Cubitruncated cuboctahedron: Difference between revisions
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{{Short description|Polyhedron with 20 faces}} |
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{{Uniform polyhedra db|Uniform polyhedron stat table|ctCO}} |
{{Uniform polyhedra db|Uniform polyhedron stat table|ctCO}} |
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[[File:Cubitruncated cuboctahedron |
[[File:Cubitruncated cuboctahedron.stl|thumb|3D model of a cubitruncated cuboctahedron]] |
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In [[geometry]], the '''cubitruncated cuboctahedron''' or '''cuboctatruncated cuboctahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>16</sub>. |
In [[geometry]], the '''cubitruncated cuboctahedron''' or '''cuboctatruncated cuboctahedron''' is a [[nonconvex uniform polyhedron]], indexed as U<sub>16</sub>. It has 20 faces (8 [[hexagon]]s, 6 [[octagon]]s, and 6 [[octagram]]s), 72 edges, and 48 vertices,<ref>{{Cite web|url=https://www.mathconsult.ch/enwiki/static/unipoly/16.html|title=16: cubitruncated cuboctahedron|last=Maeder|first=Roman|date=|website=MathConsult|url-status=live|archive-url=https://web.archive.org/web/20150329074117/http://www.mathconsult.ch:80/enwiki/static/unipoly/16.html |archive-date=2015-03-29 |access-date=}}</ref> and has a shäfli symbol of tr{4,<sup>3</sup>/<sub>2</sub>} |
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== Convex hull == |
== Convex hull == |
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: (±({{radic|2}}−1), ±1, ±({{radic|2}}+1)) |
: (±({{radic|2}}−1), ±1, ±({{radic|2}}+1)) |
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== Related polyhedra == |
== Related polyhedra == |
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=== Tetradyakis hexahedron=== |
=== Tetradyakis hexahedron=== |
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{{Uniform polyhedra db|Uniform dual polyhedron stat table|ctCO}} |
{{Uniform polyhedra db|Uniform dual polyhedron stat table|ctCO}} |
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[[File:Tetradyakis hexahedron.stl|thumb|3D model of a tetradyakis hexahedron]] |
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The '''tetradyakis hexahedron''' (or '''great disdyakis dodecahedron''') is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]]. It has 48 intersecting [[scalene triangle]] faces, 72 edges, and 20 vertices. |
The '''tetradyakis hexahedron''' (or '''great disdyakis dodecahedron''') is a nonconvex [[Isohedral figure|isohedral]] [[polyhedron]]. It has 48 intersecting [[scalene triangle]] faces, 72 edges, and 20 vertices. |
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==== Proportions ==== |
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The triangles have one angle of <math>\arccos(\frac{3}{4})\approx 41.409\,622\,109\,27^{\circ}</math>, one of <math>\arccos(\frac{1}{6}+\frac{7}{12}\sqrt{2})\approx 7.420\,694\,647\,42^{\circ}</math> and one of <math>\arccos(\frac{1}{6}-\frac{7}{12}\sqrt{2})\approx 131.169\,683\,243\,31^{\circ}</math>. The [[dihedral angle]] equals <math>\arccos(-\frac{5}{7})\approx 135.584\,691\,402\,81^{\circ}</math>. Part of each triangle lies within the solid, hence is invisible in solid models. |
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It is the [[Dual polyhedron|dual]] of the [[uniform star polyhedron|uniform]] cubitruncated cuboctahedron. |
It is the [[Dual polyhedron|dual]] of the [[uniform star polyhedron|uniform]] cubitruncated cuboctahedron. |
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== See also == |
== See also == |
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==References== |
==References== |
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{{Reflist}} |
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*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 |mr=730208 | year=1983}} p. 92 |
*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 |mr=730208 | year=1983}} p. 92 |
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Latest revision as of 05:02, 4 March 2023
| Cubitruncated cuboctahedron | |
|---|---|
| |
| Type | Uniform star polyhedron |
| Elements | F = 20, E = 72 V = 48 (χ = −4) |
| Faces by sides | 8{6}+6{8}+6{8/3} |
| Coxeter diagram |    
|
| Wythoff symbol | 3 4 4/3 | |
| Symmetry group | Oh, [4,3], *432 |
| Index references | U16, C52, W79 |
| Dual polyhedron | Tetradyakis hexahedron |
| Vertex figure |  6.8.8/3 |
| Bowers acronym | Cotco |
In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. It has 20 faces (8 hexagons, 6 octagons, and 6 octagrams), 72 edges, and 48 vertices,[1] and has a shäfli symbol of tr{4,3/2}
Convex hull
[edit]Its convex hull is a nonuniform truncated cuboctahedron.
 Convex hull |
Cubitruncated cuboctahedron |
Orthogonal projection
[edit]
Cartesian coordinates
[edit]Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of
- (±(√2−1), ±1, ±(√2+1))
Related polyhedra
[edit]Tetradyakis hexahedron
[edit]| Tetradyakis hexahedron | |
|---|---|
| |
| Type | Star polyhedron |
| Face | 
|
| Elements | F = 48, E = 72 V = 20 (χ = −4) |
| Symmetry group | Oh, [4,3], *432 |
| Index references | DU16 |
| dual polyhedron | Cubitruncated cuboctahedron |
The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.
Proportions
[edit]The triangles have one angle of , one of and one of . The dihedral angle equals . Part of each triangle lies within the solid, hence is invisible in solid models.
It is the dual of the uniform cubitruncated cuboctahedron.
See also
[edit]References
[edit]- ^ Maeder, Roman. "16: cubitruncated cuboctahedron". MathConsult. Archived from the original on 2015-03-29.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 92
External links
[edit]- Weisstein, Eric W. "Cubitruncated cuboctahedron". MathWorld.
- Weisstein, Eric W. "Tetradyakis hexahedron". MathWorld.
- http://gratrix.net Uniform polyhedra and duals
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