# Elongated triangular gyrobicupola

Elongated triangular gyrobicupola | |
---|---|

Type | JohnsonJ_{35} – – J_{36}J_{37} |

Faces | 8 triangles 12 squares |

Edges | 36 |

Vertices | 18 |

Vertex configuration | |

Symmetry group | |

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 . It is an example of Johnson solid.

## Construction

[edit]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.^{[1]} This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.^{[2]} The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in . 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 .^{[3]}

## Properties

[edit]An elongated triangular gyrobicupola with a given edge length has a surface area by adding the area of all regular faces:^{[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]}

Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group of order 12.^{[clarification needed]} 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 , and that between its base and square face is . The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately , that between each square and the hexagon is , and that between square and triangle is . 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]}

## Related polyhedra and honeycombs

[edit]The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^{[5]}

## References

[edit]**^**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.- ^
^{a}^{b}^{c}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. **^**Francis, Darryl (August 2013). "Johnson solids & their acronyms".*Word Ways*.**46**(3): 177.**^**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.**^**"J36 honeycomb".