Truncated tetrahedron

Truncated tetrahedron
Type	Archimedean solid, Uniform polyhedron, Goldberg polyhedron
Faces	4 hexagons 4 triangles
Edges	18
Vertices	12
Symmetry group	tetrahedral symmetry $\mathrm {T} _{\mathrm {h} }$
Dual polyhedron	triakis tetrahedron
Vertex figure
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 truncated tetrahedron is the Goldberg polyhedron G_III(1,1), containing triangular and hexagonal faces.

Construction

The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. A Goldberg polyhedron is one whose faces are 12 pentagons and some multiple of 10 hexagons. There are three classes of Goldberg polyhedron, one of them is constructed by truncating all vertices repeatedly, and the truncated tetrahedron is one of them, denoted as $\mathrm {G} _{\mathrm {III} }(1,1)$.

Properties

Given the edge length $a$. The surface area of a truncated tetrahedron $A$ is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume $V$ is: $${\begin{aligned}A&=7{\sqrt {3}}a^{2}&&\approx 12.124a^{2},\\V&={\tfrac {23}{12}}{\sqrt {2}}a^{3}&&\approx 2.711a^{3}.\end{aligned}}$$

The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°.

The densest packing of the truncated tetrahedron is believed to be ${\textstyle \Phi ={\frac {207}{208}}}$, as reported by two independent groups using Monte Carlo methods by Damasceno, Engel & Glotzer (2012) and Jiao & Torquato (2013). Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independence 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 a truncated tetrahedron, this new shape can be used to completely fill space.

3D model of a truncated tetrahedron

The truncated tetrahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. The truncated tetrahedron has the same three-dimensional group symmetry as the regular tetrahedron, the tetrahedral symmetry $\mathrm {T} _{\mathrm {h} }$. The polygonal faces that meet for every vertex are one equilateral triangle and two regular hexagons, and the vertex figure is denoted as $3\cdot 6^{2}$. Its dual polyhedron is triakis tetrahedron, a Catalan solid, shares the same symmetry as the truncated tetrahedron.

Related polyhedrons

A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D_2d 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. It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu₂".

Truncated tetrahedral graph

Truncated tetrahedral graph
3-fold symmetry
Vertices	12
Edges	18
Radius	3
Diameter	3
Girth	3
Automorphisms	24 (S4)
Chromatic number	3
Chromatic index	3
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. It is a connected cubic graph, and connected cubic transitive graph.

Circular

Orthographic projections
4-fold symmetry	3-fold symmetry

Examples

• 
  drawing in De divina proportione (1509)
• 
  drawing in Perspectiva Corporum Regularium (1568)
• 
  crystal model
• 
  photos from different perspectives (Matemateca)
• 
  4-sided die
• 
  12 permutations of $(4,2,0,0)$ (brown)

See also

• Quarter cubic honeycomb – Fills space using truncated tetrahedra and smaller tetrahedra
• Truncated 5-cell – Similar uniform polytope in 4-dimensions
• Truncated triakis tetrahedron
• Triakis truncated tetrahedron
• Octahedron – a rectified tetrahedron

References

• 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)
• Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press

External links

• Weisstein, Eric W., "Truncated tetrahedron" ("Archimedean solid") at MathWorld.
  • Weisstein, Eric W. "Truncated tetrahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x3o - tut".
• Editable printable net of a truncated tetrahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra