Elongated triangular orthobicupola

Elongated triangular orthobicupola
Type	Johnson; J34 - J35 - J36
Faces	2+6 triangles; 2x3+6 squares
Edges	36
Vertices	18
Vertex configuration	6(3.4.3.4); 12(3.43)
Symmetry group	D3h
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated triangular orthobicupola is one of the Johnson solids (J₃₅). As the name suggests, it can be constructed by elongating a triangular orthobicupola (J₂₇) by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron (one of the Archimedean solids), with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]

Volume

The volume of J₃₅ can be calculated as follows:

J₃₅ consists of 2 cupolae and hexagonal prism.

The two cupolae makes 1 cuboctahedron = 8 tetrahedra + 6 half-octahedra. 1 octahedron = 4 tetrahedra, so total we have 20 tetrahedra.

What is the volume of a tetrahedron? Construct a tetrahedron having vertices in common with alternate vertices of a cube (of side $\frac{1}{\sqrt{2}}$ , if tetrahedron has unit edges). The 4 triangular pyramids left if the tetrahedron is removed from the cube form half an octahedron = 2 tetrahedra. So

$V_{tetrahedron}={\frac {1}{3}}V_{cube}={\frac {1}{3}}{\frac {1}{{\sqrt {2}}^{3}}}={\frac {\sqrt {2}}{12}}$

The hexagonal prism is more straightforward. The hexagon has area $6{\frac {\sqrt {3}}{4}}$ , so

$V_{prism}={\frac {3{\sqrt {3}}}{2}}$

Finally

$V_{J_{35}}=20V_{tetrahedron}+V_{prism}={\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}$

numerical value:

$V_{J_{35}}=4.9550988153084743549606507192748$

Related polyhedra and honeycombs

The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.^[2]

References

↑ 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, Zbl 0132.14603 .
↑ http://woodenpolyhedra.web.fc2.com/J35.html

External links

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] 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, Zbl 0132.14603 .

[2] ttp://woodenpolyhedra.web.fc2.com/J35.html

Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular bipyramid square bipyramid pentagonal bipyramid elongated triangular bipyramid elongated square bipyramid elongated pentagonal bipyramid gyroelongated square bipyramid
Modified cupolae and rotundae	elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda
Augmented prisms	augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron
Miscellaneous	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)

Elongated triangular orthobicupola

Type	Johnson J₃₄ - J₃₅ - J₃₆
Faces	2+6 triangles 2x3+6 squares
Edges	36
Vertices	18
Vertex configuration	6(3.4.3.4) 12(3.4³)
Symmetry group	D_3h
Dual polyhedron	-
Properties	convex
Net