Pythagorean Triples and the Generation of Space

The traditional formulation of the theorem of Pythagoras is that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides. This is the Greek formulation, but we know that some of the properties of Pythagorean triangles were known in earlier cultures, such as in Babylonia.
The triangles of most interest were those constructed using integral values, such as the 3,4,5 triangle. These values were commensurable, and not irrational.
If we cons...