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Truncated cuboctahedron

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Truncated cuboctahedron

Truncated cuboctahedron

(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 26, E = 72, V = 48 (χ = 2)
Faces by sides 12{4}+8{6}+6{8}
Conway notation bC or taC
Schläfli symbols tr{4,3} or t\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}
t0,1,2{4,3}
Wythoff symbol 2 3 4 |
Coxeter diagram
Symmetry group Oh, BC3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 4-6:arccos(-sqrt(6)/3)=144°44'08"
4-8:arccos(-sqrt(2)/3)=135°
6-8:arccos(-sqrt(3)/3)=125°15'51"
References U11, C23, W15
Properties Semiregular convex zonohedron

Colored faces

4.6.8
(Vertex figure)

Disdyakis dodecahedron
(dual polyhedron)

Net

In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.

Contents

  • Other names 1
  • Cartesian coordinates 2
  • Area and volume 3
  • Dissection 4
  • Uniform colorings 5
  • Orthogonal projections 6
  • Spherical tiling 7
  • Related polyhedra 8
  • Truncated cuboctahedral graph 9
  • See also 10
  • References 11
  • External links 12

Other names

Alternate interchangeable names are:

The name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles. However, the resulting figure is topologically equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.

The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.

One unfortunate point of confusion: There is a nonconvex uniform polyhedron by the same name. See nonconvex great rhombicuboctahedron.

Cartesian coordinates

The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all permutations of:

(±1, ±(1+√2), ±(1+2√2))

Area and volume

The area A and the volume V of the truncated cuboctahedron of edge length a are:

A = 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 \approx 61.7551724a^2
V = \left(22+14\sqrt{2}\right) a^3 \approx 41.7989899a^3.

Dissection

The truncated cuboctahedron can be dissected into a central rhombicuboctahedron, with 6 square cupolas above each primary square face, 8 triangular cupola above each triangular face, and 12 cubes above the secondary square faces.

A dissected truncated cuboctahedron can create a genus 5, 7 or 11 Stewart toroid by removing the central rhombicuboctahedron and either the square cupola, the triangular cupola or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupola creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry. [4][5]

Stewart toroids
Genus 3 Genus 5 Genus 7 Genus 11

Uniform colorings

There is only one uniform coloring of the faces of this polyhedron, one color for each face type.

A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons.

Orthogonal projections

The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.

Orthogonal projections
Centered by Vertex Edge
4-6
Edge
4-8
Edge
6-8
Face normal
4-6
Image
Projective
symmetry
[2]+ [2] [2] [2] [2]
Centered by Face normal
Square
Face normal
Octagon
Face
Square
Face
Hexagon
Face
Octagon
Image
Projective
symmetry
[2] [2] [2] [6] [8]

Spherical tiling

The truncated cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


square-centered

hexagon-centered

octagon-centered
Orthogonal projection Stereographic projections

Related polyhedra

The truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

Uniform octahedral polyhedra
Symmetry: [4,3], (*432) [4,3]+, (432) [3+,4], (3*2)
{4,3} t{4,3} r{4,3} t{3,4} {3,4} rr{4,3} tr{4,3} sr{4,3} s{3,4}
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V35

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
 
[12i,3]
 
[9i,3]
 
[6i,3]
 
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure

4.8.4

4.8.6

4.8.8

4.8.10

4.8.12

4.8.14

4.8.16

4.8.∞
Omnitruncated
duals

V4.8.4

V4.8.6

V4.8.8

V4.8.10

V4.8.12

V4.8.14

V4.8.16

V4.8.∞

Truncated cuboctahedral graph

Truncated cuboctahedral graph
4-fold symmetry
Vertices 48
Edges 72
Automorphisms 48
Chromatic number 2
Properties Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated cuboctahedral graph (or great rhombcuboctahedral graph) is the graph of vertices and edges of the truncated cuboctahedron, one of the Archimedean solids. It has 48 vertices and 72 edges, and is a zero-symmetric and cubic Archimedean graph.[6]

See also

References

  1. ^   (Model 15, p. 29)
  2. ^   (Section 3-9, p. 82)
  3. ^ Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). (p. 82)
  4. ^ B. M. Stewart, Adventures Among the Toroids (1970) ISBN 978-0-686-11936-4
  5. ^ http://www.doskey.com/polyhedra/Stewart05.html
  6. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269 
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids.  

External links

  • Eric W. Weisstein, Great rhombicuboctahedron (Archimedean solid) at MathWorld
  • Richard Klitzing, 3D convex uniform polyhedra, x3x4x - girco
  • Editable printable net of a truncated cuboctahedron with interactive 3D view
  • The Uniform Polyhedra
  • Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • great Rhombicuboctahedron: paper strips for plaiting
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