Truncated tetrahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Truncated tetrahedron
Truncated tetrahedron
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 8, E = 18, V = 12 (χ = 2)
Faces by sides 4{3}+4{6}
Conway notation tT
Schläfli symbols t{3,3} = h2{4,3}
t0,1{3,3}
Wythoff symbol 2 3 | 3
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry group Td, A3, [3,3], (*332), order 24
Rotation group T, [3,3]+, (332), order 12
Dihedral Angle 3-6:109°28'16"
6-6:70°31'44"
References U02, C16, W6
Properties Semiregular convex
Truncated tetrahedron.png
Colored faces
Truncated tetrahedron
3.6.6
(Vertex figure)
Triakistetrahedron.jpg
Triakis tetrahedron
(dual polyhedron)
Truncated tetrahedron flat.svg
Net

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.

A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.[1]

A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.

A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, having half of the vertices of the cantellated cube (rhombicuboctahedron), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png. There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.

Area and volume[edit]

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

A = 7\sqrt{3}a^2 \approx 12.12435565a^2
V = \frac{23}{12}\sqrt{2}a^3 \approx 2.710575995a^3.

Densest Packing[edit]

The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using Monte Carlo methods.[2][3] Although no mathematical proof exists that this is the best possible packing for those shapes, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.[2]

Cartesian coordinates[edit]

Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:

  • (+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
  • (−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
  • (−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
  • (+3,−1,−1), (+1,−3,−1), (+1,−1,−3)
Truncated tetrahedron in unit cube.png Triangulated truncated tetrahedron.png UC54-2 truncated tetrahedra.png
Orthogonal projection showing Cartesian coordinates inside it bounding box: (±3,±3,±3). The hexagonal faces of the truncated tetrahedra can be divided into 6 coplanar equilateral triangles. The 4 new vertices have Cartesian coordinates:
(-1,-1,-1), (-1,+1,+1),
(+1,-1,+1), (+1,+1,-1).
The set of vertex permutations (±1,±1,±3) with an odd number of minus signs forms a complementary truncated tetrahedron, and combined they form a uniform compound polyhedron.

Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of:

(0,0,1,2)

Orthogonal projection[edit]

Orthogonal projection
Centered by Edge normal Face normal Edge Face/vertex
Image Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01.svg 3-simplex t01 A2.svg
Dual image Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01.png Dual tetrahedron t01 A2.png
Projective
symmetry
[1] [1] [3] [4]

Spherical tiling[edit]

The truncated tetrahedron 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 332-t12.png Truncated tetrahedron stereographic projection triangle.png
triangle-centered
Truncated tetrahedron stereographic projection hexagon.png
hexagon-centered
Orthographic projection Stereographic projections

Related polyhedra and tilings[edit]

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.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 fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg POV-Ray-Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3
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 POV-Ray-Dodecahedron.svg
Dimensional family of cantic polyhedra and tilings: 3.6.n.6
Symmetry
*n32
[1+,2n,3]
= [(n,3,3)]
Spherical Planar Compact Hyperbolic Paracompact
*332
[1+,4,3]
Td
*333
[1+,6,3]
P3m1
*433
[1+,8,3]
= [(4,3,3)]
*533
[1+,10,3]
= [(5,3,3)]
*633
[1+,12,3]...
= [(6,3,3)]
*∞33
[1+,∞,3]
= [(∞,3,3)]
Cantic
figure
Uniform polyhedron-33-t12.png
3.6.2.6
Uniform tiling 333-t12.png
3.6.3.6
H2 tiling 334-6.png
3.6.4.6
H2 tiling 335-6.png
3.6.5.6
H2 tiling 336-6.png
3.6.6.6
H2 tiling 33i-6.png
3.6.∞.6
Coxeter
Schläfli
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{4,3}
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{6,3}
= CDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{8,3}
= CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{10,3}
= CDel label5.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{12,3}
= CDel label6.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{∞,3}
= CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
Dual figure Triakistetrahedron.jpg
V3.6.2.6
Rhombic star tiling.png
V3.6.3.6
Uniform dual tiling 433-t12.png
V3.6.4.6

V3.6.5.6

V3.6.6.6

V3.6.∞.6
Coxeter CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png

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.

Dimensional family of truncated polyhedra and tilings: 3.2n.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]
 
Truncated
figures
Spherical triangular prism.png
3.4.4
Uniform tiling 332-t01-1-.png
3.6.6
Uniform tiling 432-t01.png
3.8.8
Uniform tiling 532-t01.png
3.10.10
Uniform tiling 63-t01.png
3.12.12
Uniform tiling 73-t01.png
3.14.14
Uniform tiling 83-t01.png
3.16.16
H2 tiling 23i-3.png
3.∞.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
t{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
t{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
t{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
t{∞,3}
Uniform dual figures
Triakis
figures
Triangular dipyramid.png
V3.4.4
Triakistetrahedron.jpg
V3.6.6
Triakisoctahedron.jpg
V3.8.8
Triakisicosahedron.jpg
V3.10.10
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
V3.12.12
Ord7 triakis triang til.png
V3.14.14
Ord8 triakis triang til.png
V3.16.16
Ord-infin triakis triang til.png
V3.∞.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png

Friauf polyhedron[edit]

A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72 degree dihedral angle on a subset of 6-6 edges.[4] Its named after J. B. Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. [5]

Use in architecture[edit]

Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.[6]

See also[edit]

References[edit]

  1. ^ Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". theory.org. Retrieved 2013-09-02. 
  2. ^ a b "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". Dec 2011. 
  3. ^ "A Packing of Truncated Tetrahedra that Nearly Fills All of Space". Sep 2011. 
  4. ^ http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
  5. ^ Friauf, J. B. The crystal structure of the intermetallic compound MgCu2 (1927) J. Am. Chem. Soc. 19, 3107-3114.
  6. ^ http://expo67.ncf.ca/man_the_producer_p1.html

External links[edit]