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.v4

Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

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)

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.

regular convex polytopes; negative dimensional spectra; fractal dimensions

Computer Science and Mathematics, Geometry and Topology

Comment 1

Received: 20 May 2022
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author

Comment: 1. Extended introduction.
2. Short presentation of the sections of the paper at the end of the introduction.
3. References to known eqs. (4)-(10) are provided.
4. All the new basic equations introduced in the paper are summarized in Section 5.
5. Section 6 presents an advantage of the new methodologies over known relationships.
6. Section 6 presents possible future applications of the findings of the study.
7. More references added.
8. A simple and approximate formula, which determines the radius of a spherical nucleus added at the end of Section 6.

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

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