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

This study examines n -balls, n -simplices, and n -orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present in known formulas. They show that in the negative, integer dimensions, the volumes of n -balls are zero if n is even, positive if n = −4 k − 1, and negative if n = −4 k − 3, for natural k . The 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 are continuous for n ∈ ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < −1, self-dual 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 . In the negative, integer dimensions, n -orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n . Out of three regular, convex polytopes present in all natural dimensions, only n -orthoplices and n -cubes (and n -balls) are defined in the negative, integer dimensions.