Gyroelongated bicupola

In geometry, the gyroelongated bicupolae are an infinite sets of polyhedra, constructed by adjoining two n-gonal cupolas to an n-gonal Antiprism. The triangular, square, and pentagonal gyroelongated bicupola are three of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form.

Set of gyroelongated bicupolae
Example pentagonal form
Faces6n triangles
2n squares
2 n-gon
Edges16n
Vertices6n
Symmetry groupDn, [n,2]+, (n22)
Rotation groupDn, [n,2]+, (n22)
Propertiesconvex, chiral

Adjoining two triangular prisms to a square antiprism also generates a polyhedron, but not a Johnson solid, as it is not convex. The hexagonal form is also a polygon, but has coplanar faces. Higher forms can be constructed without regular faces.

Image cw Image ccw Name Faces
Gyroelongated digonal bicupola 4 triangles, 4 squares
Gyroelongated triangular bicupola (J44) 6+2 triangles, 6 squares
Gyroelongated square bicupola (J45) 8 triangles, 8+2 squares
Gyroelongated pentagonal bicupola (J46) 30 triangles, 10 squares, 2 pentagon
Gyroelongated hexagonal bicupola 12 triangles, 24 squares, 2 hexagon

See also

edit

References

edit
  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
  • Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.