What is … the Fabricius-Bjerre theorem?

This page hosts information on Felix Günther's talk "What is … the Fabricius-Bjerre theorem?" at the "What is …?" seminar. The talk will take place on Friday, June 22, 1:00pm at the BMS Loft at Urania. This talk will help you better understand the talk by Tom Banchoff, which will start at 2pm.

We will again be ordering delivery pizza. If you would like to order pizza with us, please arrive to the "What is …?" seminar by 12:45pm.

Abstract

The Fabricius-Bjerre theorem states that for a generic curve in the plane, the number of crossings plus half the number of inflections plus the number of opposite-side double tangencies is equal to the number of same-side double tangencies. We will define each of the quantities referred to in the theorem and look at some examples before we give the original (and very beautiful) proof of Fabricius-Bjerre himself.

In the case of polygons with vertices in general position, similar definitions are given for crossings, opposite-side and same-side double tagencies, and inflection edges. We will sketch the proof of Tom Banchoff for the polygonal version of the Fabricius-Bjerre theorem, which is based on a deformation argument.

In the end, we will look at generic curves in the sphere, and give the Spherical Fabricius-Bjerre formula.

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