# Difference between revisions of "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.

## Explanation

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, in case you were wondering - but see the trivia section) 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.

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.

## Transcript

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.

## Trivia

Of the three non-constructables, 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.

# Discussion

No, the comic is funny because many geometrical theorems prove something along the lines of "With a compass and straightedge you cannot construct..." (e.g. a square and a circle with the same area) If you have knowledge of this type of proof, the humor is that you think he's about to talk about something that is impossible in geometry, but really he's talking about the inapplicability of geometry to real life. This is often a difficulty with nerds and brainy people, they try to apply their theoretical knowledge to human relationships and fail. 75.103.23.206 19:53, 13 December 2012 (UTC)

And then there's the converse: people who are able to apply theoretical knowledge and succeed. 76.106.251.87 04:33, 5 June 2013 (UTC)

The explanation mentions that there are "three such constructions", but doesn't go any further. What they are should at least be addressed (or linked to), even if we're not going to elaborate on the "why" of their impossibility. For the uninitiated, they are squaring the circle, trisecting any angle, and doubling the cube. 76.106.251.87 04:33, 5 June 2013 (UTC)

If such constructions are "impossible with the use of modern algebraic techniques," then why don't we just use older algebraic techniques?  ;) 213.203.138.251 (talk) (please sign your comments with ~~~~)

Those "modern algebraic techniques" just did prove that you can't solve this constructions by using only "classical geometry".--Dgbrt (talk) 18:14, 29 June 2013 (UTC)

I tried forming a club for compasses and straight edges but no one signed up :( ~JFreund

Could the “most awsome birthday party“ bear another deeper meaning, for example be analogue to the rational polynom with rational coefficients? 162.158.202.100 04:30, 9 April 2017 (UTC)

Could this "most awesome birthday party" be a reference to Stephen Hawking's party for time travelers? Spagovir (talk) 18:33, 10 August 2017 (UTC)

I'm pretty sure you're right, it sounds like a Hawking party reference to me too. - CRGreathouse (talk) 06:17, 8 August 2018 (UTC)

Anyone think of adding a reference to 710 Collatz Conjecture? Where in a similar vein, he begins by explaining the steps for applying the conjecture, then finishes up by losing all his friends. 108.162.246.17 16:26, 8 August 2018 (UTC)