Compound of cube and octahedron: Difference between revisions

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{{Short description|Polyhedral compound}}

{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"

{| class=wikitable align="right" width="250"

!bgcolor=#e7dcc3 colspan=2|Compound of cube and octahedron

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|align=center colspan=2|[[ImageFile:Compound of cube and octahedronCompound_of_cube_and_octahedron.png|220px230px]]

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|bgcolor=#e7dcc3|Type||[[List of Wenninger polyhedron models#Stellations of cuboctahedron|stellation]] and [[Polyhedral compound|compoundCompound]]

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|nodes_10ru|split2-43|node}} ∪ {{CDD|nodes_01rd|split2-43|node}}

|bgcolor=#e7dcc3|Index||[[List_of_Wenninger_polyhedron_models#Stellations_of_cuboctahedron|W₄₃]]

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|bgcolor=#e7dcc3|[[Stellation]] core||[[Stellations of cuboctahedron|cuboctahedron]]

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|bgcolor=#e7dcc3|[[Convex hull]]||[[Rhombic dodecahedron]]

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|bgcolor=#e7dcc3|Index||W₄₃

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|bgcolor=#e7dcc3|Polyhedra||1 [[octahedron]]<BR>1 [[cube]]

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|bgcolor=#e7dcc3|Faces||8 [[triangle]]s<BR>6 [[squareSquare (geometry)|squares]]s

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|bgcolor=#e7dcc3|Edges||24

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|bgcolor=#e7dcc3|[[Symmetry group]]||[[Octahedral symmetry|octahedral]] (''O''_''h'')

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[[File:Bronze mace head from Galicia.jpg|thumb|Medieval [[Mace (bludgeon)|mace]] head]]

ThisThe '''compound of cube and octahedron''' is a [[polyhedron]] which can be seen as either a polyhedral [[stellation]] or a [[Polyhedron compound|compound]].

==Construction==

The 14 [[Cartesian coordinate]]s of the vertices of the compound are.

This polyhedron can be seen as either a polyhedral [[stellation]] or a [[Polyhedron compound|compound]].

: 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)

: 8: ( ±1, ±1, ±1)

== As a compound ==

It can be seen as the [[Polyhedron compound|compound]] of aan [[octahedron]] and a [[cube]]. It is one of four compounds constructed from a [[Platonic solid]] or [[Kepler-Poinsot solidpolyhedron]], and its dual.

It has [[octahedral symmetry]] ('''O'''_''h'') and shares the same vertices as a [[rhombic dodecahedron]].

This can be seen as the three-dimensional equivalent of the compound of two squares ({8/2} "[[octagram]]"); this series continues on to infinity, with the four-dimensional equivalent being the compound of tesseract and 16-cell.

{|

|- style="vertical-align: top;"

|{{multiple image

| align = left | total_width = 320

| image2 = Polyhedron 6.png |width2=1|height2=1

| image3 = Polyhedron 8.png |width3=1|height3=1

| footer = A cube and its [[dual polyhedron|dual]] octahedron

}}

|{{multiple image

| align = left | total_width = 320

| image2 = Polyhedron 6-8 blue.png |width2=1|height2=1

| image3 = Polyhedron 6-8 dual blue.png |width3=1|height3=1

| footer = The intersection of both solids is the [[cuboctahedron]], and their [[convex hull]] is the [[rhombic dodecahedron]].

}}

|}

{{multiple image

| align = left | total_width = 480

| image2 = Polyhedron pair 6-8 from blue.png |width2=1|height2=1

| image3 = Polyhedron pair 6-8 from yellow.png |width3=1|height3=1

| image4 = Polyhedron pair 6-8 from red.png |width4=1|height4=1

| footer = Seen from 2-fold, 3-fold and 4-fold symmetry axes<br>The hexagon in the middle is the [[Petrie polygon]] of both solids.

}}

{{multiple image

| align = right | total_width = 320

| image2 = Polyhedron pair 6-8.png |width2=1|height2=1

| image3 = Polyhedron small rhombi 6-8 dual max.png |width3=1|height3=1

| footer = If the edge crossings were vertices, the [[Spherical polyhedron|mapping on a sphere]] would be the same as that of a [[deltoidal icositetrahedron]].

}}

{{clear|left}}

== As a stellation ==

It is also the first [[stellation]] of the [[cuboctahedron]], and given as [[List of Wenninger polyhedron models#Stellations of cuboctahedron|Wenninger model index 43]].

It can be seen as a [[cuboctahedron]] with [[Square (geometry)|square]] and [[triangle|triangular]] [[Pyramid (geometry)|pyramid]]s added to each face.

The stellation facets for construction are:

:[[Image:First stellation of cuboctahedron trifacets.png|240px]][[Image:First stellation of cuboctahedron square facets.png|240px]]

== See also ==

* [[Compound of two tetrahedra]]

* [[Compound of dodecahedron and icosahedron]]

* [[Compound of small stellated dodecahedron and great dodecahedron]]

* [[Compound of great stellated dodecahedron and great icosahedron]]

==References==

* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | idisbn=ISBN 978-0-521-09859-5 }}

[[Category:Polyhedral stellation]]

[[Category:Polyhedral compounds]]

{{polyhedron-stub}}