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By L. Auslander, R. Tolimieri

ISBN-10: 0387071342

ISBN-13: 9780387071343

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Extra resources for Abelian Harmonic Analysis, Theta Functions and Functional Analysis on a Nilmanifold

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The middle of those three points is clearly closer to the other line. Figure 5. Kelly’s proof from ‘the Book’ As with Apostol’s proof of the irrationality of √2, we can see the power of the right minimal configuration. Aesthetic appeal often comes from having this characteristic: that is, its appeal stems from being able to reason about an unknown number of objects by identifying a restricted view that captures all the possibilities. This is a process that is not so very different from that powerful method of proof known as mathematical induction.

300BCE Chapter 2 – Beauty and Truth in Mathematics 45 Pappus’s theorem Let ABC be any triangle with parallelograms ACDE and ABFG constructed externally on the sides AC and AB. Let the rays DE and FG meet in point H, and construct BJ and CK equal and parallel to HA. Then the sum of the areas of the parallelograms ACDE and ABFG equals the area of the parallelogram BCKJ. Figure 5: Pappus’s theorem, Book IV, The Collection, fourth century CE Pappus had the vision to see beyond a right triangle and the insight to find the right generalization of the Pythagorean theorem.

Fear of mathematics certainly does not hasten an aesthetic response. Gauss, Hadamard and Hardy Three of my personal mathematical heroes, very different individuals from different times, all testify interestingly on the aesthetic and the nature of mathematics. Gauss Carl Friedrich Gauss is claimed to have once confessed, “I have had my results for a long time, but I do not yet know how I am to arrive at them” (in Arber, 1954, p. 47). [1] One of Gauss’s greatest discoveries, in 1799, was the relationship between the lemniscate sine function and the arithmetic–geometric mean iteration.

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