Elongated cupola

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

Set of elongated cupolae

Example pentagonal form
Facesn triangles
3n squares
1 n-gon
1 2n-gon
Edges9n
Vertices5n
Symmetry groupCnv, [n], (*nn)
Rotational groupCn, [n]+, (nn)
Dual polyhedron
Propertiesconvex

There are three elongated 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 cube also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. Higher forms can be constructed without regular faces.

Forms

namefaces
elongated digonal cupola2 triangles, 6+1 squares
elongated triangular cupola (J18)3+1 triangles, 9 squares, 1 hexagon
elongated square cupola (J19)4 triangles, 12+1 squares, 1 octagon
elongated pentagonal cupola (J20)5 triangles, 15 squares, 1 pentagon, 1 decagon
elongated hexagonal cupola6 triangles, 18 squares, 1 hexagon, 1 dodecagon

See also

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.


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