Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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 Integer Dimensions

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)

How to cite: Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Integer Dimensions. Preprints 2022, 2022040263 (doi: 10.20944/preprints202204.0263.v2). Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Integer Dimensions. Preprints 2022, 2022040263 (doi: 10.20944/preprints202204.0263.v2).

Abstract

New recurrence relations for n-balls, regular n-simplices, and n-orthoplices in integer dimensions are submitted. They remove indefiniteness present in known formulas. 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 non-negative dimensions, only n-orthoplices, n-cubes (and n-balls) are defined in negative, integer dimensions.

Keywords

regular convex polytopes; negative dimensional spectra

Subject

MATHEMATICS & COMPUTER SCIENCE, Geometry & Topology

Comments (1)

Comment 1
Received: 5 May 2022
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author
Comment: 1. Submitted recurrence relations are continuous for n in ℝ.
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.
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