Truncated cuboctahedron

The name truncated cuboctahedron, given originally by Johannes Kepler, is misleading: an actual truncation of a cuboctahedron has rectangles instead of squares; however, this nonuniform polyhedron is topologically equivalent to the Archimedean solid unrigorously named truncated cuboctahedron. Alternate interchangeable names are: Truncated cuboctahedron (Johannes Kepler), Rhombitruncated cuboctahedron (Magnus Wenninger{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Polyhedron Models |publisher=Cambridge University Press |isbn=978-0-521-09859-5 |mr=0467493 |year=1974}} (Model 15, p. 29)), Great rhombicuboctahedron (Robert Williams{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9, p. 82)), Great rhombcuboctahedron (Peter CromwellCromwell, P.; [https://books.google.com/books?id=OJowej1QWpoC&pg=PA82 Polyhedra], CUP hbk (1997), pbk. (1999). (p. 82)), Omnitruncated cube or cantitruncated cube (Norman Johnson), Beveled cube (Conway polyhedron notation). |style="padding-left:50px;"| {{multiple image | align = left | width = 130 | image1 = Polyhedron 6-8 max.png | image2 = Polyhedron nonuniform truncated 6-8 max.png | footer = Cuboctahedron and its truncation }}

There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron.

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

$$\backslash Bigl(\backslash pm\; 1,\; \backslash quad\; \backslash pm\backslash left(1\; +\; \backslash sqrt\; 2\backslash right),\; \backslash quad\; \backslash pm\backslash left(1\; +\; 2\backslash sqrt\; 2\backslash right)\; \backslash Bigr).$$

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

:$\backslash begin\{align\}$

A &= 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 &&\approx 61.755\,1724~a^2, \\

V &= \left(22+14\sqrt{2}\right) a^3 &&\approx 41.798\,9899~a^3. \end{align}

{{multiple image

| align = left | total_width = 300

| image1 = Small in great rhombi 6-8, davinci small with cubes.png |width1=3944|height1=3876

| image2 = Small in great rhombi 6-8, davinci.png |width2=3944|height2=3876

}}

The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons, and 8 triangular cupolas below the hexagons.

A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components. For example, removing 4 of the triangular cupolas creates a genus 3 toroid; if these cupolas are appropriately chosen, then this toroid has tetrahedral symmetry.B. M. Stewart, Adventures Among the Toroids (1970) {{isbn|978-0-686-11936-4}}{{cite web|url=http://www.doskey.com/polyhedra/Stewart05.html|title=Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1|first=Alex|last=Doskey|website=www.doskey.com}}

class="wikitable collapsible collapsed" !colspan="4"| Stewart toroids
Genus 3 !Genus 5 !Genus 7 !Genus 11
160px |160px |160px |160px

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.

{{multiple image

| align = right | width = 120 | direction=vertical

| image1 = Polyhedron great rhombi 6-8 from blue max.png

| image2 = Polyhedron great rhombi 6-8 from yellow max.png

| image3 = Polyhedron great rhombi 6-8 from red max.png

}}

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.

class=wikitable |+ Orthogonal projections
Centered by !Vertex !Edge 4-6 !Edge 4-8 !Edge 6-8 !Face normal 4-6
Image |100px |100px |100px |100px |100px
align=center !Projective symmetry |[2]+ |[2] |[2] |[2] |[2]
Centered by !Face normal Square !Face normal Octagon !Face Square !Face Hexagon !Face Octagon
Image |100px |100px |100px |100px |100px
align=center !Projective symmetry |[2] |[2] |[2] |[6] |[4]

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.

class=wikitable
align=center valign=top |155px |165px |152px |160px
rowspan=2|Orthogonal projection !square-centered||hexagon-centered||octagon-centered
colspan=3|Stereographic projections

File:Full octahedral group elements in truncated cuboctahedron; JF.png

Like many other solids the truncated octahedron has full octahedral symmetry - but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of its dual is a fundamental domain of the group.

The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.

The edges of the solid correspond to the 9 reflections in the group:

• Those between octagons and squares correspond to the 3 reflections between opposite octagons.
• Hexagon edges correspond to the 6 reflections between opposite squares.
• (There are no reflections between opposite hexagons.)

The subgroups correspond to solids that share the respective vertices of the truncated octahedron.

E.g. the 3 subgroups with 24 elements correspond to a nonuniform snub cube with chiral octahedral symmetry, a nonuniform rhombicuboctahedron with pyritohedral symmetry (the cantic snub octahedron) and a nonuniform truncated octahedron with full tetrahedral symmetry. The unique subgroup with 12 elements is the alternating group A₄. It corresponds to a nonuniform icosahedron with chiral tetrahedral symmetry.

class="wikitable collapsible" style="text-align: center;" !colspan="5"| Subgroups and corresponding solids
valign=top !Truncated cuboctahedron {{CDD|node_1|4|node_1|3|node_1}} tr{4,3} !Snub cube {{CDD|node_h|4|node_h|3|node_h}} sr{4,3} !Rhombicuboctahedron {{CDD|node_1|4|node_h|3|node_h}} s2{3,4} !Truncated octahedron {{CDD|node_h|4|node_1|3|node_1}} h1,2{4,3} !Icosahedron {{CDD|node_h|2|4|2|node_h|3|node_h}}
[4,3] Full octahedral ![4,3]+ Chiral octahedral ![4,3+] Pyritohedral ![1+,4,3] = [3,3] Full tetrahedral ![1+,4,3+] = [3,3]+ Chiral tetrahedral
150px | 150px | 150px | 150px | 150px
all 48 vertices |colspan="3"| 24 vertices | 12 vertices

class=wikitable align=right width=320 |160px |160px

colspan=2|Bowtie tetrahedron and cube contain two trapezoidal faces in place of each square.[http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] Craig S. Kaplan

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

{{Octahedral truncations}}

This polyhedron can be considered a member of a sequence of uniform patterns with vertex configuration (4.6.2p) and Coxeter-Dynkin diagram {{CDD|node_1|p|node_1|3|node_1}}. 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.

{{Omnitruncated table}}

{{Omnitruncated4 table}}

{{Omnitruncated34 table}}

It is first in a series of cantitruncated hypercubes:

{{Cantitruncated hypercube polytopes}}

{{Infobox graph

| name = Truncated cuboctahedral graph

| image = 240px

| image_caption = 4-fold symmetry

| namesake =

| vertices = 48

| edges = 72

| automorphisms = 48

| radius =

| diameter =

| girth =

| chromatic_number = 2

| chromatic_index =

| fractional_chromatic_index =

| 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.{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998|page=269}}

{{Clear}}

{{Commons category|Truncated cuboctahedron}}

• Cube
• Cuboctahedron
• Octahedron
• Truncated icosidodecahedron
• Truncated octahedron – truncated tetratetrahedron
• Snub cube

{{Reflist}}

• {{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 Archimedean solids|isbn=0-521-55432-2}}

• {{mathworld2 |urlname=GreatRhombicuboctahedron |title=Great rhombicuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
• {{mathworld |urlname=GreatRhombicuboctahedralGraph |title=Great rhombicuboctahedral graph}}
• {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4x - girco}}
• [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=K1RB4IgXRIYFsRCy4aHC6dnfSy7rq2q4f3v4yl4nRAbEtZul3CvUo1k8TrBjkgTqbZvaXjrwBrfylwUDWC8lfzMM1h1E0NDKFJh7udiMaOsp1DWdM1xR11UHzHVHzVTY4gCc1nRXW879CvTzcUwX9wUb4vJUXWfCEBFZvcR1dJrJyE4FpNhyWBHvw6Qyuph3rT9jgDZRKNn8FaEqBr5dZ1o9SBpnEObkMMx31INNrEf75EZOWGdEXwvDNeJIuKbtorvRJHy4iYcMHnYO1v1axRBQudr31fS0np5Jn7nuzR0Vx&name=Truncated+Cuboctahedron#applet Editable printable net of a truncated cuboctahedron with interactive 3D view]
• [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
• [http://www.faust.fr.bw.schule.de/mhb/flechten/grko/indexeng.htm Great rhombicuboctahedron: paper strips for plaiting]

{{Archimedean solids}}

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