Editing Tetrahedron (section)

==Applications==

===Numerical analysis===
[[File:Malla irregular de triángulos modelizando una superficie convexa.png|thumb|An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.]]
In [[numerical analysis]], complicated three-dimensional shapes are commonly broken down into, or [[approximate]]d by, a [[polygon mesh|polygonal mesh]] of irregular [[tetrahedra]] in the process of setting up the equations for [[finite element analysis]] especially in the [[numerical solution]] of [[partial differential equations]]. These methods have wide applications in practical applications in [[computational fluid dynamics]], [[aerodynamics]], [[electromagnetic field]]s, [[civil engineering]], [[chemical engineering]], [[naval architecture|naval architecture and engineering]], and related fields.

===Structural engineering===
A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as [[spaceframe]]s.

===Fortification===
Tetrahedrons are used in [[caltrop]]s to provide an [[area denial weapon]]. This is due to their nature of having a sharp corner that always points upwards.

Large concrete tetrahedrons have been used as [[anti-tank]] measures, or as [[Tetrapod (structure)|Tetrapods]] to break down waves at coastlines.

===Aviation===
At some [[airfield]]s, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind.  It is built big enough to be seen from the air and is sometimes illuminated.  Its purpose is to serve as a reference to pilots indicating wind direction.<ref>{{citation|title=Pilot's Handbook of Aeronautical Knowledge|author=Federal Aviation Administration|publisher=U. S. Government Printing Office|year=2009|isbn=9780160876110|page=13{{hyphen}}10<!--hyphenated page-->|url=https://books.google.com/books?id=0l8WO6Drz50C&pg=SA13-PA10}}.</ref>

===Chemistry===
<div>[[Image:Ammonium-3D-balls.png|100px|thumb|The [[ammonium]] ion is tetrahedral]]</div>
[[Image:Tetrahedral_angle_calculation.svg|thumb|216px|<!-- specify width as minus sign vanishes at most sizes --> Calculation of the central angle with a [[dot product]] ]]
{{main|Tetrahedral molecular geometry}}
The tetrahedron shape is seen in nature in [[covalent bond|covalently bonded]] molecules. All [[Orbital hybridisation|sp<sup>3</sup>-hybridized]] atoms are surrounded by atoms (or [[lone pair|lone electron pairs]]) at the four corners of a tetrahedron. For instance in a [[methane]] molecule ({{chem|CH|4}}) or an [[ammonium]] ion ({{chem|NH|4|+}}), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called ''[[Tetrahedron (journal)|Tetrahedron]]''. The [[central angle]] between any two vertices of a perfect tetrahedron is arccos(−{{sfrac|1|3}}), or approximately 109.47°.<ref name="pubs.acs.org">{{cite journal|doi=10.1021/ed022p145 | volume=22 | issue=3 | title=Valence angle of the tetrahedral carbon atom | year=1945 | journal=Journal of Chemical Education | page=145 | last1 = Brittin | first1 = W. E.| bibcode=1945JChEd..22..145B }}</ref>

[[Water]], {{chem|H|2|O}}, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds.

Quaternary [[phase diagram]]s of mixtures of chemical substances are represented graphically as tetrahedra.

However, quaternary phase diagrams in [[communication engineering]] are represented graphically on a two-dimensional plane.

There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such as [[Allotropes of phosphorus|white phosphorus]] allotrope<ref>{{Cite web |title=White phosphorus |url=https://www.acs.org/molecule-of-the-week/archive/w/white-phosphorus.html |access-date=2024-05-26 |website=American Chemical Society |language=en}}</ref> and tetra-''t''-butyltetrahedrane, known derivative of the hypothetical [[tetrahedrane]].

===Electricity and electronics===
{{main|Electricity|Electronics}}
If six equal [[resistor]]s are [[solder]]ed together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.<ref>{{cite journal |last=Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |access-date=2006-09-15 |url-status=dead |archive-url=https://web.archive.org/web/20070610165115/http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |archive-date=10 June 2007}}</ref>

Since [[silicon]] is the most common [[semiconductor]] used in [[solid-state electronics]], and silicon has a [[valence (chemistry)|valence]] of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how [[crystal]]s of silicon form and what shapes they assume.

===Color space===
{{main|Color space}}
Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).<ref>{{cite journal |last=Vondran |first=Gary L. |date=April 1998 |title=Radial and Pruned Tetrahedral Interpolation Techniques |journal=HP Technical Report |volume=HPL-98-95 |pages=1–32 |url=http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf |access-date=11 November 2009 |archive-date=7 June 2011 |archive-url=https://web.archive.org/web/20110607102757/http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf |url-status=dead }}</ref>

===Games===
[[Image:4-sided dice 250.jpg|100px|thumb|[[4-sided dice]]]]
The [[Royal Game of Ur]], dating from 2600 BC, was played with a set of tetrahedral dice.

Especially in [[roleplaying]], this solid is known as a [[4-sided die]], one of the more common [[polyhedral dice]], with the number rolled appearing around the bottom or on the top vertex. Some [[Rubik's Cube]]-like puzzles are tetrahedral, such as the [[Pyraminx]] and [[Pyramorphix]].

===Geology===
{{main|tetrahedral hypothesis}}
The [[tetrahedral hypothesis]], originally published by [[William Lowthian Green]] to explain the formation of the Earth,<ref>{{cite book |title=Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography |volume=Part I |author-link=William Lowthian Green |first=William Lowthian |last=Green |publisher=E. Stanford |place=London |year=1875 |bibcode=1875vmge.book.....G |oclc=3571917 |url=https://books.google.com/books?id=9DkDAAAAQAAJ }}</ref> was popular through the early 20th century.<ref>{{cite book |author-link=Arthur Holmes |first=Arthur |last=Holmes |title=Principles of physical geology |url=https://archive.org/details/principlesofphys0000holm |url-access=registration |year=1965 |publisher=Nelson |page=[https://archive.org/details/principlesofphys0000holm/page/32 32] |isbn=9780177612992 }}</ref><ref>{{cite news |author-link=Charles Henry Hitchcock |first=Charles Henry |last=Hitchcock |editor-first=Newton Horace |editor-last=Winchell |title=William Lowthian Green and his Theory of the Evolution of the Earth's Features |work=The American Geologist |url=https://books.google.com/books?id=_Ty8AAAAIAAJ&pg=PA1 |date=January 1900 |volume=XXV |publisher=Geological Publishing Company |pages=1–10 }}</ref>

==={{anchor|Popular Culture}}Popular culture===
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 | image1 = Tétraèdres en béton.jpg
 | image2 = Coffee cream TetraPak.jpg
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[[Stanley Kubrick]] originally intended the [[Monolith (Space Odyssey)|monolith]] in ''[[2001: A Space Odyssey (film)|2001: A Space Odyssey]]'' to be a tetrahedron, according to [[Marvin Minsky]], a cognitive scientist and expert on [[artificial intelligence]] who advised Kubrick on the [[HAL 9000]] computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.<ref>{{cite web |title=Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron |url=http://www.webofstories.com/play/53140?o=R |publisher=Web of Stories |access-date=20 February 2012 }}</ref>

The tetrahedron with regular faces is a solution to an old puzzle asking to form four equilateral triangles using six unbroken matchsticks. The solution places the matchsticks along the edges of a tetrahedron.<ref>{{cite journal
 | last = Bell | first = Alexander Graham | author-link = Alexander Graham Bell
 | date = June 1903
 | doi = 10.1038/scientificamerican06131903-22947supp
 | issue = 1432supp
 | journal = Scientific American
 | pages = s2294–22950
 | title = The tetrahedral principle in kite structure
 | url = https://scholar.archive.org/work/jstv7c4lvbcenod5l4cpldrqsi
 | volume = 55}}</ref>