The Joy of Mathematics by Alfred S. Posamentier Robert Geretschläger Charles Li Christian Spreitzer

Author:Alfred S. Posamentier,Robert Geretschläger,Charles Li,Christian Spreitzer
Language: eng
Format: mobi
ISBN: 9781633882980
Publisher: Penguin Random House LLC (Publisher Services)
Published: 2017-07-24T00:00:00+00:00

THE PYTHAGOREAN THEOREM EXTENDED TO THREE DIMENSIONS

In the concluding section regarding the Pythagorean theorem, we will take another look at it from a three-dimensional aspect. Imagine cutting a corner off a rectangular solid. The piece represented by the corner is a tetrahedron with three of its faces as right triangles. In figure 3.55 you will find such a tetrahedron for which point P is the vertex of the original rectangular solid. A lovely extension of the Pythagorean theorem can be shown on this geometric solid. Namely, the sum of the squares of the areas of the three right-triangle faces is equal to the square of the area of the triangle representing the remaining face of the tetrahedron. Using the tetrahedron pictured in figure 3.55, we have the following: (Area X)2 + (Area Y)2 + (Area Z)2 = (Area ∆ABC)2, which is a nice analog to the Pythagorean theorem. This relationship is known as De Gau's theorem, named after the French mathematician Jean Paul de Gau de Malves (1712–1785), although it appears to have been known to the German mathematician Johann Faulhaber (1580–1635) and the famous French mathematician René Descartes (1596–1650).

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