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Schröder’s Equation [The Aleph One Plex]

Introduction You probably know that, for natural number exponents anyways, exponentiation can be treated as repeated multiplication. \displaystyle b^n:=\underbrace{bb\cdots b}_n More formally, we define exponentiation recursively by b^0:=1 and b^{n+1}:=bb^n for n\in\mathbb{N}. You probably also know, however, that exponentiation isn’t solely defined for natural number exponents. Raising a number to the…

Gauss and the Regular Heptadecagon [The Aleph One Plex]

How many regular polygons are constructible? In other words, how many regular polygons can be drawn in a finite number of steps using just a compass and an unmarked straightedge? Since the time of Euclid, geometers knew how to construct the equilateral triangle, the square, and the regular pentagon. Construction of an equilateral triangle inscribed…

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