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The Higher-Dimensional Pythagorean Theorem

The Higher-Dimensional Pythagorean Theorem

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1. Introduction We have all heard of the familiar Pythagorean theorem, relating the side lengths of a right triangle. But, have you heard of De Gua’s Theorem? An interesting generalization of the Pythagorean theorem into 3 dimensions, this theorem is very similar result for “right tetrahedrons” where three of the edges are mutually perpendicular. Suppose, in such a tetrahedron we have     Denote the area of as . Then,     In today’s post, we will investigate whether a…

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Volume of a n-dimensional Ball: Part 2

Volume of a n-dimensional Ball: Part 2

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1. Introduction The first installment of this post investigated the question of computing the volume of a unit ball (and by extension any ball) of dimensions. The main result we obtained was that the volume can be computed recursively as , where     In this post, we will use this result to prove an explicit formula for . 2. The Formula A quick search online yields the following explicit formula for :     where the function extends the…

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Volume of a n-dimensional Ball: Part 1

Volume of a n-dimensional Ball: Part 1

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1. Introduction This post investigates an interesting question I had been thinking about: we all know that the area of a circle of radius equals , and the volume of a sphere of radius equals . But what if we enter the territory of even higher dimensions? We should be able to say something about the “volume” of an n-dimensional sphere, where “volume” is interpreted as the equivalent measure of space in that many dimensions (e.g. area for 2D, regular…

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