6-polytope

{{Short description|6-dimensional geometric object}} {| class="wikitable skin-invert-image" style="float:right; margin-left:10px; width:500px" |+ [[Petrie polygon|Graphs]] of three [[List of regular polytopes#Dimension 5 and higher|regular]] and five [[uniform 6-polytope]]s |- align=center valign=top |[[Image:6-simplex t0.svg|120px]]<BR>[[6-simplex]] |[[File:6-cube t5.svg|120px]]<BR>[[6-orthoplex]], '''3₁₁''' |[[Image:6-cube t0.svg|120px]]<BR>[[6-cube]] (Hexeract) |[[File:Up 2 21 t0 E6.svg|120px]]<BR>[[Gosset 2 21 polytope|2₂₁]] |- align=center valign=top |[[Image:6-simplex t05.svg|120px]]<BR>[[Expanded 6-simplex]] |[[File:6-cube t4.svg|120px]]<BR>[[Rectified 6-orthoplex]] |[[Image:6-demicube t0 D6.svg|120px]]<BR>[[6-demicube]] '''1₃₁'''<BR>(Demihexeract) |[[File:Up 1 22 t0 E6.svg|120px]]<BR>[[Gosset 1 22 polytope|1₂₂]] |} In [[Six-dimensional space|six-dimensional]] [[geometry]], a '''six-dimensional polytope''' or '''6-polytope''' is a [[polytope]], bounded by [[5-polytope]] [[Facet (mathematics)|facets]]. {{-}}

== Definition == A 6-polytope is a closed six-dimensional figure with [[Vertex (geometry)|vertices]], [[Edge (geometry)|edges]], [[face (geometry)|faces]], [[cell (mathematics)|cells]] (3-faces), 4-faces, and 5-faces. A vertex is a [[Point (geometry)|point]] where six or more edges meet. An edge is a [[line segment]] where four or more faces meet, and a face is a [[polygon]] where three or more cells meet. A cell is a [[polyhedron]]. A 4-face is a [[polychoron]], and a 5-face is a [[5-polytope]]. Furthermore, the following requirements must be met: * Each 4-face must join exactly two 5-faces (facets). * Adjacent facets are not in the same five-dimensional [[hyperplane]]. * The figure is not a compound of other figures which meet the requirements.

== Characteristics == The topology of any given 6-polytope is defined by its [[Betti number]]s and [[torsion coefficient (topology)|torsion coefficient]]s.<ref name="richeson">Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008.</ref>

The value of the [[Euler characteristic]] used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 6-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.<ref name="richeson"/>

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.<ref name="richeson"/>

== Classification == 6-polytopes may be classified by properties like "[[convex set|convexity]]" and "[[symmetry]]".

*A 6-polytope is ''[[Convex polytope|convex]]'' if its boundary (including its 5-faces, 4-faces, cells, faces and edges) does not intersect itself and the line segment joining any two points of the 6-polytope is contained in the 6-polytope or its interior; otherwise, it is ''non-convex''. Self-intersecting 6-polytope are also known as [[Star polytope|''star 6-polytopes'']], from analogy with the star-like shapes of the non-convex [[Kepler-Poinsot polyhedra]]. *A '''regular 6-polytope''' has all identical regular 5-polytope facets. All regular 6-polytope are convex.

{{Main|List of regular polytopes#Dimension 5 and higher}}

*A '''[[semiregular polytope|semi-regular]] 6-polytope''' contains two or more types of [[regular 4-polytope]] facets. There is only one such figure, called [[2 21 polytope|2₂₁]]. *A '''uniform 6-polytope''' has a [[symmetry group]] under which all vertices are equivalent, and its facets are [[uniform 5-polytope]]s. The faces of a uniform polytope must be [[regular polygon|regular]].

{{Main|Uniform 6-polytope}}

*A '''prismatic 6-polytope''' is constructed by the [[Cartesian product]] of two lower-dimensional polytopes. A prismatic 6-polytope is uniform if its factors are uniform. The [[6-cube]] is prismatic (product of a [[square (geometry)|square]]s and a [[cube]]), but is considered separately because it has symmetries other than those inherited from its factors. *A ''5-space [[tessellation]]'' is the division of five-dimensional [[Euclidean space]] into a regular grid of 5-polytope facets. Strictly speaking, tessellations are not 6-polytopes as they do not bound a "6D" volume, but we include them here for the sake of completeness because they are similar in many ways to 6-polytope. A ''uniform 5-space tessellation'' is one whose vertices are related by a [[space group]] and whose facets are [[uniform 5-polytope]]s.

== Regular 6-polytopes == Regular 6-polytopes can be generated from [[Coxeter group]]s represented by the [[Schläfli symbol]] {p,q,r,s,t} with '''t''' {p,q,r,s} 5-polytope [[Facet (mathematics)|facets]] around each [[Cell (geometry)|cell]].

There are only three such [[List of regular polytopes#Convex 5|convex regular 6-polytopes]]: * {3,3,3,3,3} - [[6-simplex]] * {4,3,3,3,3} - [[6-cube]] * {3,3,3,3,4} - [[6-orthoplex]]

There are no nonconvex regular polytopes of 5 or more dimensions.

For the three convex regular 6-polytopes, their elements are:

{| class=wikitable !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!![[Coxeter-Dynkin diagram|Coxeter<BR>diagram]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!![[Coxeter group|Symmetry]] ([[Symmetry order|order]]) |- align=center |[[6-simplex]]||{3,3,3,3,3}||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}||7||21||35||35||21||7||A₆ (720) |- align=center |[[6-orthoplex]]||{3,3,3,3,4}||{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}||12||60||160||240||192||64||[[Hyperoctahedral group|B₆]] (46080) |- align=center |[[6-cube]]||{4,3,3,3,3}||{{nowrap|{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}}}||64||192||240||160||60||12||B₆ (46080) |}

== Uniform 6-polytopes == {{Main|Uniform 6-polytope}} Here are six simple uniform convex 6-polytopes, including the ''6-orthoplex'' repeated with its alternate construction.

{| class=wikitable !Name!![[Schläfli symbol|Schläfli<BR>symbol(s)]]!![[Coxeter-Dynkin diagram|Coxeter<BR>diagram(s)]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!![[Coxeter group|Symmetry]] ([[Symmetry order|order]]) |- align=center |[[Expanded 6-simplex]]||t_0,5{3,3,3,3,3}||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1}}||42||210||490||630||434||126||2×A₆ (1440) |- align=center |[[6-orthoplex]], 3₁₁<BR>(alternate construction)||{3,3,3,3^1,1}||{{CDD|nodes|split2|node|3|node|3|node|3|node_1}}||12||60||160||240||192||64||D₆ (23040) |- align=center |[[6-demicube]]||{3,3^3,1}<BR>h{4,3,3,3,3}||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}}<BR>{{CDD|node_h|4|node|3|node|3|node|3|node|3|node}}||32||240||640||640||252||44||D₆ (23040)<BR>½B₆ |- align=center |[[Rectified 6-orthoplex]]||t₁{3,3,3,3,4}<BR>t₁{3,3,3,3^1,1}||{{CDD|node|4|node|3|node|3|node|3|node_1|3|node}}<BR>{{CDD|nodes|split2|node|3|node|3|node_1|3|node}}||60||480||1120||1200||576||76||B₆ (46080)<BR>2×D₆ |- align=center |[[2 21 polytope|2₂₁ polytope]]||{3,3,3^2,1}||{{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||27||216||720||1080||648||99||E₆ (51840) |- align=center |[[1 22 polytope|1₂₂ polytope]]||{3,3^2,2}||{{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}<BR> or {{CDD|node_1|3|node|split1|nodes|3ab|nodes}}||72||720||2160||2160||702||54||2×E₆ (103680) |}

The ''expanded 6-simplex'' is the [[vertex figure]] of the uniform [[6-simplex honeycomb]], {{CDD|node_1|split1|nodes|3ab|nodes|3ab|branch}}. The [[6-demicube honeycomb]], {{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}}, vertex figure is a ''rectified 6-orthoplex'' and [[Facet (geometry)|facets]] are the ''6-orthoplex'' and ''6-demicube''. The uniform [[2 22 honeycomb|2₂₂ honeycomb]],{{CDD|node_1|3|node|3|node|split1|nodes|3ab|nodes}}, has ''1₂₂'' polytope is the vertex figure and ''2₂₁'' facets.

== References == {{reflist}} * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900 * {{cite journal|year=1910|author=A. Boole Stott|authorlink=Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|journal=Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam.|volume=XI|number=1|publisher=Johannes Müller|location=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-url=https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf|archive-date=29 April 2025}} * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: ** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, 1954 ** 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 (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta)}}

== External links == * [http://www.steelpillow.com/polyhedra/ditela.html Polytope names] * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] * {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}} {{Polytopes}}

[[Category:6-polytopes| ]]