Truncated tetrahedron explained

Truncated tetrahedron
Faces:	4 hexagons 4 triangles
Edges:	18
Vertices:	12
Symmetry:	tetrahedral symmetry T _{h}
Vertex Figure:	Polyhedron truncated 4a vertfig.svg
Net:	Polyhedron truncated 4a net.svg

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.

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

G _{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:

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 $\backslash Phi\; =\; \backslash frac$, as reported by two independent groups using Monte Carlo methods by and . 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. If the truncation of the corners is slightly smaller than that of a truncated tetrahedron, this new shape can be used to fill space completely.

T _{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 ⋅ 6 ^{2} . Its dual polyhedron is triakis tetrahedron, a Catalan solid, shares the same symmetry as the truncated tetrahedron.

Related polyhedrons

The truncated tetrahedron can be found in the construction of polyhedrons. For example, the augmented truncated tetrahedron is a Johnson solid constructed from a truncated tetrahedron by attaching triangular cupola onto its hexagonal face. The triakis truncated tetrahedron is a polyhedron constructed from a truncated tetrahedron by adding three tetrahedrons onto its triangular faces, as interpreted by the name "triakis". It is classified as plesiohedron, meaning it can tessellate in three-dimensional space known as honeycomb; an example is triakis truncated tetrahedral honeycomb.

D _{2d} of order 8.

Truncating a truncated tetrahedron gives the resulting polyhedron 54 edges, 32 vertices, and 20 faces - 4 hexagons, 4 nonagons, and 12 trapeziums. This polyhedron was used by Adidas as the underlying geometry of the Jabulani ball designed for the 2010 World Cup.

Truncated tetrahedral graph

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.^[1] It is a connected cubic graph,^[2] and connected cubic transitive graph.^[3]

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

External links

• • • • • Editable printable net of a truncated tetrahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra

Notes and References

1. An Atlas of Graphs, page 267, truncated tetrahedral graph
2. An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105
3. An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11