Very belatedly catching up on Plus articles. I enjoy these articles - although as a Masters student the maths is familiar to me already, the authors are clearly very interested in their topics and that enthusiasm comes through! Thank-you for your enthusiasm.
In the section where Pythagorean quadruples are introduced, your geometrical explanation is slightly incorrect - the diagonal of the box is of course sqrt(a^2+b^2+c^2) and not simply a^2+b^2+c^2 as written. Similarly for the diagonal of the bottom face in the diagram.