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How to Build a Half Truncated Cube
How to Build a Half Truncated Cube
Published: 2016/09/19
Channel: NathanWilson
Assembling a Truncated Cube
Assembling a Truncated Cube
Published: 2016/02/26
Channel: Go Get Some
Truncated Cube
Truncated Cube
Published: 2011/12/31
Channel: James Olsen
Make 3D Solid Shapes - Truncated Cube / Усеченный куб
Make 3D Solid Shapes - Truncated Cube / Усеченный куб
Published: 2013/06/30
Channel: ART KOSEKOMA
Half Truncated Cube
Half Truncated Cube
Published: 2010/11/14
Channel: RedKB
Half Truncated Cube Modding Process
Half Truncated Cube Modding Process
Published: 2015/08/17
Channel: TheProgrammingCuber
Half Truncated Cube Mod Review
Half Truncated Cube Mod Review
Published: 2015/08/05
Channel: TheProgrammingCuber
Truncated Cube Video Tutorial SolidWorks
Truncated Cube Video Tutorial SolidWorks
Published: 2013/11/05
Channel: CAD Gill
Super Cool Truncated Cube Pattern!
Super Cool Truncated Cube Pattern!
Published: 2017/10/30
Channel: Creekside Cubing
Half Truncated Cube Mod Tutorial! (Only Basic Tools Required)
Half Truncated Cube Mod Tutorial! (Only Basic Tools Required)
Published: 2017/05/26
Channel: OJ Cubing Essentials
I modded a cube and so can you - Half Truncated Cube
I modded a cube and so can you - Half Truncated Cube
Published: 2016/10/12
Channel: SpeedCubeReview
5 builds 5 hours #1 | Half Truncated Cube | Give Away (closed)
5 builds 5 hours #1 | Half Truncated Cube | Give Away (closed)
Published: 2017/07/23
Channel: OGRE Cubes
Tetra Pyramid Problem? or Not + Truncated Cube
Tetra Pyramid Problem? or Not + Truncated Cube
Published: 2012/08/19
Channel: IntellivisionDude
How to Make a Half Truncated Cube
How to Make a Half Truncated Cube
Published: 2016/05/02
Channel: Portalcuber
HALF TRUNCATED CUBE ИЗ КУБИКА 3х3
HALF TRUNCATED CUBE ИЗ КУБИКА 3х3
Published: 2017/11/03
Channel: AVIZORRA
4x4 Truncated Cube Unboxing
4x4 Truncated Cube Unboxing
Published: 2012/03/31
Channel: CubingUnited1
Stickering a Half Truncated Cube | Timelapse Part 1
Stickering a Half Truncated Cube | Timelapse Part 1
Published: 2016/04/23
Channel: Spotty.
Platonic Structures Series  - 21. Archimedean Solids - Truncated Cube from Spheres
Platonic Structures Series - 21. Archimedean Solids - Truncated Cube from Spheres
Published: 2017/11/07
Channel: Platonic Structures Inc.
3x3 Half-Truncated Cube Build Video (Reupload)
3x3 Half-Truncated Cube Build Video (Reupload)
Published: 2016/08/22
Channel: Dahmon's Puzzles
Building a 3x3 Triangular Anti-prism
Building a 3x3 Triangular Anti-prism
Published: 2016/10/14
Channel: NathanWilson
Half Truncated Cube
Half Truncated Cube
Published: 2009/01/27
Channel: RedKB
Half Truncated Cube | Part 2
Half Truncated Cube | Part 2
Published: 2016/05/07
Channel: Spotty.
Half truncated 3x3
Half truncated 3x3
Published: 2013/10/24
Channel: NathanWilson
Other Half Truncated Cube Patterns!
Other Half Truncated Cube Patterns!
Published: 2017/11/05
Channel: Creekside Cubing
3x3x3 Truncated Cube Review
3x3x3 Truncated Cube Review
Published: 2009/06/29
Channel: Daniel Santiago
Half-Truncated Cube
Half-Truncated Cube
Published: 2009/04/21
Channel: LanceTheBlueKnight
Half-Truncated Cube + Barrel Cube + Fisher Cube Relay
Half-Truncated Cube + Barrel Cube + Fisher Cube Relay
Published: 2012/05/14
Channel: NoahCubes
Truncated Cube
Truncated Cube
Published: 2008/08/10
Channel: C.Cube
Half truncated cube with corners (Rubik
Half truncated cube with corners (Rubik's 6 colours)
Published: 2008/07/22
Channel: Gábor Rudas
Extended Truncated Cube
Extended Truncated Cube
Published: 2008/09/17
Channel: kylerv3
Point group symmetry of truncated cube, part 4
Point group symmetry of truncated cube, part 4
Published: 2012/02/17
Channel: bhadeshia123
Tonles solving a Half truncated cube
Tonles solving a Half truncated cube
Published: 2010/09/11
Channel: tonism987
Half-Truncated Cube
Half-Truncated Cube
Published: 2009/07/10
Channel: CustomPuzzleLibrary
Pretzangles Truncated Cube 113
Pretzangles Truncated Cube 113
Published: 2015/08/12
Channel: Matthew Kells
Volume ofTruncated Cube and Volume of An Ice Cream Cone.wmv
Volume ofTruncated Cube and Volume of An Ice Cream Cone.wmv
Published: 2012/04/12
Channel: dinkinsk
My Second Mod: Half-Truncated Cube
My Second Mod: Half-Truncated Cube
Published: 2008/09/27
Channel: BWang92
half truncated cube solve
half truncated cube solve
Published: 2009/08/30
Channel: philliescuber8
Half Truncated Cube
Half Truncated Cube
Published: 2008/12/29
Channel: Traiphum Prungtaengikt
Half-Truncated Cube (Pocket Version)
Half-Truncated Cube (Pocket Version)
Published: 2010/04/03
Channel: Flyboy Montana
Half-Truncated Cube
Half-Truncated Cube
Published: 2008/10/27
Channel: Felix Lee
Custom Half Truncated Cube with Corners MOD.
Custom Half Truncated Cube with Corners MOD.
Published: 2007/05/21
Channel: ThePerson1000
Edge Truncated Cube (Zen Magnets)
Edge Truncated Cube (Zen Magnets)
Published: 2013/05/01
Channel: Dimitri Tishchenko
Half-Truncated Cube
Half-Truncated Cube
Published: 2007/08/03
Channel: Brandon Kane
Усечённый куб, Truncated Cube
Усечённый куб, Truncated Cube
Published: 2017/04/05
Channel: Mnogogranniki Ru
Vertex-Truncation Process Between Cube And Octahedron / Усечение куба
Vertex-Truncation Process Between Cube And Octahedron / Усечение куба
Published: 2013/03/17
Channel: ART KOSEKOMA
Half Truncated Cube custom made Rubik
Half Truncated Cube custom made Rubik's Cube Puzzle
Published: 2008/12/17
Channel: Thompson Clarke
Half-Truncated Cube
Half-Truncated Cube
Published: 2010/04/03
Channel: Flyboy Montana
truncated cube
truncated cube
Published: 2011/04/24
Channel: monkey134578
Half-Truncated Cube Solve
Half-Truncated Cube Solve
Published: 2008/10/27
Channel: Felix Lee
Half Truncated Cube
Half Truncated Cube
Published: 2016/11/10
Channel: Cubiverse
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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Truncated cube
Truncatedhexahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 14, E = 36, V = 24 (χ = 2)
Faces by sides 8{3}+6{8}
Conway notation tC
Schläfli symbols t{4,3}
t0,1{4,3}
Wythoff symbol 2 3 | 4
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry group Oh, B3, [4,3], (*432), order 48
Rotation group O, [4,3]+, (432), order 24
Dihedral angle 3-8: 125°15′51″
8-8: 90°
References U09, C21, W8
Properties Semiregular convex
Truncated hexahedron.png
Colored faces
Truncated cube vertfig.png
3.8.8
(Vertex figure)
Triakisoctahedron.jpg
Triakis octahedron
(dual polyhedron)
Truncated hexahedron flat.svg
Net

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + 2.

Area and volume[edit]

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

Orthogonal projections[edit]

The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
3-8
Edge
8-8
Face
Octagon
Face
Triangle
Truncated
cube
Cube t01 v.png Cube t01 e38.png Cube t01 e88.png 3-cube t01 B2.svg 3-cube t01.svg
Triakis
octahedron
Dual truncated cube t01 v.png Dual truncated cube t01 e8.png Dual truncated cube t01 e88.png Dual truncated cube t01 B2.png Dual truncated cube t01.png
Projective
symmetry
[2] [2] [2] [4] [6]

Spherical tiling[edit]

The truncated cube 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-t01.png Truncated cube stereographic projection octagon.png
octagon-centered
Truncated cube stereographic projection triangle.png
triangle-centered
Orthographic projection Stereographic projections

Cartesian coordinates[edit]

A truncated cube with its octagonal faces divided with a central vertex into triangles and pentagons, creating a topological icosidodecahedron

The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ:

ξ, ±1, ±1),
(±1, ±ξ, ±1),
(±1, ±1, ±ξ)

where ξ = 2 − 1.

The parameter ξ can be varied between ±1. A value of 1 produces a cube, 0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces.

Truncated cube sequence.png

If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, truncated octahedrons are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.

Dissection[edit]

Dissected truncated cube, with elements expanded apart

The truncated cube can be dissected into a central cube, with six square cupola around each of the cube's faces, and 8 regular tetrahedral in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.

This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupola and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.[1][2]

Excavated truncated cube.png

Vertex arrangement[edit]

It shares the vertex arrangement with three nonconvex uniform polyhedra:

Truncated hexahedron.png
Truncated cube
Uniform great rhombicuboctahedron.png
Nonconvex great rhombicuboctahedron
Great cubicuboctahedron.png
Great cubicuboctahedron
Great rhombihexahedron.png
Great rhombihexahedron

Related polyhedra[edit]

The truncated cube is related to other polyhedra and tlings in symmetry.

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

Symmetry mutations[edit]

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry, and a series of polyhedra and tilings n.8.8.

Alternated truncation[edit]

A cube can be alternately truncated producing tetrahedral symmetry, with six hexagonal faces, and four triangles at the truncated vertices. It is one of a sequence of alternate truncations of polyhedra and tilings.

Alternate truncated cube.png

Related polytopes[edit]

The truncated cube, is second in a sequence of truncated hypercubes:

Truncated hypercubes
Regular polygon 8 annotated.svg 3-cube t01.svgTruncated hexahedron.png 4-cube t01.svgSchlegel half-solid truncated tesseract.png 5-cube t01.svg5-cube t01 A3.svg 6-cube t01.svg6-cube t01 A5.svg 7-cube t01.svg7-cube t01 A5.svg 8-cube t01.svg8-cube t01 A7.svg ...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
CDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Truncated cubical graph[edit]

Truncated cubical graph
Truncated cubic graph.png
4-fold symmetry schlegel diagram
Vertices 24
Edges 36
Automorphisms 48
Chromatic number 3
Properties Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the truncated cube, one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph.[3]

3-cube t01.svg
Orthographic

See also[edit]

References[edit]

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

External links[edit]

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