Gyroelongated cupola

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Set of gyroelongated cupolae
Gyroelongated pentagonal cupola.png
Example pentagonal form
Faces 3n triangles
n squares
1 n-gon
1 2n-gon
Edges 9n
Vertices 5n
Symmetry group Cnv, [n], (*nn)
Rotational group Cn, [n]+, (nn)
Dual polyhedron
Properties convex

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


name faces
gyroelongated triangular prism 2+8 triangles, 2+1 square
Gyroelongated triangular cupola.png gyroelongated triangular cupola (J22) 9+1 triangles, 3 squares, 1 hexagon
Gyroelongated square cupola.png gyroelongated square cupola (J23) 12 triangles, 4+1 squares, 1 octagon
Gyroelongated pentagonal cupola.png gyroelongated pentagonal cupola (J24) 15 triangles, 5 squares, 1 pentagon, 1 decagon
gyroelongated hexagonal cupola 18 triangles, 6 squares, 1 hexagon, 1 dodecagon

See also[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.