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Elongated triangular gyrobicupola
Type	Johnson J35 – J36 – J37
Faces	8 triangles 12 squares
Edges	36
Vertices	18
Vertex configuration	${\displaystyle {\begin{aligned}&6\times (3\times 4\times 3\times 4)+\\&12\times (3\times 4^{3})\end{aligned}}}$
Symmetry group	${\displaystyle D_{3h}}$
Properties	convex
Net

In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in ${\displaystyle 60^{\circ }}$. It is an example of Johnson solid.

Construction

The elongated triangular gyrobicupola is similarly can be constructed as the elongated triangular orthobicupola, started from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in ${\displaystyle 60^{\circ }}$. A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid ${\displaystyle J_{36}}$.

Properties

An elongated triangular gyrobicupola with a given edge length ${\displaystyle a}$ has a surface area by adding the area of all regular faces: $${\displaystyle \left(12+2{\sqrt {3}}\right)a^{2}\approx 15.464a^{2}.}$$ Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up: $${\displaystyle \left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.}$$

Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group ${\displaystyle D_{3d}}$ of order 12. Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon ${\displaystyle 120^{\circ }=2\pi /3}$, and that between its base and square face is ${\displaystyle \pi /2=90^{\circ }}$. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately ${\displaystyle 70.5^{\circ }}$, that between each square and the hexagon is ${\displaystyle 54.7^{\circ }}$, and that between square and triangle is ${\displaystyle 125.3^{\circ }}$. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively: $${\displaystyle {\begin{aligned}{\frac {\pi }{2}}+70.5^{\circ }&\approx 160.5^{\circ },\\{\frac {\pi }{2}}+54.7^{\circ }&\approx 144.7^{\circ }.\end{aligned}}}$$

Related polyhedra and honeycombs

The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.

References

1. Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Hindustan Book Agency. p. 84–89. doi:10.1007/978-93-86279-06-4. ISBN 978-93-86279-06-4.
2. ^ Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
3. Francis, Darryl (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
4. Johnson, Norman W. (1966). "Convex polyhedra with regular faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/cjm-1966-021-8. MR 0185507. S2CID 122006114. Zbl 0132.14603.
5. "J36 honeycomb".

External links

• Weisstein, Eric W., "Elongated triangular gyrobicupola" ("Johnson solid") at MathWorld.

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