geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the truncated tetrahedron is an Archimedean solid. It has 4 regular

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

al faces, 4

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

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 In geometry, an octahedron (plural: octahedra, octahedrons) 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 ea ...

. A ''truncated tetrahedron'' is the

Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three pro ...

containing triangular and hexagonal faces. A ''truncated tetrahedron'' can be called a cantic cube, with

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, , having half of the vertices of the cantellated cube (

rhombicuboctahedron 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 ea ...

), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.

Area and volume

The area ''A'' and the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

''V'' of a truncated tetrahedron of edge length ''a'' are: :

\begin
A &= 7\sqrta^2 &&\approx 12.124\,355\,65a^2 \\
V &= \tfrac\sqrta^3 &&\approx 2.710\,575\,995a^3. \end

Densest packing

The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = , as reported by two independent groups using

Monte Carlo methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...

. 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.

Cartesian coordinates

Cartesian coordinates for the 12 vertices of a truncated

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) Another simple construction exists in 4-space as cells of the

truncated 16-cell In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the ''truncated 16-cell''. Truncated tesserac ...

, with vertices as coordinate permutation of: :(0,0,1,2)

Orthogonal projection

Spherical tiling

The truncated tetrahedron can also be represented as a

spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...

, 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.

Friauf polyhedron

A lower symmetry version of the truncated tetrahedron (a truncated

tetragonal disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...

with order 8 D_2d symmetry) is called a Friauf polyhedron in crystals such as

complex metallic alloys Complex metallic alloys (CMAs) or complex intermetallics (CIMs) are intermetallic compounds characterized by the following structural features: #large unit cells, comprising some tens up to thousands of atoms, #the presence of well-defined atom c ...

. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

on a subset of 6-6 edges. It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu₂".

Uses

Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in

Expo 67 The 1967 International and Universal Exposition, commonly known as Expo 67, was a general exhibition from April 27 to October 29, 1967. It was a category One World's Fair held in Montreal, Quebec, Canada. It is considered to be one of the most su ...

. 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. The

Tetraminx The Pyraminx () is a regular tetrahedron puzzle in the style of Rubik's Cube. It was made and patented by Uwe Mèffert after the original 3 layered Rubik's Cube by Ernő Rubik, and introduced by Tomy Toys of Japan (then the 3rd largest toy comp ...

puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a truncated tetrahedron into 4

octahedra In geometry, an octahedron (plural: octahedra, octahedrons) 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 a ...

and 6

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

. It contains 4 central planes of rotations. : Tetraminx

Truncated tetrahedral graph

In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, 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. It is a connected cubic graph, and connected cubic transitive graph.An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11

Related polyhedra and tilings

It is also a part of a sequence of cantic polyhedra and tilings with

vertex configuration In geometry, a vertex configurationCrystallography ...

3.6.''n''.6. In this

wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

the edges between the hexagons represent degenerate

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

Symmetry mutations

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with

vertex configuration In geometry, a vertex configurationCrystallography ...

s (3.2''n''.2''n''), and 'n'',3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

symmetry.

Examples

File:Truncatedtetrahedron.gif, Truncated tetrahedron in rotation File:Tetraedro truncado (Matemateca IME-USP).jpg, Truncated tetrahedron ( Matemateca IME-USP) File:D4 truncated tetrahedron.JPG, Truncated 4-sided die

References

* (Section 3-9) *

External links

* ** *
Editable printable net of a truncated tetrahedron with interactive 3D viewThe Uniform Polyhedra
The Encyclopedia of Polyhedra {{Polyhedron navigator Archimedean solids Truncated tilings Individual graphs Planar graphs