866: Compass and Straightedge
|Compass and Straightedge|
Title text: The Greeks long suspected this, but it wasn't until April 12th of 1882 that Ferdinand von Lindemann conclusively proved it when he constructed himself the most awesome birthday party possible and nobody showed up.
Compass and straightedge constructions are a class of problems in classical geometry. They take the form "Using only a compass and a straightedge, construct X", where X is a geometric figure such as a regular pentagon. The subject is typically covered in high school mathematics. Three such constructions (squaring the circle, trisecting the angle and doubling the cube) remained unsolved for thousands of years before being shown impossible with the use of modern algebraic techniques.
The comic begins as if it were stating a problem in classical geometry but veers into an observation that no amount of technical knowledge can substitute for human companionship. An additional layer of humor is that Cueball is a stick figure so technically it is possible to create friends with a straightedge and a compass, a figure constructed like Cueball is.
Ferdinand von Lindemann was a German mathematician who showed in 1882 that pi is not a zero of any polynomial with rational coefficients, i.e. it is a transcendental number. Transcendental numbers cannot be constructed with straightedge and compass. This proves that squaring the circle (a problem where it is required to construct a square with the same area as a given circle) is impossible, being as the sides of the square would need to be √π times the radius of the circle, and pi is not constructible.
- I learned in high school what geometers discovered long ago:
- [Cueball, holding a compass and straightedge, looks sad.]
- Using only a compass and straightedge, it's impossible to construct friends.
- Of the three non-constructibles, only squaring the circle is considered truly impossible, even by nonmathematicians. Why? Because if you bend the rules a bit by marking your straight edge twice you can trisect an angle or double a cube. Similar fudges are used in non-Euclidean constructions like origami construction to solve all but squaring the circle, which, as von Lindemann proved, is impossible because of the transcendental nature of π. That did not stop Edwin J. Goodwin from proposing the Indiana Pi Bill, however. Be glad it didn't pass.