6-demicube

{{Short description|Uniform 6-polytope}} {| class="wikitable" style="float:right; margin-left:10px; width:290px" !style="background:#e7dcc3" colspan=3|Demihexeract<BR>(6-demicube) |- |style="background:#fff; text-align:center" colspan=3|280px<BR>Petrie polygon projection |- |style="background:#e7dcc3"|Type |colspan=2|Uniform 6-polytope |- |style="background:#e7dcc3"|Family |colspan=2|demihypercube |- |style="background:#e7dcc3"|Schläfli symbol |colspan=2|{3,3^3,1} = h{4,3⁴}<BR>s{2^1,1,1,1,1} |- |style="background:#e7dcc3"|Coxeter diagrams |colspan=2|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node}}<BR>{{CDD|nodes_01r|3ab|nodes|split5c|nodes}} = {{CDD|nodes|3ab|nodes|split5c|nodes_10l}} {{CDD|node_h|2x|node_h|4|node|3|node|3|node|3|node|}}<BR> {{CDD|node_h|2x|node_h|2x|node_h|4|node|3|node|3|node|}}<BR> {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|4|node|3|node|}}<BR> {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h|4|node|}}<BR> {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h|2x|node_h}} |- |style="background:#e7dcc3"|Coxeter symbol |colspan=2|1₃₁ |- |style="background:#e7dcc3"|5-faces||44||12&nbsp;{3^1,2,1} 25px|class=skin-invert<BR>32 {3⁴} 25px|class=skin-invert |- |style="background:#e7dcc3"|4-faces||252||60 {3^1,1,1} 25px|class=skin-invert<BR>192 {3³} 25px|class=skin-invert |- |style="background:#e7dcc3"|Cells||640||{{nowrap|160 {3^1,0,1} 25px|class=skin-invert}}<BR>480 {3,3} 25px|class=skin-invert |- |style="background:#e7dcc3"|Faces||640||{3} 25px|class=skin-invert |- |style="background:#e7dcc3"|Edges||colspan=2|240 |- |style="background:#e7dcc3"|Vertices||colspan=2|32 |- |style="background:#e7dcc3"|Vertex figure |colspan=2|Rectified 5-simplex<BR>40px|class=skin-invert |- |style="background:#e7dcc3"|Symmetry group |colspan=2|D₆, [3^3,1,1] = [1⁺,4,3⁴]<BR>[2⁵]⁺ |- |style="background:#e7dcc3"|Petrie polygon |colspan=2|decagon |- |style="background:#e7dcc3"|Properties |colspan=2|convex |} In geometry, a '''6-demicube''', '''demihexeract''' or '''hemihexeract''' is a uniform 6-polytope, constructed from a ''6-cube'' (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. Acronym: '''hax'''.{{r|Klitzing}}

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM₆ for a 6-dimensional ''half measure'' polytope.

Coxeter named this polytope as '''1₃₁''' from its Coxeter diagram, with a ring on one of the 1-length branches, {{CDD|node_1|3|node|split1|nodes|3a|nodea|3a|nodea}}. It can named similarly by a 3-dimensional exponential Schläfli symbol <math>\left\{3 \begin{array}{l}3, 3, 3\\3\end{array}\right\}</math> or {3,3^3,1}.

== Cartesian coordinates == Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract: : (±1,±1,±1,±1,±1,±1) with an odd number of plus signs.

== As a configuration == This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.<ref>Coxeter, Regular Polytopes, p. 12, Section 1.8 Configurations</ref>{{sfnp|Coxeter|1991|p=117}}

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.<ref name= Klitzing >{{KlitzingPolytopes|../incmats/hax.htm|x3o3o *b3o3o3o - hax}}</ref>

{| class=wikitable style="width: 890px" !D₆||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|3a|nodea}}||''k''-face ||f_''k''||f₀||f₁||f₂||colspan=2|f₃||colspan=2|f₄||colspan=2|f₅|| ''k''-figure|| Notes |- align=right |A₅ ||{{CDD|nodea_x|2|branch|3a|nodea|3a|nodea|3a|nodea}}|| ( ) !f₀ |BGCOLOR="#ffe0ff"|'''32'''||15||60||20||60||15||30||6||6||r{3,3,3,3} || D₆/A₅ = 32·6!/6! = 32 |- align=right |A₃A₁A₁ ||{{nowrap|{{CDD|nodea_1|2|nodes_x0|2|nodea|3a|nodea|3a|nodea}}}} || { } !f₁ ||2||BGCOLOR="#ffe0e0"|'''240'''||8||4||12||6||8||4||2||{}x{3,3} || {{nowrap|D₆/A₃A₁A₁ {{=}} 32·6!/4!/2/2 {{=}} 240}} |- align=right |A₃A₂ ||{{CDD|nodea_1|3a|nodes_0x|2|nodea_x|2|nodea|3a|nodea}}|| {3} !f₂ ||3||3||BGCOLOR="#ffffe0"|'''640'''||1||3||3||3||3||1||{3}v( ) || D₆/A₃A₂ = 32·6!/4!/3! = 640 |- align=right |A₃A₁ ||{{CDD|nodea_1|3a|branch|2|nodea_x|2|nodea|3a|nodea}}|| h{4,3} !rowspan=2|f₃ ||4||6||4||BGCOLOR="e0ffe0"|'''160'''||BGCOLOR="e0ffe0"|*||3||0||3||0||{3} || D₆/A₃A₁ = 32·6!/4!/2 = 160 |- align=right |A₃A₂ ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|2|nodea_x|2|nodea}}|| {3,3} ||4||6||4||BGCOLOR="e0ffe0"|*||BGCOLOR="e0ffe0"|'''480'''||1||2||2||1||{}v( ) || D₆/A₃A₂ = 32·6!/4!/3! = 480 |- align=right |D₄A₁ ||{{CDD|nodea_1|3a|branch|3a|nodea|2|nodea_x|2|nodea}}|| h{4,3,3} !rowspan=2|f₄ ||8||24||32||8||8||BGCOLOR="e0ffff"|'''60'''||BGCOLOR="e0ffff"|*||2||0||rowspan=2|{ } || D₆/D₄A₁ = 32·6!/8/4!/2 = 60 |- align=right |A₄ ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea|2|nodea_x}}|| {3,3,3} ||5||10||10||0||5||BGCOLOR="e0ffff"|*||BGCOLOR="e0ffff"|'''192'''||1||1|| D₆/A₄ = 32·6!/5! = 192 |- align=right |D₅ ||{{CDD|nodea_1|3a|branch|3a|nodea|3a|nodea|2|nodea_x}}|| h{4,3,3,3} !rowspan=2|f₅ ||16||80||160||40||80||10||16||BGCOLOR="e0e0ff"|'''12'''||BGCOLOR="e0e0ff"|*||rowspan=2|( ) || D₆/D₅ = 32·6!/16/5! = 12 |- align=right |A₅ ||{{CDD|nodea_1|3a|nodes_0x|3a|nodea|3a|nodea|3a|nodea}}|| {3,3,3,3} || 6||15||20||0||15||0||6||BGCOLOR="e0e0ff"|*||BGCOLOR="e0e0ff"|'''32'''|| D₆/A₅ = 32·6!/6! = 32 |}

== Images == {{6-demicube Coxeter plane graphs|t0|120}}

== Related polytopes == There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique: {{Demihexeract_family}}

The 6-demicube, 1₃₁ is third in a dimensional series of uniform polytopes, expressed by Coxeter as ''k''₃₁ series. The fifth figure is a Euclidean honeycomb, 3₃₁, and the final is a noncompact hyperbolic honeycomb, 4₃₁. Each progressive uniform polytope is constructed from the previous as its vertex figure. {{k 31 polytopes}}

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3''k'' series. The fourth figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄. {{1 3k polytopes}}

=== Skew icosahedron === Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the '''regular skew icosahedron'''.<ref>{{cite book |last1=Coxeter |first1=H. S. M. |title=The beauty of geometry : twelve essays |publisher=Dover Publications |isbn=9780486409191 |pages=450–451}}</ref><ref>{{cite journal |last1=Deza |first1=Michael |last2=Shtogrin |first2=Mikhael |title=Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices |journal=Advanced Studies in Pure Mathematics |series=Arrangements – Tokyo 1998 |publisher=Mathematical Society of Japan |date=2000 |volume=27 |pages=73–92[77] |doi=10.2969/aspm/02710073 |url=https://projecteuclid.org/euclid.aspm/1534788966 |doi-access=free |isbn=978-4-931469-77-8}}</ref>

== References == <references /> * H.S.M. Coxeter: ** H.S.M. Coxeter, ''Regular Polytopes'', 1973, 3rd edition, Dover, New York, p. 12, Section 1.8 Configurations, {{isbn|0-486-61480-8}} ** {{cite book | last=Coxeter | first=H.S.M. | title=Regular Complex Polytopes | publisher=Cambridge University Press | year=1991 | orig-year=1974 | isbn=0-521-39490-2 }} ** '''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] * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, Chapter 26, p. 409, Hemicubes: 1_''n''1, {{isbn|978-1-56881-220-5}} * {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms|x3o3o *b3o3o3o – hax}}

== External links == * {{GlossaryForHyperspace | anchor=half | title=Demihexeract }} * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:6-polytopes