Gyroelongated square bicupola explained

Type:	Johnson
Faces:	24 triangles 10 squares
Edges:	56
Vertices:	24
Symmetry:	D _{4}
Vertex Config:	8 x ( 3 x 4 ^{3)} + 2 x 8 x ( 3 ^{4} x 4 )
Net:	Johnson solid 45 net.png

In geometry, the gyroelongated square bicupola is the Johnson solid constructed by attaching two square cupolae on each base of octagonal antiprism. It has the property of chirality.

Construction

The gyroelongated square bicupola is constructed by attaching two square cupolae on each base of octagonal antiprism, a process known as gyroelongation. This construction involves the removal of octagons, and replacing them with cupolae. As a result, this polyhedron has twenty triangular and ten square faces. The Johnson solid is the convex polyhedron with all of its faces are regular, and the gyroelongated square bicupola is one of them, enumerated as

J _{45} .

Properties

Given that the edge length

a , the surface area is:$$\backslash left(10+6\backslash sqrt\backslash right)\; a^2\; \backslash approx\; 20.392a^2,$$the total area of twenty equilateral triangles and ten squares. Its volume is:$$\backslash left(2+\backslash frac\backslash sqrt\; +\; \backslash frac\backslash sqrt\backslash right)\; a^3\; \backslash approx\; 8.154a^3,$$the total volume of two square cupolae and an octagonal antiprism. Its dihedral angles can be calculated by adding the components of cupolae and antiprism. The dihedral angle of antiprism between two adjacent triangles is approximately 153 . 9 ^{\circ} . The dihedral angle of each cupola between two squares is 135 ^{\circ} , and that between triangle and square is 144 . 7 ^{\circ} . The dihedral angle of the cupolae and antiprism between two adjacent triangles and triangle-square is 151 . 3 ^{\circ} and 141 . 6 ^{\circ} , respectively.D _{4}