Chapter VI: Stalagmites joining with stalagtites (perturbing nodes). (Image courtesy of Talia Blum. Used with permission.)
This long chapter is a visual introduction to algebraic geometry, done entirely over the real numbers rather than the complex numbers, and postponing projective space to the end. Since we assume neither knowledge of the implicit function theorem nor any algebra background, the discussion is necessarily informal. Some of the terminology is nonstandard, such as my use of “oval,” and many statements are far from their optimal versions. Lectures 20, 24-25, and 27 are still in a bit of a raw state; please be aware of that.
The relation between mechanical linkages and algebraic curves is a classical topic, still represented in some modern textbooks (such as C. Gibson, Elementary geometry of algebraic curves, Cambridge Univ. Press, 1998; which I have also used as a source of examples of algebraic curves). Lectures 21–25 are influenced by Viro’s survey articles, in particular O. Viro, Introduction to topology of real algebraic varieties (a chapter of an unpublished book project) and O. Viro, Dequantization of real algebraic geometry on logarithmic paper. European Congress of Mathematics, Vol. I, 135-146, Birkhauser, 2001. My source for configurations of lines was the wonderful classical book of Hilbert-Cohn-Vossen.
Lecture 19: Introduction to Algebraic Curves
Lecture 20: Mechanical Linkages and Polynomial Equations
Lecture 21: Intersections of Algebraic Curves
Lecture 22: Nonsingular Curves
