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 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, BC3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral Angle 4-6:cos(-sqrt(6)/3)=144°44'08"
4-8:cos(-sqrt(2)/3)=135°
6-8:cos(-sqrt(3)/3)=125°15'51"
References U11, C23, W15
Properties Semiregular convex zonohedron
Great rhombicuboctahedron.png
Colored faces
Truncated cuboctahedron
4.6.8
(Vertex figure)
Disdyakisdodecahedron.jpg
Disdyakis dodecahedron
(dual polyhedron)
Truncated cuboctahedron flat.svg
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.

Other names[edit]

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[edit]

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[edit]

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[edit]

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
Excavated truncated cuboctahedron4.png Excavated truncated cuboctahedron2.png Excavated truncated cuboctahedron3.png Excavated truncated cuboctahedron.png

Uniform colorings[edit]

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[edit]

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 Cube t012 v.png Cube t012 e46.png Cube t012 e48.png Cube t012 e68.png Cube t012 f46.png
Projective
symmetry
[2]+ [2] [2] [2] [2]
Centered by Face normal
Square
Face normal
Octagon
Face
Square
Face
Hexagon
Face
Octagon
Image Cube t012 af4.png Cube t012 af8.png Cube t012 f4.png 3-cube t012.svg 3-cube t012 B2.svg
Projective
symmetry
[2] [2] [2] [6] [8]

Spherical tiling[edit]

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.

Uniform tiling 432-t012.png Truncated cuboctahedron stereographic projection square.png
square-centered
Truncated cuboctahedron stereographic projection hexagon.png
hexagon-centered
Truncated cuboctahedron stereographic projection octagon.png
octagon-centered
Orthogonal projection Stereographic projections

Related polyhedra[edit]

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)
[1+,4,3] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. 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.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{∞,3}
Omnitruncated
figure
Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Omnitruncated
duals
Hexagonale bipiramide.png Tetrakishexahedron.jpg Disdyakisdodecahedron.jpg Disdyakistriacontahedron.jpg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Order-3 heptakis heptagonal tiling.png Order-3 octakis octagonal tiling.png H2checkers 23i.png
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞
Dimensional family of omnitruncated polyhedra and tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h
*342
[3,4]
Oh
*442
[4,4]
P4m
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
Spherical octagonal prism2.png
4.8.4
Uniform tiling 432-t012.png
4.8.6
Uniform tiling 44-t012.png
4.8.8
H2 tiling 245-7.png
4.8.10
H2 tiling 246-7.png
4.8.12
H2 tiling 247-7.png
4.8.14
H2 tiling 248-7.png
4.8.16
H2 tiling 24i-7.png
4.8.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{2,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{3,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{4,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{5,4}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{6,4}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{7,4}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{8,4}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{∞,4}
Omnitruncated
duals
Hexagonale bipiramide.png
V4.8.4
Disdyakisdodecahedron.jpg
V4.8.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 bisected pentagonal tiling.png
V4.8.10
Hyperbolic domains 642.png
V4.8.12
Hyperbolic domains 742.png
V4.8.14
Hyperbolic domains 842.png
V4.8.16
H2checkers 24i.png
V4.8.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png

See also[edit]

References[edit]

  1. ^ Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR 0467493  (Model 15, p. 29)
  2. ^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (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
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

External links[edit]