Gyroelongated pyramid

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

In geometry, the gyroelongated pyramids (also called augmented antiprisms) are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal antiprism.

There are two gyroelongated pyramids that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar faces. Others can be constructed allowing for isosceles triangles.


Image Name Faces
Augmented octahedron.png Gyroelongated triangular pyramid
(Coplanar faces)
9+1 triangles
Gyroelongated square pyramid.png Gyroelongated square pyramid (J10) 12 triangles, 1 squares
Gyroelongated pentagonal pyramid.png Gyroelongated pentagonal pyramid (J11) 15 triangles, 1 pentagon
Augmented hexagonal antiprism flat.png Gyroelongated hexagonal pyramid
(Coplanar faces)
18 triangles, 1 hexagon

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.