Login
Login

Compound of five octahedra

{{Short description|Polyhedral compound}}

class="wikitable" style="float:right; margin-left:8px; width:250px"

!bgcolor=#e7dcc3 colspan=2|Compound of five octahedra

align=center colspan=2|240px
(see here for a 3D model)
bgcolor=#e7dcc3 width=50%|TypeRegular compound
bgcolor=#e7dcc3|IndexUC17, W23
bgcolor=#e7dcc3|Coxeter symbol[5{3,4}]2{3,5}Regular polytopes, pp.49-50, p.98
bgcolor=#e7dcc3|Elements
(As a compound)		5 octahedra:
F = 40, E = 60, V = 30
bgcolor=#e7dcc3|Dual compoundCompound of five cubes
bgcolor=#e7dcc3|Symmetry groupicosahedral (Ih)
bgcolor=#e7dcc3|Subgroup restricting to one constituentpyritohedral (Th)

File:Small-icosiicosahedron-in-icosidodecahedron.png of the icosidodecahedron. ]]

The compound of five octahedra is one of the five regular polyhedron compounds, and can also be seen as a stellation. It 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 Wenninger model index 23.

It can be constructed by a rhombic triacontahedron with rhombic-based pyramids 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
Icosahedron

|150px
Icosidodecahedron

As a compound

It can also be seen as a polyhedral compound of five octahedra arranged in icosahedral symmetry (Ih).

The spherical and stereographic projections of this compound look the same as those of the disdyakis triacontahedron.

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"| 250px

| x150px

| x150px

| x150px

x120px

| x120px

| 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 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 octahedron to the standard 4-octahedra compound.

See also

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=}} (1st Edn University of Toronto (1938))
  • H.S.M. Coxeter, 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]
  • {{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}}