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: Received: 26 April 2022 / Approved: 27 April 2022 / Online: 27 April 2022 (14:19:49 CEST)
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: Received: 3 May 2022 / Approved: 5 May 2022 / Online: 5 May 2022 (10:21:33 CEST)
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: Received: 6 May 2022 / Approved: 9 May 2022 / Online: 9 May 2022 (09:39:25 CEST)
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: Received: 19 May 2022 / Approved: 20 May 2022 / Online: 20 May 2022 (09:09:43 CEST)
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: Received: 24 May 2022 / Approved: 25 May 2022 / Online: 25 May 2022 (09:54:32 CEST)
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: Received: 25 May 2022 / Approved: 26 May 2022 / Online: 26 May 2022 (08:54:38 CEST)
How to cite:
Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions. Preprints2022, 2022040263 (doi: 10.20944/preprints202204.0263.v3).
Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions. Preprints 2022, 2022040263 (doi: 10.20944/preprints202204.0263.v3).
Cite as:
Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions. Preprints2022, 2022040263 (doi: 10.20944/preprints202204.0263.v3).
Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions. Preprints 2022, 2022040263 (doi: 10.20944/preprints202204.0263.v3).
Abstract
The aim of this study was to examine n-balls, n-simplices and n-orthoplices in real dimensions using novel recurrence relations that removed indefiniteness present in known formulas. They show that in negative, integer dimensions volumes of n-balls are zero if n is even, positive if n = -4k - 1, and negative if n = -4k - 3, for natural k. Volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for n Î ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < -1 self-dual n-simplices are undefined in negative, integer dimensions and their volumes and surfaces are imaginary in negative, fractional ones, and divergent with decreasing n. In negative, integer dimensions n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n. Out of three regular, convex polytopes present in all natural dimensions, only n-orthoplices, n-cubes (and n-balls) are defined in negative, integer dimensions.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author
2. Extended references.
3. Minor edits an corrections.