Truncated 5-simplexes

{| class="wikitable skin-invert-image" style="float:right; margin-left:1em; width:460px" |- align=center |150px 5-simplex {{CDD|node_1|3|node|3|node|3|node|3|node}} |150px Truncated 5-simplex {{CDD|node_1|3|node_1|3|node|3|node|3|node}} |150px Bitruncated 5-simplex {{CDD|node|3|node_1|3|node_1|3|node|3|node}} |- !colspan=3|Orthogonal projections in A5 Coxeter plane |} In five-dimensional geometry, a '''truncated 5-simplex''' is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

== Truncated 5-simplex == {| class="wikitable" style="float:right; margin-left:10px; width:250px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Truncated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2| t{3,3,3,3} |- |bgcolor=#e7dcc3|Coxeter-Dynkin diagram |colspan=2|{{CDD||node_1|3|node_1|3|node||3|node|3|node}} {{CDD||branch_11|3b|nodeb|3b|nodeb|3b|nodeb}} |- |bgcolor=#e7dcc3|4-faces |12 |6 {3,3,3}25px 6 t{3,3,3}25px |- |bgcolor=#e7dcc3|Cells |45 |30 {3,3}25px 15 t{3,3}25px |- |bgcolor=#e7dcc3|Faces |80 |60 {3} 20 {6} |- |bgcolor=#e7dcc3|Edges |colspan=2|75 |- |bgcolor=#e7dcc3|Vertices |colspan=2|30 |- |bgcolor=#e7dcc3|Vertex figure |colspan=2|100px ( )v{3,3} |- |bgcolor=#e7dcc3|Coxeter group |colspan=2| A5 [3,3,3,3], order 720 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex |} The '''truncated 5-simplex''' has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

=== Alternate names === * Truncated hexateron (Acronym: tix) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/tix.htm (x3x3o3o3o - tix)]}}

=== Coordinates === The vertices of the ''truncated 5-simplex'' can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) ''or'' of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.

=== Images === {{5-simplex Coxeter plane graphs|t01|120}}

== Bitruncated 5-simplex == {| class="wikitable" style="float:right; margin-left:10px; width:250px" |- |bgcolor=#e7dcc3 align=center colspan=3|'''Bitruncated 5-simplex''' |- |bgcolor=#e7dcc3|Type |colspan=2|Uniform 5-polytope |- |bgcolor=#e7dcc3|Schläfli symbol |colspan=2| 2t{3,3,3,3} |- |bgcolor=#e7dcc3|Coxeter-Dynkin diagram |colspan=2|{{CDD||node|3|node_1|3|node_1|3|node|3|node}} {{CDD||branch_11|3ab|nodes|3b|nodeb}} |- |bgcolor=#e7dcc3|4-faces |12 |6 2t{3,3,3}25px 6 t{3,3,3}25px |- |bgcolor=#e7dcc3|Cells |60 |45 {3,3}25px 15 t{3,3}25px |- |bgcolor=#e7dcc3|Faces |140 |80 {3}25px 60 {6}25px |- |bgcolor=#e7dcc3|Edges |colspan=2|150 |- |bgcolor=#e7dcc3|Vertices |colspan=2|60 |- |bgcolor=#e7dcc3|Vertex figure |colspan=2|100px { }v{3} |- |bgcolor=#e7dcc3|Coxeter group |colspan=2| A5 [3,3,3,3], order 720 |- |bgcolor=#e7dcc3|Properties |colspan=2|convex |}

=== Alternate names === * Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/bittix.htm (o3x3x3o3o - bittix)]}}

=== Coordinates === The vertices of the ''bitruncated 5-simplex'' can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) ''or'' of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.

=== Images === {{5-simplex Coxeter plane graphs|t12|120}}

== Related uniform 5-polytopes == The truncated 5-simplex is one 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.) {{Hexateron family}}

== 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}} x3x3o3o3o - tix, o3x3x3o3o - bittix {{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], Jonathan Bowers ** [http://www.polytope.net/hedrondude/truncates5.htm Truncated uniform polytera] (tix), Jonathan Bowers * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

Category:5-polytopes