Elongation
Elongation is the general process of blending two halves of a polytope with a prism. Specifically, the prism gets inserted between the two halves. The halves needn't to be identical, but they, blended together, should create the original polytope.
Visually, an elongation stretches the polytope, hence its name. Elongation is similar to expansion in how it spreads apart existing facets and adds new ones in between. Prisms themselves can be thought of an elongation of a ditope. The names of elongations typically reflect where the two halves of elongation occur. Instead of being called an elongated octahedron, its name is actually the elongated square bipyramid.
Gyroelongation is similar to elongation, but instead of a prism being inserted, an alterprism is inserted (a lacing between two of the same polytopes in different orientations). However, the common definition of gyroelongation only applies to gyroelongating polyhedra with polygonal antiprisms, as seen with the johnson solids. Some atops can act as a different higher dimensional generalization of gyroelongation, such as in the octahedron atop cube, however, not all all are.
Retroelogation is topologically equivalent to elongation, but the prism in the middle is placed backwards, such as in the retroelongated triangular tiling. This can also be thought of as the two halves facing together instead of away. Retroelongation will always lead to a non-convex polytope There are 3 more non-convex variations of elongation: retrogyroelongation, gyroretroelongation, and retrogyroretroelongation. The first and last of which use retroprisms or retroalterprisms.
| Elongation | Polytope halves are attached... | Prism type | |
|---|---|---|---|
| None | Directly | Facing away | None |
| Elongation | Same orientation | Facing away | Prism |
| Retroelongation | Same orientation | Facing together | Prism |
| Gyroelongation | At different orientations | Facing away | Antiprism / Alterprism |
| Gyroretroelongation | At different orientations | Facing together | Antiprism / Alterprism |
| Retrogyroelongation | Centrally inverted | Facing away | Retroprism |
| Retrogyroretroelongation | Centrally inverted | Facing together | Retroprism |
These elongations can also be thought of as lacings between the two halves, which lace similarly to the prism it uses.