Truncated tetrahedron

Truncated tetrahedron
Type	Archimedean solid, Uniform polyhedron, Goldberg polyhedron
Faces	4 hexagons 4 triangles
Edges	18
Vertices	12
Symmetry group	tetrahedral symmetry ${\displaystyle \mathrm {T} _{\mathrm {h} }}$
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.

Construction

The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as truncation. ^[1] The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. ^[2] With edge length 1, the Cartesian coordinates of the 12 vertices are points

$${\displaystyle {\bigl (}{\pm {\tfrac {3{\sqrt {2}}}{4}}},\pm {\tfrac {\sqrt {2}}{4}},\pm {\tfrac {\sqrt {2}}{4}}{\bigr )}}$$

that have an even number of minus signs.

Properties

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

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°. ^[3]

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) harvtxt error: no target: CITEREFJiaoTorquato2013 (help). ^{[4] [5]} 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. ^[4]

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. ^[6] The truncated tetrahedron has the same three-dimensional group symmetry as the regular tetrahedron, the tetrahedral symmetry ${\displaystyle \mathrm {T} _{\mathrm {h} }}$. ^[7] The polygonal faces that meet for every vertex are one equilateral triangle and two regular hexagons, and the vertex figure is denoted as ${\displaystyle 3\cdot 6^{2}}$. Its dual polyhedron is triakis tetrahedron, a Catalan solid, shares the same symmetry as the truncated tetrahedron. ^[8]

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. ^[9] 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. ^[10]

The Friauf polyhedron is named after J. B. Friauf in which he described it as a intermetallic structure formed by a compound of metallic elements. ^[11] It can be found in crystals such as complex metallic alloys, an example is dizinc magnesium MgZn₂. ^[12] It is a lower symmetry version of the truncated tetrahedron, interpreted as a truncated tetragonal disphenoid with its three-dimensional symmetry group as the dihedral group ${\displaystyle D_{2\mathrm {d} }}$ of order 8.^{[ citation needed ]}

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. ^[1]

Truncated tetrahedral graph

The graph of a truncated tetrahedron

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. ^[13] It is a connected cubic graph, ^[14] and connected cubic transitive graph. ^[15]

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 ${\displaystyle (4,2,0,0)}$ (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
• Truncated Triangular Pyramid Number

References

1. Kuchel, Philip W. (2012). "96.45 Can you 'bend' a truncated truncated tetrahedron?". The Mathematical Gazette . 96 (536): 317–323. JSTOR   23248575.
2. Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR   0290245.
3. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics . 18: 169–200. doi: 10.4153/cjm-1966-021-8 . MR   0185507. S2CID   122006114. Zbl   0132.14603. See line II.
4. Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". ACS Nano. 6 (2012): 609–614. arXiv: 1109.1323 . doi:10.1021/nn204012y. PMID   22098586. S2CID   12785227.
5. Jiao, Yang; Torquato, Sal (2011). "A Packing of Truncated Tetrahedra that Nearly Fills All of Space". arXiv: 1107.2300 [cond-mat.soft].
6. Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN   978-3-319-64123-2.
7. Koca, M.; Koca, N. O. (2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27–31 October 2010. World Scientific. p. 46–47.
8. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 72. ISBN   978-0-486-23729-9.
9. Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN   978-93-86279-06-4.
10. Grünbaum, Branko; Shephard, G. C. (1980). "Tilings with congruent tiles". Bulletin of the American Mathematical Society . New Series. 3 (3): 951–973. doi: 10.1090/S0273-0979-1980-14827-2 . MR   0585178.
11. Friauf, J. B. (1927). "The crystal structure of the intermetallic compounds". Journal of the American Chemical Society . 49 (12): 3107–3114. doi:10.1021/ja01411a017.
12. Lalena, John N.; Cleary, David A.; Duparc, Olivier B. (2020). Principles of Inorganic Materials Design. John Wiley & Sons. p. 150. ISBN   9781119486916.
13. An Atlas of Graphs, page 267, truncated tetrahedral graph
14. An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105
15. An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11

• 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