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Omnidimensional Convex Polytopes
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)
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 volumes and surfaces of $n$-balls, $n$-simplices, and $n$-orthoplices are holomorphic functions of $n$, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about $n$-balls reveal previously unknown properties of these geometric objects. In particular, for $0 < n < 1$, the volumes of the omnidimensional polytopes are larger than those of circumscribing $n$-balls, and both their volumes and surfaces are smaller than those of inscribed $n$-balls. The surface of an $n$-simplex circumscribing unit diameter $n$-ball is spirally convergent to zero with real $n$ approaching negative infinity but first has a local maximum at $n=-3.5$. The surface of an $n$-orthoplex circumscribing unit diameter $n$-ball is spirally divergent with real $n$ approaching negative infinity but first has a local minimum at $n=-1.5$, where its real and imaginary parts are equal to each other; similarly, as its volume, where the similar local minimum occurs at $n=-3.5$. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about $n$-balls are proposed. symmetries of products and quotients of volumes in complex dimensions $n$ and $-n$ and of surfaces in complex dimensions $n$ and $2-n$ are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between volumes and surfaces in dimensions $n = -1/2$ and $n = 1/2$.
Keywords
regular basic convex polytopes; circumscribed and inscribed polytopes; negative dimensions; fractal dimensions; complex dimensions; emergent dimensionality; mathematical physics
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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Commenter: Szymon Łukaszyk
Commenter's Conflict of Interests: Author
2. Theorem 4 concerning properties of the reflection function.
3. Clarity and grammar improvements.