VISUAL MATHEMATICS "I see dead easy proofs" Joe Hurd @gilith http://www.gilith.com/ Ignite Portland 9 23 September 2010 _________________________________________________________________________ BIOGRAPHY Joe works for Galois, Inc. in Portland, OR, where he uses functional programming languages to construct correctness proofs of software. He studied at the University of Cambridge, receiving degrees in mathematics and then computer science as he made the realization that it was easier for him to program computers to find proofs than to do them by hand. In his free time Joe contributes to open source projects, carries out independent research and regularly gives talks to audiences at all levels. _________________________________________________________________________ ABSTRACT It is a sad fact that many people believe that mathematics is about doing arithmetic and memorizing formulas. However, if these people were to ask a mathematician what their profession is about they might be surprised by the answer. "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." So said the number theorist G. H. Hardy in his 1940 essay 'A Mathematician's Apology', and in this talk I will attempt to convey this alternative point of view using a series of examples of visual mathematics. Each example will take the form of an easy-to-explain statement about numbers or geometric shapes, followed by a purely visual proof of the statement. The success of the talk may be judged by the number of 'Eureka!' moments it generates in the audience. _________________________________________________________________________ SCRIPT Hi, my name is Joe Hurd, and I'm going to show you some visual mathematics. At school mathematics is presented as calculations, algebra, and (worst of all) memorizing formulas, but in truth mathematics is no more about formulas than astronomy is about telescopes. Astronomers have to use telescopes to see and measure the cosmic events they're really interested in, and in the same way mathematicians have to use formulas to understand and prove properties of mathematical objects. In this talk I'm going to show you some visual proofs that don't need formulas. Let's warm up with a puzzle. If you have a bunch of 2x1 dominoes you can perfectly tile a regular chess board, but can you do the same if you remove two squares at opposite corners? It turns out you can't, because if you cover the board with a checker pattern then each domino must cover one light square and one dark square, but you've removed two squares of the same colour, making any tiling impossible. You can use the same kind of parity argument to prove that there is no knight's tour on an odd-sized chess board that ends on the same square it started, because when a knight moves it alternates between light and dark squares. OK, now we've warmed up, let's look at a classic proof of geometry. Suppose you have a semi-circle, and you draw two lines from the corners to a point on the circumference, they will always meet at 90 degrees. To prove this, all we have to do is rotate the whole picture 180 degrees around the centre of the circle. Each line becomes two equal parallel lines, and these form a four-sided shape. Since the two diagonals have the same length, the shape must be a rectangle, so each corner is 90 degrees. And now for something completely different: number theory! A lot of proofs begin life when someone notices a pattern and wonders if it will continue indefinitely. For example, adding the first n odd numbers together results in the nth square number. Why should this always be true? The proof involves constructing squares in layers, where each layer contains an odd number of points! In this picture we can immediately see why the pattern is true, and that it will carry on forever. What is the result of adding up the first n numbers, or alternatively, how many blocks are there in this diagram? It's hard to count them like this, but using your tetris skills you can neatly fit two such diagrams together, and then it's easy to see the result must be half of n * (n + 1). Let's finish with a proof from classical times: Pythagoras' Theorem. Given a triangle where one of the angles is 90 degrees, called a right-angled triangle, extend each side of the triangle to a square. Pythagoras' Theorem says that area of the two smaller squares is the same as the area of the big square, and we're going to prove this non-obvious fact. This is an IKEA proof, and for assembly we will need: a square box where each side has length (a + b), and four identical copies of the right-angled triangle. Now we simply put the four triangles into the box in two different ways. In the first way we arrange them into two pairs, and the left-over space is the area of the two smaller squares. In the second way we arrange them along the sides of the box, and the left-over space in the middle is the area of the big square. The left-over space must be the same in both ways, so we have proved Pythagoras' Theorem. That's all the visual mathematics I have for you today: thank you for your attention. _________________________________________________________________________ IMAGE CREDITS http://www.flickr.com/photos/thearchigeek/99162829/sizes/l/in/photostream/ http://xkcd.com/774/ http://www.flickr.com/photos/judepics/449405116/sizes/l/in/photostream/ http://www.flickr.com/photos/badastronomy/4179773665/sizes/o/in/photostream/ http://www.flickr.com/photos/ogcodes/377800888/sizes/o/in/photostream/ http://www.flickr.com/photos/dullhunk/426622486/sizes/o/in/photostream/ http://www.flickr.com/photos/myklroventine/2332789392/ http://en.wikipedia.org/wiki/File:Kapitolinischer_Pythagoras_adjusted.jpg http://commons.wikimedia.org/wiki/File:Paradoxical_decomposition_F2.png http://www.flickr.com/photos/9729909@N07/4706488737/ http://commons.wikimedia.org/wiki/File:Lorenz_attractor.svg http://cookiemagik.deviantart.com/art/Love-for-Tetris-64805505