# Pythagorean analogue [closed]

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 $a^3+b^3=c^3$

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## closed as off topic by David Roberts, Gerry Myerson, Andrés Caicedo, HJRW, David LoefflerOct 10 '11 at 7:15

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I presume you mean a 3-dimensional geometric shape with polygons for faces. Such a thing needs at minimum 4 faces (simplex). As for your first question, take $\mathbb{R}^2$ with the $\ell_3$ norm. If the previous sentence doesn't make sense to you, then MO is not the site you are looking for as it is for research-level mathematicians. math.stackexchange.com welcomes mathematical questions at all levels. –  David Roberts Oct 10 '11 at 4:40
Also posted to m.se, math.stackexchange.com/questions/71271/… –  Gerry Myerson Oct 10 '11 at 4:45

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

... 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:

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

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