elongated triangular gyrobicupola
{{Short description|36th Johnson solid}}
{{Infobox polyhedron
| image = Elongated triangular gyrobicupola.png
| type = Johnson
{{math|elongated triangular orthobicupola – J{{sub|36}} – elongated square gyrobicupola}}
| faces = 8 triangles
12 squares
| edges = 36
| vertices = 18
| symmetry =
| vertex_config =
&6 \times (3 \times 4 \times 3 \times 4) + \\
&12 \times (3 \times 4^3)
\end{align}
| properties = convex
| net = Johnson solid 36 net.png
}}
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
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.{{r|rajwade}} This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.{{r|berman}} 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 .{{r|francis}}
Properties
An elongated triangular gyrobicupola with a given edge length has a surface area by adding the area of all regular faces:{{r|berman}}
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:{{r|berman}}
Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group of order 12.{{clarification needed|date=March 2024|reason=Need to explicitly explain the meaning of this symmetry}} 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:{{r|johnson}}
\frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\
\frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ.
\end{align}
Related polyhedra and honeycombs
The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J36.html|title = J36 honeycomb}}
References
{{reflist|refs=
| last = Berman | first = Martin
| year = 1971
| title = Regular-faced convex polyhedra
| journal = Journal of the Franklin Institute
| volume = 291
| issue = 5
| pages = 329–352
| doi = 10.1016/0016-0032(71)90071-8
| mr = 290245
}}
| last = Francis | first = Darryl
| title = Johnson solids & their acronyms
| journal = Word Ways
| date = August 2013
| volume = 46 | issue = 3 | page = 177
| url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}
| last = Johnson | first = Norman W. | authorlink = Norman W. Johnson
| year = 1966
| title = Convex polyhedra with regular faces
| journal = Canadian Journal of Mathematics
| volume = 18
| pages = 169–200
| doi = 10.4153/cjm-1966-021-8
| mr = 0185507
| s2cid = 122006114
| zbl = 0132.14603| doi-access = free
}}
| last = Rajwade | first = A. R.
| title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
| series = Texts and Readings in Mathematics
| year = 2001
| url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
| page = 84–89
| publisher = Hindustan Book Agency
| isbn = 978-93-86279-06-4
| doi = 10.1007/978-93-86279-06-4
}}
}}
External links
- {{MathWorld2|title2=Johnson solid|urlname2=JohnsonSolid| urlname=ElongatedTriangularGyrobicupola | title=Elongated triangular gyrobicupola}}
{{Johnson solids navigator}}
{{Polyhedron-stub}}