In order to better understand the physical structure of space, it is important to first of all establish that space is a three dimensional construction. We can have rotation in a two dimensional plane, with the rotation axis in the third dimension. Pythagoras's theorem applies in three dimensional Euclidean space with the angles in the triangle being related to rotation. The question is, could Pythagoras's theorem hold in space of higher dimensions? The reason for the question is because it is freely used in the Minkowski four dimensional space-time continuum of Special Relativity. Pythagoras's theorem worked its way into the four dimensional Minkowski world simply through two successive applications of the ordinary three dimensional version. But does this then become a genuine four dimensional usage? In the linked article it is suggested that apart from in three dimensions, the only other possibility might be for Pythagoras's theorem to hold in seven dimensions.
The article concludes that the Pythagorean Trigonometric Identity can only hold if there is a unique dimension for the rotation axis of the involved angles. This therefore rules out all dimensions, n, except where n = 3. Even two dimensions must be ruled out, because in such a flat space we cannot have an axis of rotation. It is necessary however to take a closer look at the special case of seven dimensions, owing to the manner in which it neatly fits in with the Lagrange identity.
Frederick David Tombe