Alterprism

A closely related concept to antiprisms are alterprisms, constructed by lacing two geometrically-identical polytopes either in some gyrated orientations together (i.e. tutcup) or in the same orientation with intersecting laced facets (i.e siidcup). Examples include laces of polytopes with simplecial or demicubic symmetry in any dimension, laces with icositetrachoric symmetry, or laces with the symmetry of the E6 family. Many of these polytopes are scaliform. A simple 4D example of this would be the truncated tetrahedral cupoliprism.

The only uniform 4D cases, known as the Johnson antiprisms, are the hexadecachoron xo3oo3ox&#x, the small ditrigonary icosidodecahedral antiprism xo5/2ox3oo*a&#x, the great ditrigonary icosidodecahedral antiprism xo5/4ox3oo*a&#x, and the blend of the latter two, the ditrigonary dodecadodecahedral antiprism.