truncated tetrahedron

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

\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.

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:{{r|berman}}

$$\backslash begin\{align\}$$

A &= 7\sqrt{3}a^2 &&\approx 12.124a^2, \\

V &= \tfrac{23}{12}\sqrt{2}a^3 &&\approx 2.711a^3.

\end{align}

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°.{{r|johnson}}

The densest packing of the truncated tetrahedron is believed to be $\backslash Phi\; =\; \backslash frac\{207\}\{208\}$, as reported by two independent groups using Monte Carlo methods by {{harvtxt|Damasceno|Engel|Glotzer|2012}} and {{harvtxt|Jiao|Torquato|2011}}.{{r|damasceno|jiao}} 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.{{r|damasceno}}

File:Truncated tetrahedron.stl

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

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.{{r|rajwade}} 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.{{r|grunbaum}}

File:Truncated triakis tetrahedron.gif

A truncated triakis tetrahedron is known for its usage in chemistry as a fullerene. This solid is represented as an allotrope of carbon (C₂₈), forming the smallest stable fullerene,{{cite journal

| last = Martin | first = Jan M.L.

| date = June 1996

| doi = 10.1016/0009-2614(96)00354-5

| issue = 1–3

| journal = Chemical Physics Letters

| pages = 1–6

| title = C₂₈: the smallest stable fullerene?

| volume = 255| bibcode = 1996CPL...255....1M

}} and experiments have found it to be stabilized by encapsulating a metal atom.{{cite journal

| last1 = Dunk | first1 = Paul W.

| last2 = Kaiser | first2 = Nathan K.

| last3 = Mulet-Gas | first3 = Marc

| last4 = Rodríguez-Fortea | first4 = Antonio

| last5 = Poblet | first5 = Josep M.

| last6 = Shinohara | first6 = Hisanori

| last7 = Hendrickson | first7 = Christopher L.

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| date = May 2012

| doi = 10.1021/ja302398h

| issue = 22

| journal = Journal of the American Chemical Society

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| publisher = American Chemical Society (ACS)

| title = The Smallest Stable Fullerene, M@C₂₈ (M = Ti, Zr, U): Stabilization and Growth from Carbon Vapor

| volume = 134| pmid = 22519801

| bibcode = 2012JAChS.134.9380D

}} Geometrically, this polyhedron was studied in 1935 by Michael Goldberg as a possible solution to the isoperimetric problem of maximizing the volume for a given number of faces (16 in this case) and a given surface area.{{cite journal

| last = Goldberg | first = Michael

| journal = Tohoku Mathematical Journal

| pages = 226–236

| title = The isoperimetric problem for polyhedra

| url = https://www.jstage.jst.go.jp/article/tmj1911/40/0/40_0_226/_pdf

| volume = 40}} For this optimization problem, the optimal geometric form for the polyhedron is one in which the faces are all tangent to an inscribed sphere.{{cite journal

| last = Fejes Tóth | first = László | author-link = László Fejes Tóth

| doi = 10.2307/2371944

| journal = American Journal of Mathematics

| jstor = 2371944

| mr = 24157

| pages = 174–180

| title = The isepiphan problem for {{mvar|n}}-hedra

| volume = 70

| year = 1948| issue = 1 }}

{{anchor|Friauf polyhedron}}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.{{r|friauf}} It can be found in crystals such as complex metallic alloys, an example is dizinc magnesium MgZn₂.{{r|lcd}} 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 $D\_\{2\backslash mathrm\{d\}\}$ of order 8.{{cn|date=July 2024}}

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.{{r|kuchel}}

File:Tuncated tetrahedral graph.png

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.An Atlas of Graphs, page 267, truncated tetrahedral graph It is a connected cubic graph,An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105 and connected cubic transitive graph.An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11

File:De divina proportione - Tetraedron Abscisum Vacuum.jpg | drawing in De divina proportione (1509)

File:Perspectiva Corporum Regularium 09a.jpg | drawing in Perspectiva Corporum Regularium (1568)

File:Modell, Kristallform (Verzerrungen) Oktaeder (Spinell) -Krantz 4, 6, 7, 391- (8).jpg | crystal model

File:Tetraedro truncado (Matemateca IME-USP).jpg | photos from different perspectives (Matemateca)

File:D4 truncated tetrahedron.JPG | 4-sided die

File:Permutohedron in simplex of order 4, with truncated tetrahedron (0-based).png | 12 permutations of $(4,\; 2,\; 0,\; 0)$ (brown)

• 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

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}}

• {{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998}}

{{Commons category}}

• {{mathworld2 |urlname=TruncatedTetrahedron |title=Truncated tetrahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
• {{mathworld | urlname = TruncatedTetrahedralGraph | title = Truncated tetrahedral graph}}
• {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x3o - tut}}
• [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=3cqTmfu7gdEZ8I7kRUVvji6qxBATVSp2WpmIWGx7l7pWe7bveylFxv3piHnPNZN&name=Truncated+Tetrahedron#applet Editable printable net of a truncated tetrahedron with interactive 3D view]
• [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

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