User:Kmhkmh/sandbox

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[edit] Simplexes with an "orthogonal corner"

Orthogonal corner means here, that there is a vertex at wich all adjacent hyperfaces are pairwise orthogonal. Such simplexes are generalizations of right angle triangles and for them there exists an n-dimensional version of the Pythagorean theorem:

The sum of the squared n-dimensional volumes of the hyperfaces adjacent to the orthogonal corner equals the squared n-dimensional volume of the hyperface opposite of the orthogonal corner.

\sum_{k=1}^{n} |A_{k}|^2 = |A_{0}|^2

where A_{1} \ldots A_{n} are hyperfaces being pairwise orthogonal to each other but not orthogonal to A0, which is the hyperface opposite of the orthogonal corner.

For a 2-Simplex the theorem is the the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetraeder with a cube corner.

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