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