##
Every Sphere Eversion has a Quadruple Point

by Thomas F. Banchoff

Let *i*:S^2 -> R^3 be the inclusion of the sphere, and
let *X*:R^3 -> R^3 be the reflection in the XY plane. An
eversion of the sphere is defined to be a regular homotopy of
immersions that starts with *i* and ends with *Xi*.
In this paper, we show that every eversion must have a quadruple
point.