by wendy » Wed Aug 26, 2015 7:20 am
They do indeed. There is a catalan for each of the uniform solids in 4D.
For example, the surtegmate regular figure, consists of taking a regular figure, and a dual, whose edges pass through the 2D elements of the first. These form a "bipyramid" or tegum (it's actually a tegum-product of the two named elements), it has as many faces (3d elements) as the polytope has 2d elements. (hedra).
The various trikis and tetrakis figures are the 'apiculates'. formed by raising pyramids on the faces of the faces of regular figures. These are dual to the truncates.
There are the api-tegmate, formed by raising pyramids on the faces of the surtegmates. These are the duals of the cantelates. Where the head and bases of the pyramids are cut, you get the bi-api-tegmates. There's a lot of them, but most of them are from taking the CD form like o3o3x4o and turning the x's to m's, eg o3o3m4o.