Triakis tetrahedron
{{Short description|Catalan solid with 12 faces}}
{{infobox polyhedron
| name = Triakis tetrahedron
| image = Triakistetrahedron.jpg
| type = Catalan solid, Kleetope
| symmetry = tetrahedral symmetry
| faces = 12
| edges = 18
| vertices = 8
| dual = truncated tetrahedron
| angle = 129.52°
| properties = convex, face-transitive, Rupert property
| net = Triakis tetrahedron net.svg
}}
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 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 Kleetopes 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 symmetry of full tetrahedral . Each dihedral angle between triangular faces is .{{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.{{r|fred}}
A triakis tetrahedron is different from an augmented tetrahedron as latter is obtained by 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.{{Cite web |title=Augmented Tetrahedron |url=https://mathworld.wolfram.com/AugmentedTetrahedron.html}}
See also
References
{{reflist|refs=
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| 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
}}
| 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
}}
| 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
}}
| 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
}}
| last = Smith | first = Anthony
| title = Stellations of the Triakis Tetrahedron
| journal = The Mathematical Gazette
| volume = 49 | issue = 368 | year = 1965 | pages = 135–143
| doi = 10.2307/3612303
}}
| 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
}}
}}
External links
- {{Mathworld2 |urlname=TriakisTetrahedron |title=Triakis tetrahedron |urlname2=CatalanSolid |title2=Catalan solid}}
{{Catalan solids}}
{{Polyhedron navigator}}