Pentagonal gyrobicupola


The pentagonal gyrobicupola is a polyhedron that is constructed by attaching two pentagonal cupolas base-to-base, each of its cupolas is twisted at 36°. It is an example of a Johnson solid and a composite polyhedron.

Construction

The pentagonal gyrobicupola is a composite polyhedron: it is constructed by attaching two pentagonal cupolas base-to-base. This construction is similar to the pentagonal orthobicupola; the difference is that one of the cupolas in the pentagonal gyrobicupola is twisted at 36°, as suggested by the prefix gyro-. The resulting polyhedron has the same faces as the pentagonal orthobicupola does: those cupolas cover their decagonal bases, replacing them with ten equilateral triangles, ten squares, and two regular pentagons. A convex polyhedron in which all of its faces are regular polygons is the Johnson solid. The pentagonal gyrobicupola has these, enumerating it as the thirty-first Johnson solid.

Properties

The surface area of a pentagonal gyrobicupola is the sum of its faces' area, and its volume is twice the volume of a pentagonal cupola:
The pentagonal gyrobicupola has a three-dimensional symmetry group, the antiprismatic symmetry of. Its dihedral angles are as follows:
  • the angle between a pentagon and a square is 159.09°.
  • the angle between a square and a triangle, within one cupola, is 148.28°;
  • the dihedral angle at the plane joining the two cupolas is the sum of the dihedral angle between square-to-decagon and triangle-to-decagon, 69.09°.