|
|
| Line 1: |
Line 1: |
| [[File:Ortho solid 011-uniform polychoron 53p-t0.png|320px|thumb|The [[great grand 120-cell]], one of ten Schläfli–Hess polychora by [[orthographic projection]].]]
| | Hello. Allow me introduce the writer. Her name is Emilia Shroyer but it's not the most feminine name out there. For a while she's been in South Dakota. What I adore doing is doing ceramics but I haven't produced a dime with it. I am a meter reader.<br><br>Feel free to visit my blog post at home std testing ([http://www.article-galaxy.com/profile.php?a=165109 Suggested Internet page]) |
| In four-dimensional [[geometry]], '''Schläfli–Hess polychora''' are the complete set of 10 [[Regular polytope|regular]] self-intersecting [[Star polytope|'''star polychora''']] ([[4-polytope|four-dimensional polytopes]]). They are named in honor of their discoverers: [[Ludwig Schläfli]] and [[Edmund Hess]]. Each is represented by a [[Schläfli symbol]] {''p'',''q'',''r''} in which one of the numbers is [[pentagram|5/2]]. They are thus analogous to the regular nonconvex [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]].
| |
| | |
| Allowing for regular [[star polygon]]s as [[Cell (geometry)|cells]] and [[vertex figure]]s, these 10 polychora add to the set of six [[regular convex 4-polytope]]s. All may be derived as [[stellation]]s of the [[120-cell]] {5,3,3} or the [[600-cell]] {3,3,5}.
| |
| | |
| == History ==
| |
| Four of them were found by [[Ludwig Schläfli]] while the other six were skipped because he would not allow forms that failed the [[Euler characteristic]] on cells or vertex figures (for zero-hole tori: ''F'' − ''E'' + ''V'' = 2). That excludes cells and vertex figures as {5,5/2}, and {5/2,5}.
| |
| | |
| [[Edmund Hess]] (1843–1903) published the complete list in his 1883 German book ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder''.
| |
| | |
| == Names ==
| |
| Their names given here were given by [[John Horton Conway|John Conway]], extending [[Arthur Cayley|Cayley's]] names for the [[Kepler–Poinsot polyhedra]]: along with ''stellated'' and ''great'', he adds a ''grand'' modifier. Conway offered these operational definitions:
| |
| #'''[[stellation]]''' – replaces edges by longer edges in same lines. (Example: a [[pentagon]] stellates into a [[pentagram]])
| |
| #'''greatening''' – replaces the faces by large ones in same planes. (Example: an [[icosahedron]] greatens into a [[great icosahedron]])
| |
| #'''aggrandizement''' – replaces the cells by large ones in same 3-spaces.
| |
| | |
| == Symmetry ==
| |
| All ten polychora have [3,3,5] ([[Coxeter_group#Finite_Coxeter_groups|H<sub>4</sub>]]) [[hexacosichoric symmetry]]. They are generated from 6 related [[Goursat tetrahedron|rational-order symmetry groups]]: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], [3,3,5/2].
| |
| | |
| Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.
| |
| | |
| == Table of elements ==
| |
| Note:
| |
| * There are 2 unique [[vertex arrangement]]s, matching those of the [[120-cell]] and [[600-cell]].
| |
| * There are 4 unique [[edge arrangement]]s, which are shown as ''wireframes'' [[Orthographic projection (geometry)|orthographic projections]].
| |
| * There are 7 unique [[face arrangement]]s, shown as ''solids'' (face-colored) orthographic projections.
| |
| | |
| The cells (polyhedra), their faces (polygons), the ''polygonal [[edge figure]]s'' and ''polyhedral [[vertex figure]]s'' are identified by their [[Schläfli symbol]]s.
| |
| | |
| {| class="wikitable"
| |
| ! Name<BR>(Bowers acronym)
| |
| ! Wireframe
| |
| ! Solid
| |
| ! [[Schläfli symbol|Schläfli]]<BR>{p, q,r}<BR>[[Coxeter–Dynkin diagram|Coxeter–Dynkin]]
| |
| ! Cells<BR>{p, q}
| |
| ! Faces<BR>{p}
| |
| ! Edges<BR>{r}
| |
| ! Vertices<BR>{q, r}
| |
| ![[density (polytope)|Density]]
| |
| ! [[Euler characteristic|χ]]
| |
| ! Dual<BR>{r, q,p}
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Icosahedral 120-cell]]<BR>(or ''faceted 600-cell'')<BR>(fix)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[File:ortho solid 007-uniform polychoron 35p-t0.png|75px]]
| |
| | {3,5,5/2}<BR>{{CDD|node_1|3|node|5|node|5|rat|d2|node}}
| |
| | 120<BR>[[Icosahedron|{3,5}]]<BR>[[File:Icosahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| |
| | 4
| |
| | 480
| |
| | Small stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Small stellated 120-cell]]<BR>(sishi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| |
| | [[File:ortho solid 010-uniform polychoron p53-t0.png|75px]]
| |
| | {5/2,5,3}<BR>{{CDD|node|3|node|5|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 120<BR>[[Dodecahedron|{5,3}]]<BR>[[File:Dodecahedron.png|25px]]
| |
| | 4
| |
| | −480
| |
| | Icosahedral 120-cell
| |
| |- align=center BGCOLOR="#e0ffe0"
| |
| | [[Great 120-cell]]<BR>(gohi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[File:ortho solid 008-uniform polychoron 5p5-t0.png|75px]]
| |
| | {5,5/2,5}<BR>{{CDD|node_1|5|node|5|rat|d2|node|5|node}}
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| |
| | 6
| |
| | 0
| |
| | Self-dual
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Grand 120-cell]]<BR>(gahi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[File:ortho solid 009-uniform polychoron 53p-t0.png|75px]]
| |
| | {5,3,5/2}<BR>{{CDD|node_1|5|node|3|node|5|rat|d2|node}}
| |
| | 120<BR>[[Dodecahedron|{5,3}]]<BR>[[File:Dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[File:Great icosahedron.png|25px]]
| |
| | 20
| |
| | 0
| |
| | Great stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great stellated 120-cell]]<BR>(gishi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[File:ortho solid 012-uniform polychoron p35-t0.png|75px]]
| |
| | {5/2,3,5}<BR>{{CDD|node|5|node|3|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[File:Great stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 120<BR>[[Icosahedron|{3,5}]]<BR>[[File:Icosahedron.png|25px]]
| |
| | 20
| |
| | 0
| |
| | Grand 120-cell
| |
| |- align=center BGCOLOR="#e0ffe0"
| |
| | [[Grand stellated 120-cell]]<BR>(gashi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[File:ortho solid 013-uniform polychoron p5p-t0.png|75px]]
| |
| | {5/2,5,5/2}<BR>{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| |
| | 66
| |
| | 0
| |
| | Self-dual
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Great grand 120-cell]]<BR>(gaghi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| |
| | [[File:ortho solid 011-uniform polychoron 53p-t0.png|75px]]
| |
| | {5,5/2,3}<BR>{{CDD|node_1|5|node|5|rat|d2|node|3|node}}
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[File:Great stellated dodecahedron.png|25px]]
| |
| | 76
| |
| | −480
| |
| | Great icosahedral 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great icosahedral 120-cell]]<BR>(or ''great faceted 600-cell'')<BR>(gofix)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[File:ortho solid 014-uniform polychoron 3p5-t0.png|75px]]
| |
| | {3,5/2,5}<BR>{{CDD|node|5|node|5|rat|d2|node|3|node_1}}
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[File:Great icosahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[File:Pentagon.svg|25px]]
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| |
| | 76
| |
| | 480
| |
| | Great grand 120-cell
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Grand 600-cell]]<BR>(gax)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[File:ortho solid 015-uniform polychoron 33p-t0.png|75px]]
| |
| | {3,3,5/2}<BR>{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
| |
| | 600<BR>[[Tetrahedron|{3,3}]]<BR>[[File:Tetrahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[File:Great icosahedron.png|25px]]
| |
| | 191
| |
| | 0
| |
| | Great grand stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great grand stellated 120-cell]]<BR>(gogishi)
| |
| | [[File:Schläfli-Hess polychoron-wireframe-1.png|75px]]
| |
| | [[File:ortho solid 016-uniform polychoron p33-t0.png|75px]]
| |
| | {5/2,3,3}<BR>{{CDD|node|3|node|3|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[File:Great stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Pentagram.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[File:Triangle.Equilateral.svg|25px]]
| |
| | 600<BR>[[Tetrahedron|{3,3}]]<BR>[[File:Tetrahedron.png|25px]]
| |
| | 191
| |
| | 0
| |
| | Grand 600-cell
| |
| |}
| |
| | |
| == Existence ==
| |
| The existence of a regular polychoron <math>\{p,q,r\}</math> is constrained by the existence of the regular polyhedra <math>\{p,q\}, \{q,r\}</math> and a [[dihedral angle]] constraint:
| |
| :<math>\sin(\frac{\pi}{p}) \sin(\frac{\pi}{r}) < \cos(\frac{\pi}{q}). </math>
| |
| | |
| The six regular convex polytopes and 10 star polytopes above are the only solutions to these constraints.
| |
| | |
| There are four nonconvex [[Schläfli symbol]]s {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}.
| |
| | |
| == See also ==
| |
| * [[List of regular polytopes]]
| |
| * [[Convex regular polychoron]]
| |
| * [[Kepler-Poinsot polyhedra]] – regular [[star polyhedron]]
| |
| * [[Star polygon]] – regular star polygons
| |
| | |
| == References ==
| |
| * [[Edmund Hess]], (1883) ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' [http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].
| |
| * [[Edmund Hess]] ''Uber die regulären Polytope höherer Art'', Sitzungsber Gesells Beförderung gesammten Naturwiss Marburg, 1885, 31-57
| |
| * '''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, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
| |
| ** (Paper 10) H.S.M. Coxeter, ''Star Polytopes and the Schlafli Function f(α,β,γ)'' [Elemente der Mathematik 44 (2) (1989) 25–36]
| |
| *[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q,r} in four dimensions, pp. 292–293)
| |
| *[[Coxeter|H. S. M. Coxeter]], ''Regular Complex Polytopes'', 2nd. ed., Cambridge University Press 1991. ISBN 978-0-521-39490-1. [http://www.amazon.com/dp/0521394902]
| |
| * Peter McMullen and Egon Schulte, ''Abstract Regular Polytopes'', 2002, [http://assets.cambridge.org/052181/4960/sample/0521814960ws.pdf PDF]
| |
| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular Star-polytopes, pp. 404–408)
| |
| | |
| == External links == | |
| *{{Mathworld | urlname=RegularPolychoron | title=Regular polychoron }}
| |
| * {{GlossaryForHyperspace | anchor=Hecatonicosachoron | title=Hecatonicosachoron}}
| |
| ** {{GlossaryForHyperspace | anchor=Hexacosichoron | title= Hexacosichoron}}
| |
| ** {{GlossaryForHyperspace | anchor=Stellation | title=Stellation}}
| |
| ** {{GlossaryForHyperspace | anchor=Greatening | title=Greatening}}
| |
| ** {{GlossaryForHyperspace | anchor=Aggrandizement | title=Aggrandizement}}
| |
| * [http://www.polytope.net/hedrondude/regulars.htm Jonathan Bowers, 16 regular polychora]
| |
| * [http://mathforum.org/library/drmath/view/54786.html Discussion on names]
| |
| * [http://www.mathematik.uni-regensburg.de/Goette/sterne Reguläre Polytope]
| |
| * [http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
| |
| * [http://www.software3d.com/Stella.php#stella4D Stella4D] [[Stella (software)]] produces interactive views of all 1849 known uniform polychora including the 64 convex forms and the infinite prismatic families. Was used to create images for this page.
| |
| | |
| {{DEFAULTSORT:Schlafli-Hess polychoron}}
| |
| [[Category:Polychora]]
| |
| [[Category:Four-dimensional geometry]]
| |