Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions

Szymon Łukaszyk

doi:10.20944/preprints202204.0263.v10

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preprints.org > computer science and mathematics > geometry and topology > doi: 10.20944/preprints202204.0263.v10

Preprint Article Version 10 Preserved in Portico This version is not peer-reviewed

Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Real Dimensions

Szymon Łukaszyk

Version 1 : Received: 26 April 2022 / Approved: 27 April 2022 / Online: 27 April 2022 (14:19:49 CEST)
Version 2 : Received: 3 May 2022 / Approved: 5 May 2022 / Online: 5 May 2022 (10:21:33 CEST)
Version 3 : Received: 6 May 2022 / Approved: 9 May 2022 / Online: 9 May 2022 (09:39:25 CEST)
Version 4 : Received: 19 May 2022 / Approved: 20 May 2022 / Online: 20 May 2022 (09:09:43 CEST)
Version 5 : Received: 24 May 2022 / Approved: 25 May 2022 / Online: 25 May 2022 (09:54:32 CEST)
Version 6 : Received: 25 May 2022 / Approved: 26 May 2022 / Online: 26 May 2022 (08:54:38 CEST)
Version 7 : Received: 28 May 2022 / Approved: 30 May 2022 / Online: 30 May 2022 (11:28:08 CEST)
Version 8 : Received: 1 June 2022 / Approved: 1 June 2022 / Online: 1 June 2022 (09:43:58 CEST)
Version 9 : Received: 7 June 2022 / Approved: 8 June 2022 / Online: 8 June 2022 (12:29:03 CEST)
Version 10 : Received: 10 June 2022 / Approved: 10 June 2022 / Online: 10 June 2022 (16:13:45 CEST)
Version 11 : Received: 14 June 2022 / Approved: 16 June 2022 / Online: 16 June 2022 (10:39:40 CEST)
Version 12 : Received: 18 June 2022 / Approved: 20 June 2022 / Online: 20 June 2022 (09:40:53 CEST)
Version 13 : Received: 23 June 2022 / Approved: 27 June 2022 / Online: 27 June 2022 (11:17:06 CEST)

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.

Abstract

The study examines n-balls, regular n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that removed indefiniteness present in known formulas. They show that in the 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 the negative, integer dimensions, and their volumes and surfaces are imaginary in the 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 the negative, integer dimensions.

Keywords

regular convex polytopes; negative dimensions; fractal dimensions

Subject

Computer Science and Mathematics, Geometry and Topology

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.

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Comments (1)

Comment 1

Received: 10 June 2022
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author

Comment: 1. Proof for the recurrence relation (11) for natural n, along with its algebraic forms (14), (15) for even and odd n.
2. Proof for the algebraic form (18) for the recurrence relation (15) in negative, odd n.
3. Proof of the symmetry relation (19).
4. Novel symmetry relation (21) along with the proof.

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