Question

Is there any consistent geometry in which $a^3+b^3=c^3$ holds for right-angled triangles with 'hypothenuse' $c$?

Is there any natural generalization of pythagoras theorem to higher dimensions?

Is there any geometric shape with $3$ sides, with areas $a,b,c$ and $a^3+b^3=c^3$

Answer 1

Is there any natural generalization of pythagoras theorem to higher dimensions?

From http://www.cs.bc.edu/~alvarez/NDPyt.pdf :

... we will restrict our attention to the analogs of right triangles. We will call these orthogonal tetrahedrons. An orthogonal tetrahedron is any tetrahedron which has a vertex at which three faces meet at right angles to each other ... we refer to [these faces] as the orthogonal faces, and we refer to the remaining face as the opposing face ...

Let $A$ be the area of the opposing face, and let $A_1 , A_2 , A_3$ be the areas of the orthogonal faces of a given orthogonal tetrahedron. Then the following relation holds: $A^2 = A_1^2 + A_2^2 + A_3^2$

Next we address the n-dimensional case ...

Let $A$ be the area of the opposing face, and let $A_1, A_2, ..., A_n$ be the areas of the orthogonal faces of a given orthogonal n-simplex. Then the following relation holds:

$$A^2 = \sum_{j=1}^n A_j^2$$