Truncated tetrahedron | |
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(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 | = |
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 | U 02, C 16, W 6 |
Properties | Semiregular convex |
Colored faces | 3.6.6 (Vertex figure) |
Triakis tetrahedron (dual polyhedron) | 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, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.
The area A and the volume V of a truncated tetrahedron of edge length a are:
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 the truncated tetrahedron, 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 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:
Orthogonal projection showing Cartesian coordinates inside its bounding box: (±3,±3,±3). | The hexagonal faces of the truncated tetrahedra can be divided into six coplanar equilateral triangles. The four new vertices have Cartesian coordinates: (−1,−1,−1), (−1,+1,+1), (+1,−1,+1), (+1,+1,−1). As a solid, this can represent a 3D dissection, making four red octahedra and six yellow tetrahedra. | 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:
Centered by | Edge normal | Face normal | Edge | Face |
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Wireframe | ||||
Wireframe | ||||
Dual | ||||
Projective symmetry | [1] | [1] | [4] | [3] |
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.
triangle-centered | hexagon-centered | ||
Orthographic projection | Stereographic projections |
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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] It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2". [5]
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]
The Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a truncated tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations.[ citation needed ]
Truncated tetrahedral graph | |
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Vertices | 12 [7] |
Edges | 18 |
Radius | 3 |
Diameter | 3 [7] |
Girth | 3 [7] |
Automorphisms | 24 (S4) [7] |
Chromatic number | 3 [7] |
Chromatic index | 3 [7] |
Properties | Hamiltonian, regular, 3-vertex-connected, planar graph |
Table of graphs and parameters |
In the mathematical field of graph theory, a truncated tetrahedral graph is an Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges. [8] It is a connected cubic graph, [9] and connected cubic transitive graph. [10]
Circular |
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Orthographic projections | |
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4-fold symmetry | 3-fold symmetry |
Family of uniform tetrahedral polyhedra | |||||||
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Symmetry: [3,3], (*332) | [3,3]+, (332) | ||||||
{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 | |||||||
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 |
It is also a part of a sequence of cantic polyhedra and tilings with vertex configuration 3.6.n.6. In this wythoff construction the edges between the hexagons represent degenerate digons.
Orbifold *n32 | Spherical | Euclidean | Hyperbolic | Paracompact | |||
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*332 | *333 | *433 | *533 | *633... | *∞33 | ||
Cantic figure | |||||||
Vertex | 3.6.2.6 | 3.6.3.6 | 3.6.4.6 | 3.6.5.6 | 3.6.6.6 | 3.6.∞.6 |
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.
*n32 symmetry mutation of truncated spherical tilings: t{n,3} | |||||||||||
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Symmetry *n32 [n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | |||||||
*232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3]... | *∞32 [∞,3] | ||||
Truncated figures | |||||||||||
Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | |||
Triakis figures | |||||||||||
Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ |
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. 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, the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.
In geometry, a tetrahedrally diminished dodecahedron is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces.
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.