De Gua's theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search
tetrahedron with a right-angle corner in O

De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves.

If a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces.

 A_{ABC}^2 = A_{\color {blue} ABO}^2+A_{\color {green} ACO}^2+A_{\color {red} BCO}^2

History[edit]

Jean Paul de Gua de Malves (1713–85) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Tinseau d'Amondans (1746–1818), as well. However the theorem had been known much earlier to Johann Faulhaber (1580–1635) and René Descartes (1596–1650).[1][2]

Generalisations[edit]

The Pythagorean theorem and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplices with a right-angle corner. This, in turn, is a special case of a yet more general theorem, which can be stated as follows.[3]

Let P be a k-dimensional plane in \mathbb{R}^n (so k \le n) and let C be a compact subset of P. For any subset I \subseteq \{ 1, \ldots, n \} with exactly k elements, let C_I be the orthogonal projection of C onto the linear span of e_{i_1}, \ldots, e_{i_k}, where I = \{i_1, \ldots, i_k\} and e_1, \ldots, e_n is the standard basis for \mathbb{R}^n. Then

\mbox{vol}_k^2(C) = \sum_I \mbox{vol}_k^2(C_I),

where \mbox{vol}_k(C) is the k-dimensional volume of C and the sum is over all subsets I \subseteq \{ 1, \ldots, n \} with exactly k elements.

This theorem is essentially the inner-product-space version of Pythagoras’ theorem applied to the kth exterior power of n-dimensional Euclidean space. De Gua's theorem and its generalisation (above) to n-simplices with right-angle corners correspond to the special case where k = n−1 and C is an (n−1)-simplex in \mathbb{R}^n with vertices on the co-ordinate axes.

Notes[edit]

  1. ^ Weisstein, Eric W., "de Gua's theorem", MathWorld.
  2. ^ Hans-Bert Knoop: Ausgewählte Kapitel zur Geschichte der Mathematik. Lecture Notes (University of Düsseldorf), p. 55 (§ 4 Pythagoreische n-Tupel, p. 50-65) (German)
  3. ^ Theorem 9 of James G. Dowty (2014). Volumes of logistic regression models with applications to model selection. arXiv:1408.0881v3 [math.ST ]

References[edit]

Further reading[edit]

  • Kheyfits, Alexander (2004). "The Theorem of Cosines for Pyramids". The College Mathematics Journal (Mathematical Association of America) 35 (5): 385–388. JSTOR 4146849.  Proof of de Gua's theorem and of generalizations to arbitrary tetrahedra and to pyramids.