Gyroelongated pyramid

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
Faces	3n triangles 1 n-gon
Edges	5n
Vertices	2n + 1
Symmetry group	Cnv, [n], (*nn)
Rotation group	Cn, [n]+, (nn)
Properties	convex

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

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

• Gyroelongated bipyramid
• Elongated bipyramid
• Elongated pyramid
• Diminished trapezohedron

References

• 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.