On the Basic Convex Polytopes and n-Balls in Complex Dimensions

This paper 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. The main result of this paper is a proof that these recurrence relations are continuous for complex n, wherein the volume of an n-simplex is a multivalued function for n < 0, and thus the surfaces of n-simplices and n-orthoplices are also multivalued functions for n < 1. Applications of these formulas to n-simplices, n-orthoplices, and n-cubes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects in negative, real dimensions. In particular, it is shown that the volume and surface of a regular n-simplex inscribed in an n-ball are complex for −1 < n < 0, imaginary for n < −1, and divergent with decreasing n; the volume and surface of a regular n-simplex circumscribed about an n-ball is complex for n < 0 and left-handedly respectively convergent to zero or divergent towards infinity; and the volume and surface of an n-orthoplex circumscribed about an n-ball is complex for n < 0 and oscillatory divergent towards infinity with decreasing n.