Triakis tetrahedron

Triakis tetrahedron
Triakis tetrahedron
Type	Catalan solid, Kleetope
Faces	12
Edges	18
Vertices	8
Symmetry group	tetrahedral symmetry
Dihedral angle (degrees)	129.52°
Dual polyhedron	truncated tetrahedron
Properties	convex, face-transitive, Rupert property
Net

In geometry, a triakis tetrahedron (or tristetrahedron^[1], or kistetrahedron^[2]) is a solid constructed by attaching four triangular pyramids onto the triangular faces of a regular tetrahedron, a Kleetope of a tetrahedron.^[3] This replaces the triangular faces with three, so there are twelve in total; eight vertices and eighteen edges form them.^[4] This interpretation is also expressed in the name, triakis, which is used for the Kleetopes of polyhedra with triangular faces.^[2]

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 $\mathrm {T} _{\mathrm {d} }$ . Each dihedral angle between triangular faces is $\arccos(-7/11)\approx 129.52^{\circ }$ .^[4] Unlike its dual, the truncated tetrahedron has no vertex-transitive, but rather face-transitive, meaning its solid appearance is unchanged by any transformation like reflecting and rotation between two triangular faces.^[5] Whenever a triakis tetrahedron has a hole, it is possible for a polyhedron to exist with the same or larger size passing through it.^[6]

References

^ Smith, Anthony (1965), "Stellations of the Triakis Tetrahedron", The Mathematical Gazette, 49 (368): 135–143, doi:10.2307/3612303
^ ^a ^b Conway, John H.; Burgiel, Heidi (2008), The Symmetries of Things, Chaim Goodman-Strauss, p. 284, ISBN 978-1-56881-220-5
^ Brigaglia, Aldo; Palladino, Nicla; Vaccaro, Maria Alessandra (2018), "Historical notes on star geometry in mathematics, art and nature", in Emmer, Michele; Abate, Marco (eds.), Imagine Math 6: Between Culture and Mathematics, Springer International Publishing, pp. 197–211, doi:10.1007/978-3-319-93949-0_17, ISBN 978-3-319-93948-3
^ ^a ^b 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
^ Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazon (2010), "Catalan Solids Derived From 3D-Root Systems and Quaternions", Journal of Mathematical Physics, 51 (4), arXiv:0908.3272, doi:10.1063/1.3356985
^ Fredriksson, Albin (2024), "Optimizing for the Rupert property", The American Mathematical Monthly, 131 (3): 255–261, arXiv:2210.00601, doi:10.1080/00029890.2023.2285200