For more proofs of the Pythagorean theorem, including the one created by former U.S. President James Garfield, visit this site. Another resource, The Pythagorean Proposition, by Elisha Scott Loomis, contains an impressive collection of 367 proofs of the Pythagorean theorem.

The Pythagorean theorem can be extended in its breadth and usage in many ways. For example, the theorem can be extended to 3 dimensions: the squared distance between diagonal corners of a cube is equal to the squared distance of the length, width, and height of the cube. In the same way, though perhaps difficult to visualize, the theorem can be extended to any number of dimensions. As well, the theorem can be extended to apply to a trirectangular tetrahedron, as outlined in de Gua’s theorem.

One of the implications that results from the Pythagorean theorem is the inevitability of irrational numbers, numbers that cannot be expressed as a ratio of two integers. For example, a triangle with side lengths of 1 would have a hypotenuse of √2. This TED Ed video explores more about the shattering impact of this discovery.

The Pythagorean theorem is based on the propositions of Euclidean geometry, the geometry of planes or flat surfaces. In fact, Pythagorean theorem is shown to be synonymous with the parallel postulate, the proposition that only one line can be drawn through a certain point so that it is parallel to a given line that does not contain the point. When the parallel postulate is altered, geometries of surfaces with positive and negative curvatures emerge, and the Pythagorean theorem no longer holds true. Modifications to the Pythagorean theorem are then needed for both elliptical and hyperbolic surfaces.

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