Duals of cantellation, runcination, omnitruncation...

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

Duals of cantellation, runcination, omnitruncation...

Postby Vector_Graphics » Thu Nov 09, 2023 5:45 pm

Do you guys have any terms for these? The same way ...kis is the dual of truncation
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Re: Duals of cantellation, runcination, omnitruncation...

Postby Klitzing » Tue Nov 14, 2023 5:15 pm

omnitruncation (xxx...) is not a single operation, in fact it means s.th. quite different within each dimension.

runcination (xooxo...) only occurs from 4D onwards. However the set of Conway operators (which well includes the duals) is defined for 3D only.

cantellation (xoxo...) indeed can be applied within 3D and happens to be there nothing but the e operator (expand). Its dual (there) is the o operator (ortho).

However, have in mind that Stott's expansion allows for every single node to be expanded from unringed to ringed (or conversely contracted from ringed to unringed), whereas Conway's "expansion" only applies to the very end nodes of a linear CD, even if extrapolated to higher dimensions.

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Re: Duals of cantellation, runcination, omnitruncation...

Postby wendy » Wed Nov 22, 2023 1:41 pm

the general dual for truncation is to raise a peak on the face. Imagine the base is covered by a rubber sheet. The dual expand in edge size from within.
The vertex cause a peak, so you get apiculated Dual.
When the peaks are high egough, the edge of the dual is visible. this gives a edge surtegmated Dual. the facets becom tegumd of the original edges and the original margis(boundary between facets). an so the dual of an ntruncate, becomse a n-appiculate, where the nd surtope of the dual stands above the surface.
The dual of a n-rectate, is a n-surtegmate. These have facets, that are tegum products of n-d surtope of the dual, and s-n surtope of the s-d solid.
the dual of the omnitruncate is to divede the original surface into flags, or vanes, making a vaniate figure,
the dual of the runcinate (pg sence) is the strombiated. The surface becomes antitegums of the marginal.
see eg http://www.ostfan2.com/gloss/poltytope.pdf.
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