You&#x27;ve made the same error as the OP.  The volume of a unit n-dimensional sphere decreases exponentially in n.  But that doesn&#x27;t mean you need to increase r exponentially to compensate - in the volume formula, the r also has an exponent of n.   The distance between opposite corners of a unit cube in n-dimensional space is root(n).  That&#x27;s hardly exponential.