Elongated triangular orthobicupola

Elongated triangular orthobicupola
Type	Johnson J34 - J35 - J36
Faces	2+6 triangles 2.3+6 squares
Edges	36
Vertices	18
Vertex configuration	6(3.4.3.4) 12(3.43)
Symmetry group	D3h
Dual polyhedron	-
Properties	convex

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.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

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:

$V_{J_{35}}=4.9550988153084743549606507192748$

External links

• Johnson Solid -- from MathWorld