Truncated octahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=14, E=36, V=24 (χ=2)
Faces by sides	6{4}+8{6}
Schläfli symbol	t{3,4}
Wythoff symbol	2 4 | 3 3 3 2 |
Coxeter-Dynkin
Symmetry	Oh
References	U08, C20, W7
Properties	Semiregular convex zonohedron
Colored faces	4.6.6 (Vertex figure)
Tetrakis hexahedron (dual polyhedron)	Net

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 regular square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... // List of symmetry groups on the sphere Spherical symmetry groups are also called point groups (in 3D). ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... H.S.M. Coxeter. ... This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ... A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Truncated_octahedron_vertfig. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Download high resolution version (767x737, 72 KB)Tetrakis hexahedron, made by me using POV-Ray, see image:poly. ... A tetrakis hexahedron is a Catalan solid which looks a bit like an overinflated cube. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... Image File history File links Flattened polyhedron, source is image:makepoly. ... Categories: Polyhedra | Stub ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... The symmetry group of an object (e. ... A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic...

Coordinates and Permutations

All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes. Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... A truncated cube - faces double in sides, and vertices replaced by new faces. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1,2,3,4) form the vertices of a truncated octahedron in the three-dimensional subspace x+y+z+w=10. For this reason the truncated octahedron is also sometimes known as the permutohedron. The permutohedron of order 4. ...

Area and volume

The area A and the volume V of a truncated octahedron of edge length a are: The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

Uniform colorings

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry: The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Chiral and achiral tetrahedral symmetry and pyritohedral symmetry are discrete point symmetries (or equivalently, symmetries on the sphere). ... The octahedral rotation group O with fundamental domain Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. ...

122 coloring Oh symmetry Wythoff: 2 4 | 3

123 coloring Th symmetry Wythoff: 3 3 2 |

Image File history File links Download high-resolution version (800x800, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated octahedron I, the creator of this work, hereby release it into the public domain. ... In geometry, a Wythoff construction, named after mathematician Willem Abraham Wijthoff, is a method for constructing a uniform polyhedron or plane tiling. ... Image File history File links Download high-resolution version (800x800, 20 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated octahedronahedron as omnitruncated tetrahedron I, the creator of this work, hereby release it into the public domain. ...

Related polyhedra

The truncated octahedron exists within the set of truncated forms between a cube and octahedron: A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Cube

Truncated cube

cuboctahedron

Truncated octahedron

Octahedron

Image File history File links Download high-resolution version (800x800, 14 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Cube I, the creator of this work, hereby release it into the public domain. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (799x800, 16 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated cube I, the creator of this work, hereby release it into the public domain. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (800x800, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Cuboctahedron I, the creator of this work, hereby release it into the public domain. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (800x800, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated octahedron I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (800x800, 18 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks octahedron as dual to cube I, the creator of this work, hereby release it into the public domain. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Tessellations

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations): In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ...

Bitruncated cubic

Cantitruncated cubic

Truncated alternated cubic

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centred cubic lattice. Image File history File links Download high-resolution version (868x848, 43 KB) Image for Bitruncated_cubic_honeycomb - 2 colored truncated octahedron cells for original cells and vertices. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... Image File history File links Cantitruncated_cubic_honeycomb. ... The cantitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Truncated_alternated_cubic_honeycomb. ... The truncated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... The rhombic dodecahedral honeycomb is an example of a cell-transitive space-filling tessellation. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... This is the Voronoi diagram of a random set of points in the plane (all points lie within the image). ... Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...

References

• 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)
• Freitas, Robert A., Jr. Uniform space-filling using only truncated octahedra. Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. Retrieved on 2006-09-08.

• Gaiha, P., and Guha, S. K. (1977). "Adjacent vertices on a permutohedron". SIAM Journal on Applied Mathematics 32 (2): 323–327. 

• Hart, George W. VRML model of truncated octahedron. Virtual Polyhedra: The Encyclopedia of Polyhedra. Retrieved on 2006-09-08.

• Mäder, Roman. The Uniform Polyhedra: Truncated Octahedron. Retrieved on 2006-09-08.

• Weisstein, Eric W. Permutohedron. MathWorld–A Wolfram Web Resource. Retrieved on 2006-09-08.

This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 251st day of the year (252nd in leap years) in the Gregorian calendar. ...

External links

• Eric W. Weisstein, Truncated octahedron at MathWorld.