Examine individual changes

'{{Short description|Polyhedral compound}} {| class=wikitable width=250 align=right !bgcolor=#e7dcc3 colspan=2|Compound of five octahedra |- |align=center colspan=2|[[Image:Compound of five octahedra.png|240px]]<br>[[:File:Compound of five octahedra (full).stl|(see here for a 3D model)]] |- |bgcolor=#e7dcc3 width=50%|Type||[[Regular polyhedral compound|Regular compound]] |- |bgcolor=#e7dcc3|Index||UC₁₇, W₂₃ |- |bgcolor=#e7dcc3|Coxeter symbol|| [5{3,4}]2{3,5}<ref>Regular polytopes, pp.49-50, p.98</ref> |- |bgcolor=#e7dcc3|[[Euler characteristic|Elements]]<BR>(As a compound)||5 [[Octahedron|octahedra]]:<BR>''F'' = 40, ''E'' = 60, ''V'' = 30 |- |bgcolor=#e7dcc3|[[Dual polyhedron|Dual compound]]||[[Compound of five cubes]] |- |bgcolor=#e7dcc3|[[Symmetry group]]||[[Icosahedral symmetry|icosahedral]] (''I''_h) |- |bgcolor=#e7dcc3|[[Subgroup]] restricting to one constituent||[[Tetrahedral symmetry|pyritohedral]] (''T''_h) |} [[File:Small-icosiicosahedron-in-icosidodecahedron.png|thumb|right|It is also a [[faceting]] of the icosidodecahedron. ]] The '''[[polyhedral compound|compound]] of five octahedra''' is one of the five regular polyhedron compounds. This polyhedron can be seen as either a polyhedral [[stellation]] or a [[Polyhedron compound|compound]]. This compound was first described by [[Edmund Hess]] in 1876. It is unique among the regular compounds for not having a regular convex hull. == As a stellation == It is the second [[stellation]] of the [[icosahedron]], and given as [[List of Wenninger polyhedron models#Stellations of icosahedron|Wenninger model index 23]]. It can be constructed by a [[rhombic triacontahedron]] with rhombic-based [[Pyramid (geometry)|pyramid]]s added to all the faces, as shown by the five colored model image. (This construction does not generate the ''regular'' compound of five octahedra, but shares the same topology and can be smoothly deformed into the regular compound.) It has a density of greater than 1. {| class=wikitable ![[Stellation diagram]]!![[Stellation]] core||[[Convex hull]] |- valign=top align=center |[[File:Compound of five octahedra stellation facets.svg|150px|Stellation facets]] |[[File:Icosahedron.png|150px]]<BR>[[Stellations of icosahedron|Icosahedron]] |[[File:Icosidodecahedron.png|150px]]<BR>[[Icosidodecahedron]] |} == As a compound == It can also be seen as a [[polyhedral compound]] of five [[Octahedron|octahedra]] arranged in [[icosahedral symmetry]] ('''I'''_h). The [[spherical polyhedron|spherical]] and [[stereographic projection|stereographic]] projections of this compound look the same as those of the [[disdyakis triacontahedron]].<br> But the convex solid's vertices on 3- and 5-fold symmetry axes (gray in the images below) correspond only to edge crossings in the compound. {| class="wikitable" style="text-align: center;" !rowspan="2"| Spherical polyhedron !colspan="3"| Stereographic projections |- ! 2-fold ! 3-fold ! 5-fold |- |rowspan="2"| [[File:Spherical disdyakis triacontahedron as compound of five octahedra.png|250px]] | [[File:Disdyakis triacontahedron stereographic d2 colored.svg|x150px]] | [[File:Disdyakis triacontahedron stereographic d3 colored.svg|x150px]] | [[File:Disdyakis triacontahedron stereographic d5 colored.svg|x150px]] |- | [[File:Disdyakis triacontahedron stereographic d2 colored crop.svg|x120px]] | [[File:Disdyakis triacontahedron stereographic d3 colored crop.svg|x120px]] | [[File:Disdyakis triacontahedron stereographic d5 colored crop.svg|x120px]] |- |colspan="4" style="font-size: small;"| The area in the black circles below corresponds to the frontal hemisphere of the spherical polyhedron. |} Replacing the octahedra by [[Tetrahemihexahedron|tetrahemihexahedra]] leads to the [[compound of five tetrahemihexahedra]]. == Other 5-octahedra compounds == A second 5-octahedra compound, with octahedral symmetry, also exists. It can be generated by adding a fifth octahedra to the [[Compound of four octahedra|standard 4-octahedra compound]]. ==See also== *[[Compound of three octahedra]] *[[Compound of four octahedra]] *[[Compound of ten octahedra]] *[[Compound of twenty octahedra]] ==References== {{reflist}} * [[Peter R. Cromwell]], ''Polyhedra'', Cambridge, 1997. * {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }} *{{Cite book | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Du Val | first2=P. | last3=Flather | first3=H. T. | last4=Petrie | first4=J. F. | title=The fifty-nine icosahedra | publisher=Tarquin | edition=3rd | isbn=978-1-899618-32-3 | mr=676126 | year=1999 | postscript=<!--None-->}} (1st Edn University of Toronto (1938)) * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}}, 3.6 ''The five regular compounds'', pp.47-50, 6.2 ''Stellating the Platonic solids'', pp.96-104 * [[E. Hess]] 1876 ''Zugleich Gleicheckigen und Gleichflächigen Polyeder'', Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg 11 (1876) pp 5–97. == External links == * [http://mathworld.wolfram.com/Octahedron5-Compound.html MathWorld: Octahedron5-Compound] * [http://www.korthalsaltes.com/model.php?name_en=compound%20of%20five%20octahedra Paper Model Compound of Five Octahedra] * [[VRML]] model: [http://www.interocitors.com/polyhedra/UCs/17%__5_Octahedra.wrl]{{dead link|date=August 2017 |bot=InternetArchiveBot |fix-attempted=yes }} * {{KlitzingPolytopes|../incmats/se.htm|3D compound|}} * [https://www.facebook.com/media/set?vanity=charles.stevens.1848&set=a.1013884028662829 Octahedron5-Compound as Gauss Pentagramma Mirificum] {{Icosahedron stellations}} [[Category:Polyhedral stellation]] [[Category:Polyhedral compounds]] {{polyhedron-stub}}'