Set of gyroelongated cupolae

Example pentagonal form
Faces3n triangles
n squares
1 n-gon
1 2n-gon
Edges9n
Vertices5n
Symmetry groupCnv, [n], (*nn)
Rotational groupCn, [n]+, (nn)
Dual polyhedron
Propertiesconvex

In geometry, the gyroelongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal antiprism.

There are three gyroelongated cupolae that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. Adjoining a triangular prism to a square antiprism also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. The hexagonal form can be constructed from regular polygons, but the cupola faces are all in the same plane. Topologically other forms can be constructed without regular faces.

Forms

namefaces
gyroelongated digonal cupola2+8 triangles, 2+1 square
gyroelongated triangular cupola (J22)9+1 triangles, 3 squares, 1 hexagon
gyroelongated square cupola (J23)12 triangles, 4+1 squares, 1 octagon
gyroelongated pentagonal cupola (J24)15 triangles, 5 squares, 1 pentagon, 1 decagon
gyroelongated hexagonal cupola18 triangles, 6 squares, 1 hexagon, 1 dodecagon

See also

References

  • Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169200. 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.


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