Cubitruncated cuboctahedron: Difference between revisions

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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	O_h, [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

Its convex hull is a nonuniform truncated cuboctahedron.

Convex hull	Cubitruncated cuboctahedron

Orthogonal projection

Cartesian coordinates

Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of

(±(√2−1), ±1, ±(√2+1))

Related polyhedra

Tetradyakis hexahedron

Tetradyakis hexahedron

Type	Star polyhedron
Face
Elements	F = 48, E = 72 V = 20 (χ = −4)
Symmetry group	O_h, [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

The triangles have one angle of $\arccos({\frac {3}{4}})\approx 41.409\,622\,109\,27^{\circ }$ , one of $\arccos({\frac {1}{6}}+{\frac {7}{12}}{\sqrt {2}})\approx 7.420\,694\,647\,42^{\circ }$ and one of $\arccos({\frac {1}{6}}-{\frac {7}{12}}{\sqrt {2}})\approx 131.169\,683\,243\,31^{\circ }$ . The dihedral angle equals $\arccos(-{\frac {5}{7}})\approx 135.584\,691\,402\,81^{\circ }$ . Part of each triangle lies within the solid, hence is invisible in solid models.

It is the dual of the uniform cubitruncated cuboctahedron.

References

^ 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

This polyhedron-related article is a stub. You can help Wikipedia by expanding it.

[1] Maeder, Roman. "16: cubitruncated cuboctahedron". MathConsult. Archived from the original on 2015-03-29.

[1]

@@ Line 1: / Line 1: @@
+{{Short description|Polyhedron with 20 faces}}
 {{Uniform polyhedra db|Uniform polyhedron stat table|ctCO}}
+[[File:Cubitruncated cuboctahedron.stl|thumb|3D model of a cubitruncated cuboctahedron]]
-In [[geometry]], the '''cubitruncated 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>}
 == Convex hull ==
@@ Line 10: / Line 12: @@
 |[[File:Cubitruncated cuboctahedron.png|100px]]<BR>Cubitruncated cuboctahedron
 |}
+== Orthogonal projection==
+[[File:Cubitruncated cuboctahedron ortho wireframes.png|480px]]
 == Cartesian coordinates ==
 [[Cartesian coordinates]] for the vertices of a cubitruncated cuboctahedron are all the permutations of
-: (±(&radic;2−1), ±1, ±(&radic;2+1))
+: (±({{radic|2}}−1), ±1, ±({{radic|2}}+1))
+{{clear}}
+== Related polyhedra ==
+=== Tetradyakis hexahedron===
+{{Uniform polyhedra db|Uniform dual polyhedron stat table|ctCO}}
+[[File:Tetradyakis hexahedron.stl|thumb|3D model of a tetradyakis hexahedron]]
+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.
+==== Proportions ====
+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.
+It is the [[Dual polyhedron|dual]] of the [[uniform star polyhedron|uniform]] cubitruncated cuboctahedron.
 == See also ==
 * [[List of uniform polyhedra]]
+==References==
+{{Reflist}}
+*{{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.&nbsp;92
 == External links ==
 * {{mathworld | urlname = CubitruncatedCuboctahedron| title = Cubitruncated cuboctahedron}}
+* {{mathworld | urlname = TetradyakisHexahedron| title =Tetradyakis hexahedron}}
+* [http://gratrix.net/polyhedra/uniform/summary/ http://gratrix.net Uniform polyhedra and duals]
-{{Polyhedron-stub}}
 [[Category:Uniform polyhedra]]
+{{Polyhedron-stub}}