A solid right angle is one which includes the eighth part of the sphere, even if it is not made up of three plane right angles. The sum of all the plane angles which bound it is equal to three right angles.

Then comes this theorem:

Just as in a plane polygon the sum of all the exterior angles is 4 plane right angles: so in a polyhedron the sum of all the exterior solid angles is 8 solid right angles.

with

By exterior angle I understand the curvature & bending of the planes with respect to one another, which can be measured from the plane angles which bound the solid angle. For that part by which the sum of all the plane angles making up a solid angle is less than 4 plane right angles, represents the exterior solid angle.

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:

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

At each corner there are three face angles, two from rectangles (90 degrees each) and one from a hexagon (120 degrees). The bending is thus 360 - (2 x 90 + 120) = 60 degrees. Since there are twelve corners the total bending is 12 x 60 = 720 degrees = 8 right angles.

This remarkable theorem was lost for over 200 years!

In this column we will examine this theorem and its remarkable history, including what happened while it was lost: Euler's Theorem, the Euler Characteristic, and the Gauss-Bonnet Theorem.

• The strange history of Descartes' Theorem

• Euler's Theorem

• The Euler characteristic

• Curvature and polyhedral curvature

• The polyhedral Gauss-Bonnet Theorem

--Tony Phillips

 
Comments: webmaster@ams.org
© copyright 1999, American Mathematical Society.