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I am asking about the Platonic solids in all dimension, some reference about the proofs of many of the statement made in here.

I would like to here about how to think about higher dimensions mainly the solids.

I could only understand the famous "tesseract" (famous thanks to the marvel) which is a simple 4-cube or a simple ball in the maximum norm.

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Everything you might possibly want to know is proved in H.S.M.Coxeter's Regular polytopes. –  Mariano Suárez-Alvarez May 4 '12 at 23:03
    
in stock! store.doverpublications.com/0486614808.html –  Will Jagy May 4 '12 at 23:06
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