9: Proof Techniques IV - Magic
If you can keep your head when all about you are losing theirs, it’s just possible you haven’t grasped the situation.
–Jean Kerr
The famous mathematician Paul Erdös is said to have believed that God has a Book in which all the really elegant proofs are written. The greatest praise that a collaborator 1 could receive from Erdös was that they had discovered a “Book proof.” It is not easy or straightforward for a mere mortal to come up with a Book proof but notice that, since the Book is inaccessible to the living, all the Book proofs of which we are aware were constructed by ordinary human beings. In other words, it’s not impossible!
The title of this final chapter is intended to be whimsical – there is no real magic involved in any of the arguments that we’ll look at. Nevertheless, if you reflect a bit on the mental processes that must have gone into the development of these elegant proofs, perhaps you’ll agree that there is something magical there.
At a minimum we hope that you’ll agree that they are beautiful – they are proofs from the Book 2 .
Acknowledgment: Several of the topics in this section were unknown to the author until he visited the excellent mathematics website maintained by Alexander Bogomolny at http://www.cut-the-knot.org/
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- 9.1: Morley’s Miracle
- Probably you have heard of the impossibility of trisecting an angle. Because of the central place of Euclid’s Elements in mathematical training throughout the centuries, and thereby, a very strong predilection towards that which possible via compass and straight-edge alone, it is perhaps not surprising that a perfectly beautiful result that involved trisecting angles went undiscovered until 1899, when Frank Morley stated his Trisector Theorem.
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- 9.2: Five Steps Into the Void
- In this section we’ll talk about another Book proof also due to John Conway. This proof serves as an introduction to a really powerful general technique – the idea of an invariant. An invariant is some sort of quantity that one can calculate that itself doesn’t change as other things are changed. Of course different situations have different invariant quantities.
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- 9.3: Monge’s Circle Theorem
- There’s a nice sequence of matchstick puzzles that starts with "Use nine non-overlapping matchsticks to form 4 triangles (all of the same size).’’ It’s not that hard, and after a while most people come up with. The kicker comes when you next ask them to “use six matches to form 4 (equal sized) triangles.” The answer involves thinking three-dimensionally. Monge’s circle theorem has nothing to do with matchsticks, but it is a sweet example of a proof that works by moving to a higher dimension.
Notes
1. The collaborators of Paul Erdös were legion. His collaborators, and their collaborators, and their collaborators, etc. are organized into a tree structure according to their so-called Erd¨os number, see [5].
2. There is a lovely book entitled “Proofs from the Book” [2] that has a nice collection of Book proofs.