Thursday, October 24, 2019 — 4:00 PM EDT

Dan Ursu, Department of Pure Mathematics, University of Waterloo

"Ruler-compass constructions"

Start with two distinct points on the plane. Between any two points, you can draw a straight line passing through them, or a circle centered at one and passing through the other. New points are given by intersections of distinct lines and/or circles. Repeat. What can you construct?

This is a problem dating back to Euclid and the gang. They were able to bisect an arbitrary angle, but not trisect one. They were able to construct regular polygons with a certain number of sides, but not others (ex: regular pentagons but not regular heptagons). They were also unable to construct a square and a circle having the same area. These problems remained unanswered for two millennia until the 19th century.

In this talk, we will view ruler-compass constructions in the framework of field theory, and see how answering these questions suddenly becomes quite possible.

MC 5479

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