Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Integer Dimensions
Version 2 : Received: 3 May 2022 / Approved: 5 May 2022 / Online: 5 May 2022 (10:21:33 CEST)
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A peer-reviewed article of this Preprint also exists.
Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions. Mathematics 2022, 10, 2212. Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions. Mathematics 2022, 10, 2212.
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Commenter: Szymon Łukaszyk
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2. Volumes and surfaces of n-simplices are imaginary in negative, fractional dimensions for n<-1 (surfaces also for n<0) and divergent with decreasing n.
3. Volumes of n-orthoplices are real in negative, fractional dimensions; surfaces are imaginary. Both volumes and surfaces are divergent in negative, fractional dimensions with decreasing n.