Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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)

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.v1). 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.v1).

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 show that the constant of π is absent in 0 and 1 integer dimensions. It is shown that self-dual n-simplices are undefined for n < -1, while n-orthoplices reduce to the empty set for n ≤ -1. Out of three regular, convex polytopes (and n-balls) present in all non-negative dimensions, only n-orthoplices, n-cubes and n-balls are defined in negative dimensions.

Keywords

regular; convex polytopes; negative dimensional spectra

Subject

MATHEMATICS & COMPUTER SCIENCE, Geometry & Topology

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