On the Omnidimensional Convex Polytopes and n-Balls in Negative, Fractional and Complex Dimensions

Szymon Łukaszyk

doi:10.20944/preprints202209.0089.v4

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preprints.org > computer science and mathematics > geometry and topology > doi: 10.20944/preprints202209.0089.v4

Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

On the Omnidimensional Convex Polytopes and n-Balls in Negative, Fractional and Complex Dimensions

Szymon Łukaszyk

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

The study shows that the recurrence relations defining volumes and surfaces of omnidimensional convex polytopes and n-balls are continuous and defined for complex n, whereas in the indefinite points their values are given in the sense of a limit of a function. The volume of an n-simplex is a bivalued function for n < 0, and thus the surfaces of n-simplices and n-orthoplices are also bivalued functions for n < 1. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects in negative, real dimensions. In particular for 0 < n < 1 the volumes of the omnidimensional polytopes are larger than the volumes of circumscribing n-balls, while their volumes and surfaces are smaller than the volumes of inscribed n-balls. Specific products and quotients of volumes and surfaces of the omnidimensional polytopes and n-balls are shown to be independent of the gamma function.

Keywords

regular basic convex polytopes; circumscribed and inscribed polytopes; negative dimensions; fractal dimensions; complex 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: 7 October 2022
Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author

Comment: 1. Amended introduction (concept of a dimension as a point on a semiaxis).
2. Shortened and linked sections 2 and 3.
3. New formulas (40), (41), (43), (44), (52), (54).
4. Novel relations (62), (73).
5. Amended relations (75), (76).
6. Novel relations (77), (80).
7. New and amended drawings.

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