compound of cube and octahedron
{{Short description|Polyhedral compound}}
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!bgcolor=#e7dcc3 colspan=2|Compound of cube and octahedron | |
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| bgcolor=#e7dcc3|Type | Compound |
| bgcolor=#e7dcc3|Coxeter diagram | {{CDD|nodes_10ru|split2-43|node}} ∪ {{CDD|nodes_01rd|split2-43|node}} |
| bgcolor=#e7dcc3|Stellation core | cuboctahedron |
| bgcolor=#e7dcc3|Convex hull | Rhombic dodecahedron |
| bgcolor=#e7dcc3|Index | W43 |
| bgcolor=#e7dcc3|Polyhedra | 1 octahedron 1 cube |
| bgcolor=#e7dcc3|Faces | 8 triangles 6 squares |
| bgcolor=#e7dcc3|Edges | 24 |
| bgcolor=#e7dcc3|Vertices | 14 |
| bgcolor=#e7dcc3|Symmetry group | octahedral (Oh) |
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The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound.
Construction
The 14 Cartesian coordinates of the vertices of the compound are.
: 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)
: 8: ( ±1, ±1, ±1)
As a compound
It can be seen as the compound of an octahedron and a cube. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot polyhedron and its dual.
It has octahedral symmetry (Oh) 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.
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|{{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 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
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| 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
The hexagon in the middle is the Petrie polygon of both solids.
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{{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 mapping on a sphere would be the same as that of a deltoidal icositetrahedron.
}}
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As a stellation
It is also the first stellation of the cuboctahedron and given as Wenninger model index 43.
It can be seen as a cuboctahedron with square and triangular pyramids added to each face.
The stellation facets for construction are:
See also
References
- {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=978-0-521-09859-5 }}
Category:Polyhedral stellation
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