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 | |
| Faces | n triangles 3n 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 |
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
| name | faces | |
|---|---|---|
 | elongated digonal cupola | 2 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 cupola | 6 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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