Gyroelongated triangular bicupola

Gyroelongated triangular bicupola
Type	Johnson J43 - J44 - J45
Faces	2+3.6 triangles 6 squares
Edges	42
Vertices	18
Vertex configuration	6(3.4.3.4) 2.6(3^4.4)
Symmetry group	D3
Dual polyhedron	-
Properties	convex, chiral
Net

In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (J₄₄). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either J₂₇ or the cuboctahedron) by inserting a hexagonal antiprism between its congruent halves.

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]

The gyroelongated triangular bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of J₄₄ are not considered different Johnson solids.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:^[2]

${\displaystyle V={\sqrt {2}}({\frac {5}{3}}+{\sqrt {1+{\sqrt {3}}}})a^{3}\approx 4.69456...a^{3}}$

${\displaystyle A=(6+5{\sqrt {3}})a^{2}\approx 14.6603...a^{2}}$

References

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. Stephen Wolfram, "Gyroelongated triangular bicupola" from Wolfram Alpha. Retrieved July 30, 2010.

External links

• Eric W. Weisstein, Gyroelongated triangular bicupola (Johnson solid) at MathWorld.