Novel Recurrence Relations for Volumes and Surfaces of N-Balls, Regular N-Simplices, and N-Orthoplices in Integer Dimensions

New recurrence relations for n-balls, regular n-simplices, and n-orthoplices in integer dimensions are submitted. They remove indefiniteness present in known formulas. In negative, integer dimensions volumes of n-balls are zero if n is even, positive if n = -4k - 1, and negative if n = -4k - 3, for natural k. Volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations show that the constant of π is absent in 0 and 1 integer dimensions. It is shown that self-dual n-simplices are undefined for n < -1, while n-orthoplices reduce to the empty set for n ≤ -1. Out of three regular, convex polytopes (and n-balls) present in all non-negative dimensions, only n-orthoplices, n-cubes and n-balls are defined in negative dimensions.