Gyroelongated square bicupola
Type	Johnson J44 – J45 – J46
Faces	3×8 triangles 2+8 squares
Edges	56
Vertices	24
Vertex configuration	8(3.43) 2.8(34.4)
Symmetry group	D4
Dual polyhedron	-
Properties	convex, chiral
Net

In geometry, the gyroelongated square bicupola is one of the Johnson solids (J₄₅). As the name suggests, it can be constructed by gyroelongating a square bicupola (J₂₈ or J₂₉) by inserting an octagonal antiprism between its congruent halves.

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (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 square 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 left half of the figure is connected by a path of two triangular faces to a square face below it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each square on the left would be connected to a square face above it and to the right. The two chiral forms of J₄₅ are not considered different Johnson solids.

Area and Volume

With edge length a, the surface area is

${\displaystyle A=\left(10+6{\sqrt {3}}\right)a^{2}\approx 20.392304845...a^{2},}$

and the volume is

${\displaystyle V=\left(2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\approx 8.153574834...a^{3}.}$

References

External links

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