Continuous Recurrence Relations for Basic Convex Polytopes and n-Balls in Complex Dimensions
The results of previous research on n-balls, regular n-simplices, and n-orthoplices in real dimensions are extended in this study by employing recurrence relations that eliminated the ambiguity contained in established formulas. In the prior study, it was demonstrated that the volumes of n-balls in the negative, integer dimensions are zero if n is even, positive if n = 4k 1, and negative if n = 4k 3 for natural k. Additionally, it was demonstrated that the volumes and surfaces of n-cubes inscribed in n-balls in the negative, integer dimensions are complex and related to integral powers of the imaginary unit. It was further demonstrated that, for n 1, n-simplices are undefined in the negative, integer dimensions, and their real volumes and imaginary surfaces are divergent in the negative, fractional ones with decreasing n. In contrast, n-orthoplices reduce to the empty set in the negative, integer dimensions. Probabilistic fractal metrics take into consideration negative dimensions. Out of the three regular, convex polytopes that exist in all natural dimensions, only the n-orthoplices and n-cubes (and n-balls) are defined in the negative, integer dimensions. This work demonstrates that for complex n, these recurrence relations are continuous.
Łukaszyk Patent Attorneys, Ul. Głowackiego 8, 40-052 Katowice, Poland.
Please see the link here: https://stm.bookpi.org/NFPSR-V2/article/view/8205
Keywords: Regular convex polytopes, negative dimensions, fractal dimensions, complex dimensions, recurrence relations