Introduction to the Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

This theorem can be written as an equation relating the lengths of the sides a, b, and c, often called the Pythagorean equation: a² + b² = c², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

The theorem is named after the ancient Greek mathematician Pythagoras, who lived in the 6th century BC. Although the theorem had been known earlier in various cultures, including Babylonian and Indian mathematics, Pythagoras and his followers are credited with providing one of the earliest rigorous proofs.

Historical Context

Evidence of knowledge of the relationship appears on clay tablets from ancient Mesopotamia dating back more than a thousand years before Pythagoras. In China, the Zhoubi Suanjing text discusses the theorem, and in India it appears in the Baudhayana Sulba Sutra.

Many different proofs of the theorem exist. Euclid provided a geometric proof in his Elements. Algebraic proofs and proofs based on similar triangles are also common in modern textbooks.

Applications

Beyond pure geometry, the Pythagorean theorem is used extensively in trigonometry, coordinate geometry, physics, engineering, architecture, and computer graphics. It forms the basis for the distance formula in the Cartesian plane and appears in the definitions of the trigonometric functions sine and cosine.

In three dimensions the theorem generalizes to the relationship between the space diagonal of a rectangular box and its edge lengths. Further generalizations lead to the law of cosines and to notions of distance in higher-dimensional Euclidean spaces.

Let X and Y be smooth oriented manifolds of dimension m and n, and let f:X→Y a proper smooth map. There is a theorem called the "Disintegration Theorem" which says roughly that a measure on X induces a measure on the fibers of f (except for a set of measure 0) such that integrating over fibers and then over Y is the same as integrating over X.

Now, since X is a smooth manifold, most measures that anyone would care about are given by m-forms Ω, so disintegration of such a measure would hopefully give us n−m-forms on the regular fibers of f.

In fact, I sort of see how one would define this: the pushforward of Ω to Y is a top form there, and there is a dual m-vector field on Y that pairs to 1 with this top form (maybe with some singularities at non-regular fibers). At any regular point in X, we can choose a subspace in TxX which maps isomorphically to Tf(x)Y, and transfer the m-vector field to this subspace, and contract with Ω and pull back to f−1(f(x)). This should be uniquely defined because the components where you've contracted with vectors parallel to the fiber drop out when you do the pull-back.

I don't especially like this phrasing, since it is quite non-canonical, but I'm confident it must work because the disintegration theorem is true.

I have to imagine that I am not the first person to have had this thought. Is there some better way of describing this process and making computations with it? There should be a text book or something that covers this, but I cannot find it. Anyone have suggestions?

Abstract:We prove a generalisation of the disintegration theorem to the setting of multifunctions between Polish probability spaces. Whereas the classical disintegration theorem guarantees the disintegration of a probability measure along the partition of the underlying space by the fibres of a measurable function, our theorem gives necessary and sufficient conditions for the measure to disintegrate along a cover of the underlying space defined by the fibres of a measurable multifunction. Building on this theorem, we introduce a new statistical notion: We declare a metric Polish probability space to be asymptotically disintegrable if n i.i.d.-centred balls of decreasing radius carry a disintegration of the measure with probability tending to unity as n→∞. We give a number of both 1-dimensional and higher-dimensional examples of asymptotically disintegrable spaces with associated quantitative rates, as well as a strong counterexample. Finally, we give two applications of the notion of asymptotic disintegrability. First, we prove that asymptotically disintegrable spaces admit an easy high-probability quantification of the law of large numbers in Wasserstein space, which in all dimensions either recovers or improves upon the best known rates in some regimes, and is never any worse than existing rates by more than a factor of 2 in the exponent of n, where n is the number of sample points. Second, we prove that any asymptotically disintegrable space admits a high-probability bound on the error in approximating the expectation of any Lipschitz function by its empirical average over an i.i.d.\ sample. The bound is average-case in the sense that it depends only on the empirical average of the local Lipschitz constants of the function, rather than the global Lipschitz constant as obtained by Kantorovich-Rubinstein duality.
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