0,1,1,1,2,3)

This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[steritruncated 6-orthoplex]].

=== Images ===
{{5-simplex Coxeter plane graphs|t014|100}}

==Stericantellated 5-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericantellated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|rr2r{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD||node_1|3|node||3|node_1|3|node|3|node_1}} or {{CDD|node_1|split1|nodes|3ab|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces
| 62
|12 [[Cantellated 5-cell|rr{3,3,3}]] 30 [[Cantellated tetrahedron|rr{3,3}]]x{} 20 [[Duoprism|{3}×{3}]]
|-
|bgcolor=#e7dcc3|Cells
|420
|60 [[Cantellated tetrahedron|rr{3,3}]] 240 [[Triangular prism|{}×{3}]] 90 [[Cube|{}×{}×{}]] 30 [[Rectified tetrahedron|r{3,3}]]
|-
|bgcolor=#e7dcc3|Faces
|900
|360 [[Triangle|{3}]] 540 [[Square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|720
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|180
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericantellated 5-simplex verf.png|100px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A5×2, [[3,3,3,3]], order 1440
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Stericantellated hexateron
* Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)<ref>Klitizing, (x3o3x3o3x - card)</ref>

=== Coordinates ===
The coordinates can be made in 6-space, as permutations of:
: (0,1,1,2,2,3)

This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[stericantellated 6-orthoplex]].

=== Images ===
{{5-simplex2 Coxeter plane graphs|t024|100}}

==Stericantitruncated 5-simplex==

{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Stericantitruncated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|t0,1,2,4{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|3|node_1||3|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|4-faces
|62
|
|-
|bgcolor=#e7dcc3|Cells
|480
|
|-
|bgcolor=#e7dcc3|Faces
|1140
|
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|1080
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|360
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Stericanitruncated 5-simplex verf.png|100px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A5 [3,3,3,3], order 720
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Stericantitruncated hexateron
* Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)<ref>Klitizing, (x3x3x3o3x - cograx)</ref>

=== Coordinates ===
The coordinates can be made in 6-space, as 360 permutations of:
: (0,1,1,2,3,4)

This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[stericantitruncated 6-orthoplex]].

=== Images ===
{{5-simplex Coxeter plane graphs|t0124|100}}

==Steriruncitruncated 5-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Steriruncitruncated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|2t2r{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
|colspan=2|{{CDD||node_1|3|node_1||3|node|3|node_1|3|node_1}} or {{CDD|node|split1|nodes_11|3ab|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces
|62
|12 [[Runcitruncated 5-cell|t0,1,3{3,3,3}]] 30 {}×[[Truncated tetrahedron|t{3,3}]] 20 [[Duoprism|{6}×{6}]]
|-
|bgcolor=#e7dcc3|Cells
|450
|
|-
|bgcolor=#e7dcc3|Faces
|1110
|
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|1080
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|360
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Steriruncitruncated 5-simplex verf.png|100px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2|A5×2, [[3,3,3,3]], order 1440
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[Isogonal figure|isogonal]]
|}

=== Alternate names ===
* Steriruncitruncated hexateron
* Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)<ref>Klitizing, (x3x3o3x3x - captid)</ref>

=== Coordinates ===
The coordinates can be made in 6-space, as 360 permutations of:
: (0,1,2,2,3,4)

This construction exists as one of 64 [[orthant]] [[Facet (geometry)|facets]] of the [[steriruncitruncated 6-orthoplex]].

=== Images ===
{{5-simplex2 Coxeter plane graphs|t0134|100}}

== Omnitruncated 5-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
|-
|bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated 5-simplex'''
|-
|bgcolor=#e7dcc3|Type
|colspan=2|[[Uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]
|colspan=2|tr2r{3,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]]
|colspan=2|{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} or {{CDD|node_1|split1|nodes_11|3ab|nodes_11}}
|-
|bgcolor=#e7dcc3|4-faces
|62
|12 [[Omnitruncated 5-cell|t0,1,2,3{3,3,3}]][[Image:Schlegel half-solid omnitruncated 5-cell.png|25px]] 30 [[Truncated octahedral prism|{}×tr{3,3}]][[Image:Truncated octahedral prism.png|25px]] 20 [[Duoprism|{6}×{6}]][[Image:6-6 duoprism.png|25px]]
|-
|bgcolor=#e7dcc3|Cells
|540
|360 [[truncated octahedron|t{3,4}]][[Image:Truncated octahedron.png|25px]] 90 [[cube|{4,3}]][[Image:Tetragonal prism.png|25px]] 90 [[hexagonal prism|{}×{6}]][[Image:Hexagonal prism.png|25px]]
|-
|bgcolor=#e7dcc3|Faces
|1560
|480 [[Hexagon|{6}]] 1080 [[Square (geometry)|{4}]]
|-
|bgcolor=#e7dcc3|Edges
|colspan=2|1800
|-
|bgcolor=#e7dcc3|Vertices
|colspan=2|720
|-
|bgcolor=#e7dcc3|[[Vertex figure]]
|colspan=2|[[File:Omnitruncated 5-simplex verf.png|80px]] [[Irregular 5-cell]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]
|colspan=2| A5×2, [[3,3,3,3]], order 1440
|-
|bgcolor=#e7dcc3|Properties
|colspan=2|[[Convex polytope|convex]], [[isogonal]], [[zonotope]]
|}
The '''omnitruncated 5-simplex''' has 720 [[vertex (geometry)|vertices]], 1800 [[Edge (geometry)|edge]]s, 1560 [[Face (geometry)|faces]] (480 [[hexagon]]s and 1080 [[Square (geometry)|squares]]), 540 [[Cell (geometry)|cells]] (360 [[truncated octahedron]]s, 90 [[cube]]s, and 90 [[hexagonal prism]]s), and 62 [[4-face]]s (12 [[omnitruncated 5-cell]]s, 30 [[truncated octahedral prism]]s, and 20 6-6 [[duoprism]]s).

=== Alternate names ===
* Steriruncicantitruncated 5-simplex (Full description of [[omnitruncation]] for 5-polytopes by Johnson)
* Omnitruncated hexateron
* Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)<ref>Klitizing, (x3x3x3x3x - gocad)</ref>

=== Coordinates ===
The vertices of the ''truncated 5-simplex'' can be most simply constructed on a [[hyperplane]] in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive [[orthant]] [[Facet (geometry)|facet]] of the [[steriruncicantitruncated 6-orthoplex]], t0,1,2,3,4{34,4}, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}.

=== Images ===

{| class=wikitable width=640
|- valign=top align=center
|
{{5-simplex2 Coxeter plane graphs|t01234|120}}
|[[Stereographic projection]] [[Image:Omnitruncated Hexateron.png|160px]]
|}

=== Permutohedron ===

The omnitruncated 5-simplex is the permutohedron of order 6. It is also a [[zonotope]], the [[Minkowski sum]] of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.

{| class=wikitable
|[[Image:Omnitruncated Hexateron as Permutohedron.svg|480px]] [[Orthogonal projection]], vertices labeled as a [[permutohedron]].
|}

=== Related honeycomb ===
The [[omnitruncated 5-simplex honeycomb]] is constructed by '''omnitruncated 5-simplex''' facets with 3 [[Facet (geometry)|facets]] around each [[Ridge (geometry)|ridge]]. It has [[Coxeter-Dynkin diagram]] of {{CDD|branch_11|3ab|nodes_11|3ab|branch_11}}.

{| class=wikitable
|- align=center
![[Coxeter group]]
!<math>{\tilde{I}}_{1}</math>
!<math>{\tilde{A}}_{2}</math>
!<math>{\tilde{A}}_{3}</math>
!<math>{\tilde{A}}_{4}</math>
!<math>{\tilde{A}}_{5}</math>
|- align=center
![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
|{{CDD|node_1|infin|node_1}}
|{{CDD||branch_11|split2|node_1}}
|{{CDD|branch_11|3ab|branch_11}}
|{{CDD|branch_11|3ab|nodes_11|split2|node_1}}
|{{CDD|branch_11|3ab|nodes_11|3ab|branch_11}}
|-
!Picture
|[[File:Uniform apeirogon.png|100px]]
|[[File:Uniform tiling 333-t012.png|100px]]
|[[File:Bitruncated cubic honeycomb4.png|100px]]
|
|
|-
!Name
|[[Apeirogon]]
|[[Hextille]]
|[[Bitruncated cubic honeycomb|Omnitruncated 3-simplex honeycomb]]
|[[Omnitruncated 4-simplex honeycomb|Omnitruncated 4-simplex honeycomb]]
|[[Omnitruncated 5-simplex honeycomb|Omnitruncated 5-simplex honeycomb]]
|-
!Facets
|[[File:Segment definition.svg|100px]]
|[[File:Omnitruncated 2-simplex graph.png|100px]]
|[[File:Omnitruncated 3-simplex.png|100px]]
|[[File:Omnitruncated 4-simplex.png|100px]]
|[[File:Omnitruncated 5-simplex.png|100px]]
|}

== Related uniform polytopes ==

These polytopes are a part of 19 [[Uniform_polyteron#The_A5_.5B3.2C3.2C3.2C3.5D_family_.285-simplex.29|uniform polytera]] based on the [3,3,3,3] [[Coxeter group]], all shown here in A5 [[Coxeter plane]] [[orthographic projection]]s. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

{{Hexateron family}}

==Notes==
{{reflist}}

== References ==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
** '''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]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

== External links ==
* {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:5-polytopes]]

Uniformly distributed measure - Revision history

en>Novn: Added link

en>Sullivan.t.j: ←Created page with 'In mathematics — specifically, in geometric measure theory — a '''uniformly distributed measure''' on a metric space is one for which the me…'