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Difference between revisions of "Pythagorean theorem, multi-dimensional"

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Consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102801.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102802.png" /> (with the usual metric and measure). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102803.png" /> be a point on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102804.png" />th coordinate axis and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102805.png" /> be the origin. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102806.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102807.png" />-dimensional volume of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102808.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p1102809.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028010.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028011.png" />-dimensional volume of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028012.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028013.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110280/p11028014.png" />.
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Consider the $n$-dimensional space $\mathbf R^n$ (with the usual metric and measure). Let $A_i$ be a point on the $i$th coordinate axis and let $O$ be the origin. Let $s$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $A_1\ldots A_n$ and let $s_i$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $OA_1\ldots A_{i-1}A_{i+1}\ldots A_n$. Then $s^2=\sum_{i=1}^ns_i^2$.
  
 
For other and further generalizations of the classical [[Pythagoras theorem|Pythagoras theorem]], see [[#References|[a2]]] and the references therein.
 
For other and further generalizations of the classical [[Pythagoras theorem|Pythagoras theorem]], see [[#References|[a2]]] and the references therein.

Latest revision as of 14:17, 14 August 2014

Consider the $n$-dimensional space $\mathbf R^n$ (with the usual metric and measure). Let $A_i$ be a point on the $i$th coordinate axis and let $O$ be the origin. Let $s$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $A_1\ldots A_n$ and let $s_i$ be the $(n-1)$-dimensional volume of the $(n-1)$-dimensional simplex $OA_1\ldots A_{i-1}A_{i+1}\ldots A_n$. Then $s^2=\sum_{i=1}^ns_i^2$.

For other and further generalizations of the classical Pythagoras theorem, see [a2] and the references therein.

References

[a1] Etsua Yoshinaga, Shigeo Akiba, "Very simple proofs of the generalized Pythagorean theorem" Sci. Reports Yokohama National Univ. Sect. I , 42 (1995) pp. 45–46
[a2] D.R. Conant, W.A. Beyer, "Generalized Pythagorean theorem" Amer. Math. Monthly , 81 (1974) pp. 262–265
How to Cite This Entry:
Pythagorean theorem, multi-dimensional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_theorem,_multi-dimensional&oldid=32914
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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