Suppose we want to obtain a figure from the sum of three numbers..
Suppose we want to obtain a figure from the sum of three numbers such that a^n+b^n=c^n. If we consider n dimensions starting from n=1, we obtain:
n=1: (It's impossible to obtain a figure even if, for example, a^1+b^1=c^1.)
n=2: Possible! (Right-angled triangle).
n=3 (Impossible?) The figure would be a triangular prism, but it turns out that a^3+b^3<>c^3. That is, the sum of the volumes of the cubes built on the legs is not equal to the volume of the cube built on the hypotenuse!
n=4 ?
Why does it seem to be possible only for n=2?