4d catalans

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

4d catalans

Postby Starmute » Tue Aug 25, 2015 8:19 pm

I've never seen any pictures of the 4d catalan solids - do any images exist?
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Re: 4d catalans

Postby 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.
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Re: 4d catalans

Postby quickfur » Wed Mar 16, 2016 9:29 pm

You can find a partial image of a 4D catalan in this post. It's the dual of the rectified 600-cell o5o3x3o. It doesn't show the entire polytope, only a few cells, but it should give a general idea.
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