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.

Gyroelongated pyramid
Example pentagonal form
Faces3n triangles
1 n-gon
Edges5n
Vertices2n + 1
Symmetry groupCnv, [n], (*nn)
Rotation groupCn, [n]+, (nn)
Propertiesconvex

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.

Forms

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

See also

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References

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  • 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.