

For once, an economist is talking about an "equation"—which makes it worth a closer look



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The midpoint of two points and the center of gravity of a triangle are familiar concepts from middle school onward. What do they have in common? How can they be generalized? In the early 19th century, a new approach to geometry put them on a firm mathematical footing through the concept of the barycenter.

Three sides completely determine a triangle, up to orientation. With four sides, infinitely many polygons can be constructed. Nevertheless, many general properties can be established for arbitrary convex quadrilaterals.

What could a quadratic polynomial possibly have in common with a vector in three-dimensional space? At first glance, nothing: they are different kinds of objects. Yet both have the same form—each is described by a triple of numbers. Better still, calculations with one correspond to calculations with the other!

Because Augustin-Louis Cauchy did not take a direct interest in solving algebraic equations, he is an overlooked figure in the history of group theory. Yet his research on permutations provided valuable tools for those who worked on Galois theory.
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