View source for Triakis tetrahedron

{{Short description|Catalan solid with 12 faces}}
{{infobox polyhedron
 | name = Triakis tetrahedron
 | image = Triakistetrahedron.jpg
 | type = [[Catalan solid]], [[Kleetope]]
 | symmetry = [[tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{d} </math>
 | faces = 12
 | edges = 18
 | vertices = 8
 | dual = [[truncated tetrahedron]]
 | angle = 129.52°
 | properties = [[convex set|convex]], [[face-transitive]], [[Rupert property]]
 | net = Triakis tetrahedron net.svg
}}
[[File:Triakis tetrahedron.stl|thumb|3D model of a triakis tetrahedron]]
In [[geometry]], a '''triakis tetrahedron''' (or '''tristetrahedron'''{{r|smith}}, or '''kistetrahedron'''{{r|conway}}) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a [[regular tetrahedron]], a [[Kleetope]] of a tetrahedron.{{r|bpv}} This replaces the equilateral triangular faces of the regular tetrahedron with three [[Isosceles triangle|isosceles triangles]] at each face, so there are twelve in total; eight vertices and eighteen edges form them.{{r|williams}} This interpretation is also expressed in the name, triakis, which is used for [[Kleetope]]s of polyhedra with triangular faces.{{r|conway}}

The triakis tetrahedron is a [[Catalan solid]], the [[dual polyhedron]] of a [[truncated tetrahedron]], an [[Archimedean solid]] with four hexagonal and four triangular faces, constructed by cutting off the vertices of a regular tetrahedron; it shares the same [[Tetrahedral symmetry|symmetry of full tetrahedral]] <math> \mathrm{T}_\mathrm{d} </math>. Each [[dihedral angle]] between triangular faces is <math> \arccos(-7/11) \approx 129.52^\circ</math>.{{r|williams}} Unlike its dual, the truncated tetrahedron is not [[vertex-transitive]], but rather [[face-transitive]], meaning its solid appearance is unchanged by any transformation like reflecting and rotation between two triangular faces.{{r|koca}} The triakis tetrahedron has the [[Rupert property|Rupert property]].{{r|fred}}

A triakis tetrahedron is different from an ''augmented tetrahedron'' as latter is obtained by [[Augmentation (geometry)|augmenting]] the four faces of a tetrahedron with four ''regular'' tetrahedra (instead of nonuniform triangular pyramids) resulting in an equilateral polyhedron which is a concave [[deltahedron]] (whose all faces are congruent equilateral triangles). The convex hull of an augmented tetrahedron is a triakis tetrahedron.<ref>{{Cite web |title=Augmented Tetrahedron |url=https://mathworld.wolfram.com/AugmentedTetrahedron.html}}</ref>

==See also==
*[[Truncated triakis tetrahedron]]

==References==
{{reflist|refs=

<ref name="bpv">{{citation
 | last1 = Brigaglia | first1 = Aldo
 | last2 = Palladino | first2 = Nicla
 | last3 = Vaccaro | first3 = Maria Alessandra
 | editor1-last = Emmer | editor1-first = Michele
 | editor2-last = Abate | editor2-first = Marco
 | contribution = Historical notes on star geometry in mathematics, art and nature
 | doi = 10.1007/978-3-319-93949-0_17
 | pages = 197–211
 | publisher = Springer International Publishing
 | title = Imagine Math 6: Between Culture and Mathematics
 | year = 2018| isbn = 978-3-319-93948-3
| hdl = 10447/325250
 | hdl-access = free
 }}</ref>

<ref name="conway">{{citation
 | last1 = Conway | first1 = John H. | author-link1 = John H. Conway
 | last2 = Burgiel | first2 = Heidi 
 | title = The Symmetries of Things
 | year = 2008
 | publisher = Chaim Goodman-Strauss
 | isbn = 978-1-56881-220-5
 | url = https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA284
 | page = 284
}}</ref>

<ref name=fred>{{citation
 | last = Fredriksson | first = Albin
 | title = Optimizing for the Rupert property
 | journal = [[The American Mathematical Monthly]]
 | pages = 255–261
 | volume = 131
 | issue = 3
 | year = 2024
 | doi = 10.1080/00029890.2023.2285200
 | arxiv = 2210.00601
}}</ref>

<ref name="koca">{{citation
 | title = Catalan Solids Derived From 3D-Root Systems and Quaternions
 | last1 = Koca | first1 = Mehmet
 | last2 = Ozdes Koca | first2 = Nazife
 | last3 = Koc | first3 = Ramazon
 | year = 2010
 | journal = Journal of Mathematical Physics
 | volume = 51 | issue = 4 | doi = 10.1063/1.3356985 |arxiv=0908.3272
}}</ref>

<ref name="smith">{{citation
 | last = Smith | first = Anthony
 | title = Stellations of the Triakis Tetrahedron
 | journal = [[The Mathematical Gazette]]
 | volume = 49 | issue = 368 | year = 1965 | pages = 135&ndash;143
 | doi = 10.2307/3612303
}}</ref>

<ref name="williams">{{citation
 | last = Williams | first = Robert | author-link = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | url = https://archive.org/details/geometricalfound00will/page/72
 | page = 72
 | isbn = 978-0-486-23729-9
}}</ref>

}}

==External links==
*{{Mathworld2 |urlname=TriakisTetrahedron |title=Triakis tetrahedron |urlname2=CatalanSolid |title2=Catalan solid}}

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]