Snub Augmented Prisms

A family of chiral prismatic polyhedra. The first that I came across (although not the simplest, this honour goes to the cingulated anti-prisms).  They are generated by taking a prism, performing a "kis" operation on the square prismatic faces by replacing them with the caps of square pyramids, and performing a snub operation on the resulting polyhedron.  The pyramids themselves are not snubbed.  Note that the "kis" triangles are always equilateral.

An example is the square prism or cube, which is after the kis operation appears thus and after the subsequent snub operation thus.  The pyramidical faces are highlighted in yellow on this model.  This is a unique member of the family in that the pyramids stay regular, and in this case the kis operation need not have taken place.  Without the kis operation the result is a snub cube.  Apart from the square prism, the kis operation is required because the snub operation either distorts the prismatic faces or requires non-equilateral triangles.  A example is given of an snub pentagonal prism.  The distortion in this case has been confined to the triangular faces.  Viewing the source of the VRML model will give an indication of the amount of distortion required (up to 3% on some edges)..

For prisms other than the square prism, the kis operation allows the pyramidical caps to distort, allowing all faces to remain regular.

The family is 4-regular with vertices (A): n/d-3-3-3-3, (B): 3-3-3-3-3-3, (C): 3-3-3-3-3-3 and (D): 3-3-3-3.  The (B) and (C) vertices are distinguished as they are not interchangeable.  The (D) vertices being the apices of the pyramids.  The existence of these "3^6" vertices means that the family is not convex (globally or locally) at any n/d.

Models have been generated for a number of values of n and n/d.  Some retrograde models with n/d < 2 have also been generated.  I have so far been unable to generate any models with n>6.  Whether this is a fixed upper limit of this family or whether this is merely a problem of discovering the correct way to 'fold' the (A) vertices is as yet undetermined.  Links to "highlight" models are to ones with the pyramidical faces highlighted.

 2 highlight 3 highlight 4 highlight 5 highlight 6 highlight 5/2 highlight 7/2 highlight 7/3 highlight 8/3 highlight 9/2 highlight 9/4 highlight 10/3 highlight 11/2 highlight 11/3 highlight 11/4 highlight 11/5 highlight 12/5 highlight 5/3 highlight 7/5 highlight 7/4 highlight 8/5 highlight 9/5 highlight 10/7 highlight 11/6 highlight

Unlike many of the other prismatic families, I have been unable to generate any isomeric forms.  This is not altogether surprising as the snub cube itself has no isomers amongst the uniform polyhedra.