![Truncated tetrahedron](https://upload.wikimedia.org/wikipedia/commons/1/13/Truncated_tetrahedron.stl)
In
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 c ...
, the truncated tetrahedron is an
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
. It has 4 regular
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
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 othe ...
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 the o ...
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
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
. 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 to ...
, , 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 eac ...
), . 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). The de ...
''V'' of a truncated tetrahedron of edge length ''a'' are:
:
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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the 12 vertices of a
truncated 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 the o ...
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
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. 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 Face (geometry), faces are Congruence (geometry), congruent acute-angled triangles. It can also be described as a tetrahedron in which every two Edge (geometry), edges that are opposite ea ...
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 cl ...
. 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 uni ...
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 MgCu
2".
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 compa ...
puzzle has a truncated tetrahedral shape. This puzzle shows a
dissection
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause o ...
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 at ea ...
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.
:
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 conne ...
, a truncated tetrahedral graph is an
Archimedean graph
In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vert ...
, the
graph of vertices and edges of the truncated tetrahedron, one of the
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s. 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
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 visua ...
s.
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform
truncated polyhedra with
s (3.2''n''.2''n''), and
'n'',3Coxeter 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 refl ...
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
See also
*
Quarter cubic honeycomb
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is c ...
– Fills space using truncated tetrahedra and smaller tetrahedra
*
Truncated 5-cell
In geometry, a truncated 5-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 5-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 5-cell
The truncated 5-cell, tr ...
– Similar uniform polytope in 4-dimensions
*
Truncated triakis tetrahedron
In geometry, the truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps.
Cons ...
*
Triakis truncated tetrahedron
In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb.
The triakis truncate ...
*
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 rectified tetrahedron
References
* (Section 3-9)
*
External links
*
**
*
Editable printable net of a truncated tetrahedron with interactive 3D viewThe Uniform PolyhedraThe Encyclopedia of Polyhedra
{{Polyhedron navigator
Archimedean solids
Truncated tilings
Individual graphs
Planar graphs