B4 polytope

{{DISPLAYTITLE:B₄ polytope}} {{No footnotes|date=March 2026}} {| class=wikitable style="float:right; margin-left:10px; width:360px" |+ Orthographic projections in the B₄ Coxeter plane |- align=center |120px|class=skin-invert<BR>Tesseract<BR>{{CDD|node_1|4|node|3|node|3|node}} |120px|class=skin-invert<BR>16-cell<BR>{{CDD|node_1|3|node|3|node|4|node}} |} In 4-dimensional geometry, there are 15 uniform 4-polytopes with B₄ symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively.

== Visualizations == They can be visualized as symmetric orthographic projections in Coxeter planes of the B₅ Coxeter group, and other subgroups.

Symmetric orthographic projections of these 32 polytopes can be made in the B₅, B₄, B₃, B₂, A₃, Coxeter planes. A_k has ''[k+1]'' symmetry, and B_k has ''[2k]'' symmetry.

These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

{| class="wikitable" !rowspan=2|# !rowspan=2|Name<!--<br>(Bowers style acronym)--> !colspan=4| Coxeter plane projections !colspan=2| Schlegel<br>diagrams !rowspan=2|Net |- !B₄<br>[8] !B₃<br>[6] !B₂<br>[4] !A₃<br>[4] !Cube<br>centered !Tetrahedron<br>centered |- BGCOLOR="#f0e0e0" !1 ||8-cell or tesseract<!-- (tes)--><BR>{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node}}}} = {4,3,3} |80px |80px |80px |80px |80px | |70px |- BGCOLOR="#f0e0e0" !2 ||rectified 8-cell<!--] (rit)--><BR>{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node}}}} = r{4,3,3} |80px |80px |80px |80px |70px | |70px |- BGCOLOR="#e0e0f0" !3 ||16-cell<!-- (hex)--><BR>{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1}}}} = {3,3,4} |80px |80px |80px |80px | |80px |70px |- BGCOLOR="#f0e0e0" !4 ||truncated 8-cell<!-- (tat)--><BR>{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node}}}} = t{4,3,3} |80px |80px |80px |80px |80px | |70px |- BGCOLOR="#f0e0e0" !5 ||cantellated 8-cell<!-- (srit)--><BR>{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node}}}} = rr{4,3,3} |80px |80px |80px |80px |80px | |70px |- BGCOLOR="#e0f0e0" !6 ||''runcinated 8-cell''<br>(also ''runcinated 16-cell'')<!-- (sidpith)--><BR>{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1}}}} = t03{4,3,3} |80px |80px |80px |80px |80px |80px |70px |- BGCOLOR="#e0f0e0" !7 ||''bitruncated 8-cell''<br>(also ''bitruncated 16-cell'')<!-- (tah)--><BR>{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node}}}} = 2t{4,3,3} |80px |80px |80px |80px | 80px |80px |70px |- BGCOLOR="#e0e0f0" !8 ||truncated 16-cell<!-- (thex)--><BR>{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1}}}} = t{3,3,4} |80px |80px |80px |80px | |80px |70px |- BGCOLOR="#f0e0e0" !9 ||cantitruncated 8-cell<!-- (grit)--><BR>{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1}}}} = tr{3,3,4} |80px |80px |80px |80px | 80px | |70px |- BGCOLOR="#f0e0e0" !10 ||runcitruncated 8-cell<!-- (proh)--><BR>{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1}}}} = t013{4,3,3} |80px |80px |80px |80px | 80px | |70px |- BGCOLOR="#e0e0f0" !11 ||runcitruncated 16-cell<!-- (prit)--><BR>{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1}}}} = t013{3,3,4} |80px |80px |80px |80px | | 80px |70px |- BGCOLOR="#e0f0e0" !12 ||''omnitruncated 8-cell''<br>(also ''omnitruncated 16-cell'')<!-- (gidpith)--><BR>{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}}} = t0123{4,3,3} |80px |80px |80px |80px |80px |80px |70px |}

{| class="wikitable" !rowspan=2|# !rowspan=2|Name<!--<br>(Bowers style acronym)--> !colspan=5| Coxeter plane projections !colspan=2| Schlegel<br>diagrams !rowspan=2|Net |- !F₄<br>[12] !B₄<br>[8] !B₃<br>[6] !B₂<br>[4] !A₃<br>[4] !Cube<br>centered !Tetrahedron<br>centered

|- BGCOLOR="#e0e0f0" !13 |*rectified 16-cell<br>(Same as ''24-cell'')<!-- (ico)--><BR>{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node}}}} = {{dark mode invert|{{CDD|node_1|3|node|4|node|3|node}}}}<BR>r{3,3,4} = {3,4,3} |80px |80px |80px |80px |80px | |80px |70px |- BGCOLOR="#e0e0f0" !14 ||*cantellated 16-cell<br>(Same as ''rectified 24-cell'')<!-- (rico)--><BR>{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1}}}} = {{dark mode invert|{{CDD|node|3|node_1|4|node|3|node}}}}<BR>rr{3,3,4} = r{3,4,3} |80px |80px |80px |80px |80px | | 80px | 70px |- BGCOLOR="#e0e0f0" !15 ||*cantitruncated 16-cell<br>(Same as ''truncated 24-cell'')<!-- (tico)--><BR>{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1}}}} = {{dark mode invert|{{CDD|node_1|3|node_1|4|node|3|node}}}}<BR>tr{3,3,4} = t{3,4,3} |80px |80px |80px |80px |80px | |80px |70px |}

{| class="wikitable" !rowspan=2|# !rowspan=2|Name<!--<br>(Bowers style acronym)--> !colspan=5| Coxeter plane projections !colspan=2| Schlegel<br>diagrams !rowspan=2|Net |- !F₄<br>[12] !B₄<br>[8] !B₃<br>[6] !B₂<br>[4] !A₃<br>[4] !Cube<br>centered !Tetrahedron<br>centered |- BGCOLOR="#d0f0f0" !16 ||''alternated cantitruncated 16-cell''<br>(Same as the snub 24-cell)<!-- (sadi)--><BR>{{dark mode invert|{{CDD|node|4|node_h|3|node_h|3|node_h}}}} = {{dark mode invert|{{CDD|node_h|3|node_h|4|node|3|node}}}}<BR>sr{3,3,4} = s{3,4,3} |80px |80px |80px |80px | | |80px |70px |}

== Coordinates == The tesseractic family of 4-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 4-polytopes. All coordinates correspond with uniform 4-polytopes of edge length 2.

{|class="wikitable" |+Coordinates for uniform 4-polytopes in Tesseract/16-cell family |- !# !Base point ! Name<!--<br>Bowers Name (Bowers style acronym)--> !Coxeter diagram !colspan=2|Vertices

|- BGCOLOR="#f0e0e0" !3 |(0,0,0,1){{radic|2}} |16-cell |{{dark mode invert|{{CDD|node|4|node|3|node|3|node_1}}}} |8||2^4-34!/3!

|- BGCOLOR="#e0e0f0" !1 |(1,1,1,1) |Tesseract |{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node}}}} |16||2⁴4!/4!

|- BGCOLOR="#f0e0e0" !13 |(0,0,1,1){{radic|2}} |Rectified 16-cell (24-cell) |{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node}}}} |24||2^4-24!/(2!2!)

|- BGCOLOR="#e0e0f0" !2 |(0,1,1,1){{radic|2}} |Rectified tesseract |{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node}}}} |32||2⁴4!/(3!2!)

|- BGCOLOR="#f0e0e0" !8 |(0,0,1,2){{radic|2}} |Truncated 16-cell |{{dark mode invert|{{CDD|node|4|node|3|node_1|3|node_1}}}} |48||2^4-24!/2!

|- BGCOLOR="#e0f0e0" !6 |(1,1,1,1) + (0,0,0,1){{radic|2}} |Runcinated tesseract |{{dark mode invert|{{CDD|node_1|4|node|3|node|3|node_1}}}} |64||2⁴4!/3! |- BGCOLOR="#e0e0f0" !4 |(1,1,1,1) + (0,1,1,1){{radic|2}} |Truncated tesseract |{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node}}}} |64||2⁴4!/3!

|-BGCOLOR="#f0e0e0" !14 |(0,1,1,2){{radic|2}} |Cantellated 16-cell (rectified 24-cell) |{{dark mode invert|{{CDD|node|4|node_1|3|node|3|node_1}}}} |96||2⁴4!/(2!2!) |- BGCOLOR="#e0f0e0" !7 |(0,1,2,2){{radic|2}} |Bitruncated 16-cell |{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node}}}} |96||2⁴4!/(2!2!) |- BGCOLOR="#e0e0f0" !5 |(1,1,1,1) + (0,0,1,1){{radic|2}} |Cantellated tesseract |{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node}}}} |96||2⁴4!/(2!2!)

|- BGCOLOR="#f0e0e0" !15 |(0,1,2,3){{radic|2}} |cantitruncated 16-cell (truncated 24-cell) |{{dark mode invert|{{CDD|node|4|node_1|3|node_1|3|node_1}}}} |192||2⁴4!/2! |- BGCOLOR="#f0e0e0" !11 |(1,1,1,1) + (0,0,1,2){{radic|2}} |Runcitruncated 16-cell |{{dark mode invert|{{CDD|node_1|4|node|3|node_1|3|node_1}}}} |192||2⁴4!/2! |- BGCOLOR="#e0e0f0" !10 |(1,1,1,1) + (0,1,1,2){{radic|2}} |Runcitruncated tesseract |{{dark mode invert|{{CDD|node_1|4|node_1|3|node|3|node_1}}}} |192||2⁴4!/2! |- BGCOLOR="#e0e0f0" !9 |(1,1,1,1) + (0,1,2,2){{radic|2}} |Cantitruncated tesseract |{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node}}}} |192||2⁴4!/2!

|- BGCOLOR="#e0f0e0" !12 |(1,1,1,1) + (0,1,2,3){{radic|2}} |Omnitruncated 16-cell |{{dark mode invert|{{CDD|node_1|4|node_1|3|node_1|3|node_1}}}} |384||2⁴4! |}

== References == {{reflist}} * J.H. Conway and M.J.T. Guy: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, pp. 38–39, 1965 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26) * 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 Ivic 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] * N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966

== External links == * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)}} * [http://www.polytope.de Uniform, convex polytopes in four dimensions], Marco Möller {{in lang|de}} ** {{Cite thesis |url=https://ediss.sub.uni-hamburg.de/volltexte/2004/2196/pdf/Dissertation.pdf |title=Vierdimensionale Archimedische Polytope |last=Möller |first=Marco |date=2004 |publisher=University of Hamburg |type=Doctoral dissertation |language=de }} *{{PolyCell | urlname = uniform.html| title = Uniform Polytopes in Four Dimensions}} ** {{PolyCell | urlname = section2.html| title = Convex uniform polychora based on the tesserract/16-cell}}

{{Polytopes}}

Category:Uniform 4-polytopes