[math-fun] Pythagorean relation in arithmetic progression in higher dimensions

The (3,4,5) triangle has sides in arithmetic progression.  I was curious in what dimensions can the sides and main diagonal of a rectangular box be in arithmetic progression.  Here's what I found. In 2 dimensions, all solutions are multiples of (3,4,5). In 3, 4, and 5 dimensions, arithmetic progression is possible only when the lengths are allowed to be quadratic surds. In 6 or more dimensions, arithmetic progression requires complex lengths.   --  Gene