Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

On the Basic Convex Polytopes and n-Balls in Complex Dimensions

Version 1 : Received: 5 September 2022 / Approved: 6 September 2022 / Online: 6 September 2022 (10:13:02 CEST)
Version 2 : Received: 13 September 2022 / Approved: 14 September 2022 / Online: 14 September 2022 (08:37:54 CEST)
Version 3 : Received: 16 September 2022 / Approved: 19 September 2022 / Online: 19 September 2022 (10:21:05 CEST)
Version 4 : Received: 6 October 2022 / Approved: 7 October 2022 / Online: 7 October 2022 (08:30:43 CEST)
Version 5 : Received: 24 December 2022 / Approved: 26 December 2022 / Online: 26 December 2022 (11:08:05 CET)
Version 6 : Received: 24 February 2023 / Approved: 24 February 2023 / Online: 24 February 2023 (15:41:43 CET)

A peer-reviewed article of this Preprint also exists.

Łukaszyk, S.; Tomski, A. Omnidimensional Convex Polytopes. Symmetry 2023, 15, 755. https://doi.org/10.3390/sym15030755 Łukaszyk, S.; Tomski, A. Omnidimensional Convex Polytopes. Symmetry 2023, 15, 755. https://doi.org/10.3390/sym15030755

Abstract

This paper extends the findings of the prior research concerning n-balls, regular n-simplices, and n-orthoplices in real dimensions using recurrence relations that removed the indefiniteness present in known formulas. The main result of this paper is a proof that these recurrence relations are continuous for complex n, wherein the volume of an n-simplex is a multivalued function for n < 0, and thus the surfaces of n-simplices and n-orthoplices are also multivalued functions for n < 1. Applications of these formulas to n-simplices, n-orthoplices, and n-cubes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects in negative, real dimensions. In particular, it is shown that the volume and surface of a regular n-simplex inscribed in an n-ball are complex for −1 < n < 0, imaginary for n < −1, and divergent with decreasing n; the volume and surface of a regular n-simplex circumscribed about an n-ball is complex for n < 0 and left-handedly respectively convergent to zero or divergent towards infinity; and the volume and surface of an n-orthoplex circumscribed about an n-ball is complex for n < 0 and oscillatory divergent towards infinity with decreasing n.

Keywords

regular basic convex polytopes; negative dimensions; fractal dimensions; complex dimensions; circumscribed and inscribed polytopes

Subject

Computer Science and Mathematics, Geometry and Topology

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