Continuous Recurrence Relations for Basic Convex Polytopes and n-Balls in Complex Dimensions

Abstract

This study 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. It was shown in the previous study that in the negative, integer dimensions, the volumes of n-balls are zero if n is even, positive if n = -4k - 1, and negative if n = -4k - 3, for natural k. It was also shown that the volumes and the surfaces of n-cubes inscribed in n-balls in negative dimensions are complex and associated with integral powers of the imaginary unit in the negative, integer dimensions. It was further shown that for n < -1 n-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones and divergent with decreasing n, whereas in the negative, integer dimensions, n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in the negative, fractional ones with decreasing n. Negative dimensions are considered in probabilistic fractal measures. Only the n-orthoplices and n-cubes (and n-balls) are defined in the negative, integer dimensions out of the three regular, convex polytopes that exist in all natural dimensions. This study shows that these recurrence relations are continuous for complex n.