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Compare higher degree rectifications.

What are higher degree cantellations?

I didn't find any definition on internet, at least no operational description.

You can find constructions by coordinates for "bicantellated 7-cube" or "tricantellated 8-simplex", so please don't tell me that the term is "not useful" or meaningless.

From the term itself, I can imagine many possible meaning of "bicantellated". What is then the correct notion of higher degree cantellations?

Hao Chen
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    I did not find definition of "higher degree cantellations" either. Note that just because higher degree rectifications are "defined" does not imply that the phrase you ask about is necessarily useful. For example, "higher order derivatives" is a well-worn phrase, but it would not be immediately clear to a Reader or Listener what "higher order integrals" should encompass. – hardmath Oct 29 '14 at 12:27
  • @hardmath You can find on http://en.wikipedia.org/wiki/Cantellated_8-simplexes notions like "bicantellated 8-simplex" or "tricantellated 8-simplex". So this notion is used, and you can find examples of them. My question is: what is the general definition of it! – Hao Chen Oct 29 '14 at 13:08
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    It would likely be simpler to say what "bicantellated" or "tricantellated" mean than to provide the "general definition". I believe @Wendy.Krieger might have some insight into the terminology. "Cantellated" and "Rectified" are (in this context) words coined by Norman Johnson. – hardmath Oct 29 '14 at 15:22
  • I will hazard a guess: A cantellation of a $d$-polytope (or its dual) can be constructed beginning with a $d+1$-polytope with two disjoint, parallel facets taking up all the vertices of the $d+1$-polytope, one facet being your $d$-polytope, the other being its dual. The cantellation is the cross section of the $d+1$-polytope with the mid-hyperplane between the two parallel facets. For a bicantellation, the facets are the cantellation and its dual, etc. – Dan Moore Oct 29 '14 at 15:24
  • @DanMoore, so the bicantellation is the cantellation of cantellation. That isn't the case of bi-rectification. – Hao Chen Oct 29 '14 at 15:40
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    @hardmath, Johnson's book on uniform polytopes is (still) not published (after years). Thanks for the link, it answers my question: the $k$-cantellation is the rectification of the $k$-rectification. – Hao Chen Oct 29 '14 at 15:43

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