Elongated triangular orthobicupola

Elongated triangular orthobicupola
Type	Johnson J34 – J35 – J36
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 orthobicupola or cantellated triangular prism^{[ citation needed ]} is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid.

Construction

The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. ^[1] This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. ^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid ${\displaystyle J_{35}}$. ^[3]

Properties

An elongated triangular orthobicupola with a given edge length ${\displaystyle a}$ has a surface area, by adding the area of all regular faces: ^[2]

$${\displaystyle \left(12+2{\sqrt {3}}\right)a^{2}\approx 15.464a^{2}.}$$

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: ^[2]

$${\displaystyle \left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.}$$

It has the same three-dimensional symmetry groups as the triangular orthobicupola, the dihedral group ${\displaystyle D_{3h}}$ 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: ^[4]

$${\displaystyle {\begin{aligned}{\frac {\pi }{2}}+70.5^{\circ }&\approx 160.5^{\circ },\\{\frac {\pi }{2}}+54.7^{\circ }&\approx 144.7^{\circ }.\end{aligned}}}$$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids. ^[5]

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. "J35 honeycomb".

External links

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