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I know how to compute the content of orthoschemes in 3- and 4-dimensional spherical space from dihedral angles using Schlafli series computations. Can anyone direct me to a textbook description of the general computation in 5- or higher dimensional spherical space? I understand it probably involves iterated integrals, but I would like to see a detailed example such as might be given in a textbook.

It would also be helpful to know of any off-the-shelf software that performs such computations. Also, any tabulated listing of contents of 5-D and/or higher spherical orthoschemes for various dihedral angles would be helpful.

I am a self studying enthusiast rather than a mathemetician and have taken no courses on this subject. A pointer to a comprehensive textbook would be perfect.

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After asking this question, I have had considerable success making these computations using methods of my own invention through 7-dimensional spherical space (spherical orthoschematic octotopes). The method involves iterated numerical integration in Excel spreadsheets. This is a work in progress as I'm working toward improved efficiency and accuracy. Interested parties can contact me at Thank you, Igor, for your advice and giving a darn. – user32121 Mar 11 '13 at 20:57

There is no software I am aware of. Nor is there any textbook. For hyperbolic orthoschemes (a very similar subject) you should check out Ruth Kellerhals' articles (there is a very detailed one in GAFA in 1995).

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There is no software? Is that definitive? Is there a resonably accessible paper? – Jim Sather Oct 23 '12 at 16:18
Ruth's papers are accessible. As for software, I am pretty sure, though life is full of surprises. – Igor Rivin Nov 7 '12 at 21:41

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