Descartes' Lost Theorem

1. The sum of the plane angles of a polyhedron

René Descartes 1596-1650

Somewhere in the years 1619-1621 René Descartes wrote an Elementary Treatise on Polyhedra in which, very near the beginning, the following statement appears: If four plane right angles are multiplied by the number of solid angles & from the product are subtracted 8 plane right angles, there remains the sum of all the plane angles which exist on the surface of that polyhedron.

For example, a right prism on a regular hexagonal base has twelve solid angles. Continuing with Descartes' terminology, but converting to degrees for the calculation (one plane right angle = 90 degrees):

_____ / \ / /| |\ _____ / | | | | | | | | / \|_____|/ Hexagonal prism has twelve solid angles.

Four plane right angles multiplied by the number of solid angles = 48 plane right angles ( = 4320 degrees). Subtracting 8 plane right angles (720 degrees) gives 3600 degrees. The prism has two hexagonal and six rectangular faces. Each of the six face angles of a regular hexagon measures 120 degrees; total for both hexagons = 12 x 120 = 1440 degrees. Each of the four face angles of a rectangle measures 90 degrees; total for all six rectangles = 24 x 90 = 2160 degrees. The sum of all the plane angles which exist on the surface = 1440 + 2160 = 3600 degrees.

This remarkable theorem is not part of the history of mathematics. It was never published, and lay hidden for over 200 years.

In this column we will examine Descartes Treatise as a mathematical time capsule, giving us a look back at the beginning of the XVII century, and showing us how its author's geometrical intuition allowed him to anticipate some of the more interesting mathematical discoveries of the next two centuries.

• 1. The sum of the plane angles of a polyhedron

• 2. The strange history of Descartes' Treatise

• 3. Euler's Theorems

• 4. The Euler characteristic

• 5. Descartes' curvatura

• 6. Curvature and polyhedral curvature

• 7. The polyhedral Gauss-Bonnet Theorem

--Tony Phillips