Stericated 5-simplexes

{|class="wikitable skin-invert-image" style="float:right; margin-left:8px; width:520px" |- align=center valign=top |120px 120px 5-simplex {{CDD|node_1|3|node|3|node|3|node|3|node}} |120px 120px '''Stericated 5-simplex''' {{CDD|node_1|3|node|3|node|3|node|3|node_1}} |- align=center valign=top |120px 120px '''Steritruncated 5-simplex''' {{CDD|node_1|3|node_1|3|node|3|node|3|node_1}} |120px 120px '''Stericantellated 5-simplex''' {{CDD|node_1|3|node|3|node_1|3|node|3|node_1}} |- align=center valign=top |120px 120px '''Stericantitruncated 5-simplex''' {{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1}} |120px 120px '''Steriruncitruncated 5-simplex''' {{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1}} |- align=center valign=top |colspan=2|180px 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 projections in A5 and A4 Coxeter planes |}

In five-dimensional geometry, a '''stericated 5-simplex''' is a convex uniform 5-polytope with fourth-order 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 operation applied to the regular 5-simplex. The highest form, the ''steriruncicantitruncated 5-simplex'' is more simply called an omnitruncated 5-simplex with all of the nodes ringed. {{clear}}

== Stericated 5-simplex == {|class="wikitable" style="float:right; margin-left:8px; width:280px" |- |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} 2r{32,2} = <math>2r\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\}</math> |- |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 {3,3,3} 25px 15+15 {}×{3,3} 25px 20 {3}×{3} 25px |- |bgcolor=#e7dcc3|Cells |180 |60 {3,3} 25px 120 {}×{3} 25px |- |bgcolor=#e7dcc3|Faces |210 |120 {3} 90 {4} |- |bgcolor=#e7dcc3|Edges |colspan=2|120 |- |bgcolor=#e7dcc3|Vertices |colspan=2|30 |- |bgcolor=#e7dcc3|Vertex figure |colspan=2|80px Tetrahedral antiprism |- |bgcolor=#e7dcc3|Coxeter group |colspan=2|A5×2, {{brackets|3,3,3,3}}, order 1440 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal, isotoxal |}

A '''stericated 5-simplex''' can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an '''expanded 5-simplex'''. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).

=== Alternate names === * Expanded 5-simplex * Stericated hexateron * Small cellated dodecateron (Acronym: scad) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/scad.htm (x3o3o3o3x - scad)]}}

=== Cross-sections === The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 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 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|120}}

{|class="wikitable skin-invert" |- align=center |160px Orthogonal projection with [6] symmetry |}

== Steritruncated 5-simplex == {|class="wikitable" style="float:right; margin-left:8px; width:280px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Steritruncated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2|t0,1,4{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 t{3,3,3} 15 {}×t{3,3} 20 {3}×{6} 15 {}×{3,3} 6 t0,3{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|100px |- |bgcolor=#e7dcc3|Coxeter group |colspan=2|A5 [3,3,3,3], order 720 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal |}

=== Alternate names === * Steritruncated hexateron * Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/cappix.htm (x3x3o3o3x - cappix)]}}

=== 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 facets of the steritruncated 6-orthoplex.

=== Images === {{5-simplex Coxeter plane graphs|t014|120}}

== Stericantellated 5-simplex == {| class="wikitable" style="float:right; margin-left:8px; width:280px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Stericantellated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2|t0,2,4{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 rr{3,3,3} 30 rr{3,3}x{} 20 {3}×{3} |- |bgcolor=#e7dcc3|Cells |420 |60 rr{3,3} 240 {}×{3} 90 {}×{}×{} 30 r{3,3} |- |bgcolor=#e7dcc3|Faces |900 |360 {3} 540 {4} |- |bgcolor=#e7dcc3|Edges |colspan=2|720 |- |bgcolor=#e7dcc3|Vertices |colspan=2|180 |- |bgcolor=#e7dcc3|Vertex figure |colspan=2|100px |- |bgcolor=#e7dcc3|Coxeter group |colspan=2|A5×2, {{brackets|3,3,3,3}}, order 1440 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal |}

=== Alternate names === * Stericantellated hexateron * Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/card.htm (x3o3x3o3x - card)]}}

=== 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 facets of the stericantellated 6-orthoplex.

=== Images === {{5-simplex2 Coxeter plane graphs|t024|120}}

== Stericantitruncated 5-simplex == {|class="wikitable" style="float:right; margin-left:8px; width:280px" |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|100px |- |bgcolor=#e7dcc3|Coxeter group |colspan=2|A5 [3,3,3,3], order 720 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal |}

=== Alternate names === * Stericantitruncated hexateron * Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/cograx.htm (x3x3x3o3x - cograx)]}}

=== 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 facets of the stericantitruncated 6-orthoplex.

=== Images === {{5-simplex Coxeter plane graphs|t0124|120}}

== Steriruncitruncated 5-simplex == {|class="wikitable" style="float:right; margin-left:8px; width:280px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Steriruncitruncated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2|t0,1,3,4{3,3,3,3} 2t{32,2} |- |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 t0,1,3{3,3,3} 30 {}×t{3,3} 20 {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|100px |- |bgcolor=#e7dcc3|Coxeter group |colspan=2|A5×2, {{brackets|3,3,3,3}}, order 1440 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal |}

=== Alternate names === * Steriruncitruncated hexateron * Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/captid.htm (x3x3o3x3x - captid)]}}

=== 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 facets of the steriruncitruncated 6-orthoplex.

=== Images === {{5-simplex2 Coxeter plane graphs|t0134|120}}

== Omnitruncated 5-simplex == {|class="wikitable" style="float:right; margin-left:8px; width:280px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Omnitruncated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2|t0,1,2,3,4{3,3,3,3} 2tr{32,2} |- |bgcolor=#e7dcc3|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 t0,1,2,3{3,3,3} 25px 30 {}×tr{3,3} 25px 20 {6}×{6} 25px |- |bgcolor=#e7dcc3|Cells |540 |360 t{3,4} 25px 90 {4,3} 25px 90 {}×{6} 25px |- |bgcolor=#e7dcc3|Faces |1560 |480 {6} 1080 {4} |- |bgcolor=#e7dcc3|Edges |colspan=2|1800 |- |bgcolor=#e7dcc3|Vertices |colspan=2|720 |- |bgcolor=#e7dcc3|Vertex figure |colspan=2|80px Irregular 5-cell |- |bgcolor=#e7dcc3|Coxeter group |colspan=2| A5×2, {{brackets|3,3,3,3}}, order 1440 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex, isogonal, zonotope |} The '''omnitruncated 5-simplex''' has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedra, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

=== Alternate names === * Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson) * Omnitruncated hexateron * Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gocad.htm (x3x3x3x3x - gocad)]}}

=== Coordinates === The vertices of the ''omnitruncated 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 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 === {{5-simplex2 Coxeter plane graphs|t01234|120}}

[[Image:Omnitruncated Hexateron.png|thumb|Stereographic projection]]

=== 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 skin-invert" |480px Orthogonal projection, vertices labeled as a permutohedron. |}

=== Related honeycomb === The omnitruncated 5-simplex honeycomb is constructed by '''omnitruncated 5-simplex''' facets with 3 facets around each 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 |{{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 |100px |100px |100px | | |- !Name |Apeirogon |Hextille |Omnitruncated 3-simplex honeycomb |Omnitruncated 4-simplex honeycomb |Omnitruncated 5-simplex honeycomb |- !Facets |100px|class=skin-invert |100px|class=skin-invert |100px|class=skin-invert |100px|class=skin-invert |100px|class=skin-invert |}

=== Full snub 5-simplex === The '''full snub 5-simplex''' or '''omnisnub 5-simplex''', defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram {{CDD|node_h|3|node_h|3|node_h|3|node_h|3|node_h}} and symmetry {{brackets|3,3,3,3}}+, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular 5-cells filling the gaps at the deleted vertices.

== Related uniform polytopes == These polytopes are a part of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple, magenta having progressively more vertices)

== Notes == {{reflist}}

== References == * H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd edition, Dover, New York, 1973 ** Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}} *** (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 ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. * {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera) with acronyms}} x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad {{sfn whitelist| CITEREFKlitzing}}

== 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]

Category:5-polytopes