%\documentclass[trans]{beamer} \documentclass{beamer} \usepackage{beamerthemesplit, verbatim} \usepackage{movie15} \useoutertheme{infolines} \newcommand{\fpause}{\pause\vfill} \title[Elegant ideas]{ Elegant ideas, and why you should love them} \author[Staecker]{P. Christopher Staecker} \institute[Fairfield U.]{Fairfield University} \date[]{Karim Faroud memorial lecture, 2011} \begin{document} \frame{\titlepage} \frame{ \frametitle{Introduction} Mathematical beauty is not just for mathematicians \pause, and not very different from other types of beauty. \fpause You don't need a beautiful mind. \fpause You probably already have a beautiful mind. \fpause We'll talk about: \begin{itemize} \item Some elegant ideas (``Green noses'') \item Similar types of beauty in other arts \end{itemize} } \section{An elegant sandwich} \frame{ \frametitle{An elegant sandwich} \begin{center} \includegraphics[scale=.06]{sandwichuncut.JPG} \end{center} \vfill My lunch box. } \frame{ \begin{center} \includegraphics[scale=.15]{sandwichoutline.JPG} \end{center} \vfill It should fit, but it's the wrong shape. } \frame{ How to make it fit? \fpause Here's one method: \begin{center} \includegraphics[scale=.075]{sandwichsmashed.JPG} \end{center} \fpause Not elegant. } \frame{ A better solution: \fpause \begin{center} \includegraphics[scale=.12]{sandwichendcut1.JPG} \quad\pause \includegraphics[scale=.12]{sandwichendcut2.JPG} \end{center} \fpause Better. } \frame{ An elegant solution: \fpause \begin{center} \includegraphics[scale=.15]{sandwichdiagcut1.JPG} \quad\pause \includegraphics[scale=.15]{sandwichdiagcut2.JPG} \end{center} } \frame{ An elegant solution: \begin{center} \includegraphics[scale=.2]{sandwichdiagcut3.JPG} \end{center} (hold your applause) \fpause ``Elegance'' is simple, effortless beauty. } \frame{ If you can be even slightly impressed by this example, mathematical beauty is for you. } \section{Green noses} \frame{ \frametitle{Green noses} ``Up'' Series (1964 -- 2005 -- ?) by Michael Apted. \fpause \emph{``Give me the child when he is seven, and I will give you the man.''} -- Ignatius of Loyola \fpause \begin{center} \includegraphics[scale=.015]{7up.jpg} \end{center} } \frame{ ``Still looking for the green noses.'' \fpause Most people disappointed by mathematics. \fpause But there are lots of green noses. \fpause Here's three examples: } \begin{comment} \subsection{An elegant fact} \frame{ \frametitle{Green nose \#1: An elegant fact} Here's a map of global temperatures: \begin{center} \includegraphics[scale=.3]{globetemp.jpg} \end{center} \fpause Colors represent numerical measurements on the surface. } \frame{ Here's another map: \begin{center} \includegraphics[scale=4]{colorball.jpg} \end{center} \fpause {\bf The Borsuk--Ulam Theorem}: there is a pair of opposite points where the colors match. } \frame{ So there is a pair of opposite points on the earth right now where the temperatures are the same. \fpause Hard to believe? \pause Maybe not. \fpause But this isn't about temperature. \pause This works for \emph{any} continuously varying numerical quantity. } \frame{ So there's a pair of opposite points on the earth where the humidity is equal. \fpause A pair of opposite points where the wind speed is equal. \fpause A pair of opposite points where the groundhog population density is equal. \fpause A pair of opposite points where the average twinkie consumption per sq. mile is equal. } \frame{ But wait there's more! the real statement of the theorem is this: \fpause {\bf Borsuk--Ulam Theorem:} Given any \emph{two} numerical functions on the sphere, there is a pair of opposite points where \emph{both} functions match! \fpause That means that there is a pair of opposite points where both the groundhog population density \emph{and} the average twinkie consumption are equal. } \frame{ How do the groundhogs know how to make that happen? \fpause Does the Hostess Corporation know about this? \fpause Why is this true? \fpause I could show you the proof \pause, but that's probably not what you mean. \fpause The deeper question is: ``why are things like this true?'' (unexpected facts about ordinary things) \fpause These facts are beautiful. } \end{comment} \subsection{An elegant shape} \frame{ \frametitle{Green nose \#1: An elegant shape} Here's a picture you may have seen before: \fpause \begin{center} \includegraphics{mandlebrot.jpg} \end{center} \fpause The Mandlebrot set. (It's a fractal.) \fpause ``The most complicated object in mathematics'' \emph{Scientific American, 1986} } \frame{ So what? \pause I can do that. \fpause \begin{center} \includegraphics[scale=.2]{staeckerbrot.jpg} \end{center} \fpause The Staeckerbrot! \pause Not really a ``mathematical'' shape. } \frame{ The Mandlebrot set can be described by an equation. \fpause And the equation is very very simple: \pause \[ f(z) = z^2 + c \] } \frame{ \[ f(z) = z^2 + c \] \vfill \begin{center} \includegraphics{mandlebrot.jpg} \end{center} \fpause Boneheadedly simple, but shockingly complex. } \frame{ Why does such complexity exist in mathematics? \fpause Math was created to solve complex problems, so we get complex answers. \fpause But what about things that weren't meant to be complex? \fpause $f(z)=z^2 + c$ isn't really supposed to be that complicated. \fpause If we created mathematics, then why does it surprise us? \fpause These surprises are beautiful. } \subsection{An elegant way of thinking} \frame{ \frametitle{Green nose \#2: An elegant way of thinking} Let's try a little arithmetic: \vfill \begin{center}\begin{tabular}{lr} &\only<2->{{\tiny 1}\phantom{1}} \\ &\tt 344 \\ +& \tt 217 \\ \hline &\only<4->{\tt5}\only<3->{\tt6}\only<2->{\tt1} \end{tabular}\end{center} \fpause\pause\pause\pause You could probably even do this in your head, the same way. \fpause Awesome, right? } \frame{ Imagine you're a Roman centurion, and you want to add these numbers: \pause \[ \mathtt{CCCXXXIV} + \mathtt{CCXVII} = ? \] \fpause Try: \begin{center}\begin{tabular}{lr} & \tt CCCXXXIV \\ + & \tt CCXVII \\ \hline \end{tabular}\end{center} } \frame{ How about this: \[ 2355 \div 3 \] \fpause Here goes: \begin{center}\begin{tabular}{r} \only<3-6>{\tt 7\phantom{88}}% \only<7-10>{\tt 78\phantom{8}}% \only<11->{\tt 785} \\ \tt 3 $\mathtt{\overline{ \big) 2355}}$ \\ \only<4->{$\mathtt{\underline{21}}$\phantom{88}}\\ \only<5>{\tt 2\phantom{88}}% \only<6->{\tt 25\phantom{8}}% \\ \only<8->{$\mathtt{\underline{24}}$\phantom{8}}\\ \only<9>{\tt 1\phantom{8}}% \only<10->{\tt 15}\\ \only<12->{$\mathtt{\underline{15}}$}\\ \only<13->{\tt 0} \end{tabular}\end{center} } \frame{ Ask the centurion: \[ \mathtt{III \overline{\big) MMCCCLV}} \] \fpause The centurion probably will be unable to do this without mechanical help. \fpause Your advantage is having an elegant way of thinking. } \frame{ Roman numerals are the problem and Hindu-Arabic numerals are the solution. \fpause They don't just make things easier, they actually change the way you think about numbers. \fpause Roman numerals are a bit like counting change: everything in groups of 1, 5, 10, etc. \fpause What's 3 quarters, 4 dimes, 1 nickel and 2 pennies plus 2 quarters, 3 dimes, and 4 pennies? } \frame{ Hindu-Arabic numbers are more like ... \pause numbers. \fpause It's hard to describe simply because our use of the notation forms the way we think about them. \fpause Elegant ideas transform the way we see the world, and transform the way we think. \fpause The most powerful ways of thinking completely dominate our thoughts, and we don't even recognize it. \fpause `The best design is an invisible one' } \frame{ The method for addition:\fpause Not clear who originally invented it, but it's credited to Al-Kwarizmi who is best known for: \fpause \emph{Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala} (820 AD). ``The Compendious Book on Calculation by Completion and Balancing'' \fpause Book includes the quadratic formula, and lots of applications to geometry and Islamic inheritance. } \frame{ Al-Kwarizmi also wrote \emph{Kitab al-Jam' wa-l-tafriq bi-hisab al-Hind} ``The Book of Addition and Subtraction According to the Hindu Calculation'' \fpause This describes the method for arithmetic with Hindu-Arabic numerals. \fpause This (and \emph{Al-Jabr}) was translated into Latin (12th century), and the method was referred to as ``Al-Kwarizmi's method'' \pause or ``Algorism's method'' \pause or eventually just ``The Algorism'' \pause or ``The Algorithm''. } \frame{ It took a long time for the arabic numerals to become popular in Europe. \begin{center} \includegraphics[scale=.3]{algorist.jpg} \end{center} Gregor Reisch, \emph{Typus Arithmeticae}, 1525 } \frame{ Fibbonacci (13th century) helped spread them to Europe. \fpause Elegant ways of thinking are not always recognized, even by very smart people. } \subsection{An elegant formula} \frame{ \frametitle{Green nose \#3: An elegant formula} A common candidate for ``most beautiful formula in mathematics'' is Euler's identity (1748) involving $\pi$, $e$, and $i$. \fpause The number $\pi$ comes from ancient geometry (Egypt/Babylon 1900BC, Archimedes 250BC), the ratio of a circle's circumpherence and diameter: \[ \pi = 3.1415\dots \] \fpause The number $e$ comes from calculus (Bernoulli, etc. 1600s AD): \[ e = 1 + \frac1{1!} + \frac1{2!} + \frac1{3!} + \dots = 2.7182\dots \] \pause Bernoulli was working on compound interest. } \frame{ The number $i$ is the ``imaginary number'' \[ i = \sqrt{-1} \] which comes from algebra (Italians, 1500s AD), useful to solve polynomial equations. \fpause Cardano: this method is ``truly sophisticated.'' \fpause So we have $\pi$ from geometry, $e$ from analysis, and $i$ from algebra. \fpause Each concept was developed by different people to solve completely different problems. There should be no relationship between them. } \frame{ Euler's identity is: \pause \[ e^{i\pi} = -1 \] \fpause or even more provocative: \[ e^{i\pi} + 1 = 0 \] \fpause It is shocking that there is such a relationship. \fpause It is beautiful. } \subsection{Why?} \frame{ \frametitle{From whence the green noses?} Why should the green noses exist? \fpause Mathematics is a human tool to solve human problems, right? \fpause We didn't invent mathematics so that these things would be true, so why are they true? \fpause Reminds me of Scooby Doo. \fpause Maybe there is something else at work here... } \frame{ Maybe mathematics is just part of the universe. \fpause A deep part. \fpause More fundamental than the laws of physics. \fpause But why should it be beautiful? } \frame{ Maybe God made it that way. \fpause This could explain why it's beautiful: \fpause God values beauty, and so God made the creation beautiful at the most fundamental level possible. \pause This can be inspiring, and worthy of our attention. \pause Discovering this creation and sharing it with others can be an act of worship. \fpause But ``because God did it'' or ``because God said so'' shouldn't be satisfying answers. \fpause In other contexts, these answers are dangerous. \fpause With or without God in the picture, the existence of mathematical beauty is a fundamental mystery that should inspire us and humble us. } \section{Beauty} \frame{ \frametitle{Mathematical beauty and other types of beauty} The characteristics of mathematical beauty appear in other arts. \fpause First: what exactly is mathematics about? } \subsection{Deeper magic} \frame{ \frametitle{What is it?} Mathematics is not really about numbers. \fpause Certainly numbers are mathematical, but they are only a part of mathematics in general. \fpause Mathematics generally is about patterns and structured reasoning. \fpause About learning how to think appropriately. \fpause Certainly numbers display patterns and require structured reasoning, but this is only one setting. } \frame{ \frametitle{Deeper magic} Mathematics is a deeper magic. \fpause \emph{The Lion, The Witch, and the Wardrobe}, C.S. Lewis } \frame{ \frametitle{Beauty} The basic themes of structure and patterns are universal. \fpause Let's look at some other beautiful arts which are beautiful in similar ways to mathematics. } \begin{comment} \frame{ \frametitle{Unique properties of mathematical beauty} It is fundamentally about \emph{truth}. Facts and ideas are the medium of the mathematical art. \fpause ``Beauty is truth, truth beauty.'' (Keats) \fpause Not all facts are beautiful-- just because you use the medium, doesn't make what you're doing into art. \fpause All artists know this. } \end{comment} \subsection{Structured beauty} \frame{ \frametitle{Structured beauty} Mathematical research is creative, but strongly \emph{structured}. \fpause All mathematics must lie within the rules of logical reasoning. \fpause ``Physics is imagination in a straightjacket'' -- Moffat (1939--) } \frame{ Is true artistry possible within strict structures? \fpause Poetry \fpause Would Shakespeare's works have been better if he hadn't written in meter? \fpause In the hands of the artist, the structure becomes a strength rather than a weakness. } \frame{ Often, restrictions often make the art better. \fpause Star Wars episode IV (1977) vs. \pause Star Wars episode I (1999) \fpause Films of Lars von Trier, \pause \emph{The Five Obstructions} \fpause Cinema Verit\'e, etc. } \frame{ To a lesser extent, any visual art which incorporates its surroundings is like this. \fpause Cave art flows with the contours of the walls. \fpause Architecture and graffiti art use the existing landscape. } \frame{ \frametitle{Deterministic beauty} Art in the landscape: \begin{center} \only<1>{\includegraphics[scale=.3]{gettyframe.jpg}} \only<2>{\includegraphics[scale=.3]{gettywindow.jpg}} \end{center} Getty Center Museum, Los Angeles. \tiny{(photo: http://academic.reed.edu/getty/)} } \frame{ \frametitle{Deterministic beauty} Art in the landscape: \begin{center} \includegraphics[scale=.25]{banksy.jpg} \end{center} Israeli West Bank barrier. \tiny{(photo: Wikipedia)} } \frame{ Mathematics research is about building onto and into a vast pre-existing landscape of knowledge. \fpause Like mural-making. \fpause The most beautiful facts will touch the surrounding landscape in new and unexpected ways. (Euler's identity) } \subsection{Deterministic beauty} \frame{ \frametitle{Deterministic beauty} Mathematics isn't just structured, it's \emph{deterministic}. \fpause New concepts are created, but can only be created in a specific way. \fpause Like negative numbers. \fpause We begin with the rules of logic, and play them out. \pause We do not personally ``intervene'' to decide what the truth is-- the truth reveals itself to the us as we work. \fpause Like Michelangelo: the sculpture already exists inside the block, we just need to ``free the idea'' by chipping away. } \frame{ Can something fundamentally deterministic be truly beautiful? \fpause Determinism is sometimes used in music: \fpause Ligeti, \emph{Po¸me Symphonique for 100 metronomes}, 1962. \pause(Not conventionally beautiful to listen to.) \fpause Steve Reich, \emph{It's Gonna Rain} (1965), \emph{Come Out} (1966) } \frame{ P\"art, \emph{Cantus in Memoriam Benjamin Britten}, 1977. \fpause First violin part: \begin{center} \only<1-2>{\includegraphics[scale=.6]{violin1}} \only<3->{\includegraphics[scale=.6]{violin1groups}} \only<1-2>{\includegraphics[scale=.6]{violin2}} \only<3->{\includegraphics[scale=.6]{violin2groups}} \end{center} \pause\pause a few pages later... \begin{center} \includegraphics[scale=.6]{violin3} \end{center} } \frame{ P\"art, \emph{Cantus in Memoriam Benjamin Britten}, 1977. Second violin part: \begin{center} \includegraphics[scale=.6]{2violin1} \includegraphics[scale=.6]{2violin2} \end{center} \fpause Same pattern, half speed } \frame{ P\"art, \emph{Cantus in Memoriam Benjamin Britten}, 1977. Viola part: \begin{center} \includegraphics[scale=.6]{viola1} \includegraphics[scale=.6]{viola2} \end{center} Same pattern, one-fourth speed } \frame{ The cello plays the same pattern at one-eighth speed, \pause the bass at one-sixteenth speed. \fpause But it sounds beautiful, and not at all artificial. \fpause It is very creative. } \frame{ Big philosophical question: \pause are mathematicians \emph{discovering} their truths \pause or \emph{creating} them? \fpause Certainly Britten created his music. \pause He chose the rules so that it would sound good. \fpause Mathematicians don't even get to choose their own rules. \fpause It is still profoundly creative. \fpause This is a beautiful mystery. } \frame{ That's all! \vfill \url{http://faculty.fairfield.edu/cstaecker} } \end{document} \begin{comment} \frame{ \frametitle{Wonderfully effective tools: FTC} A very common problem in calculus: \[ \int_0^1 x^2 \, dx \] \pause This is the shaded area. Picture of graph. \pause What is the area? It's probably just some obscure real number. Actually it's $1/3$. Here's how you compute that: } \frame{ \frametitle{Wonderfully effective tools: FTC} \begin{eqnarray*} \int_0^1 x^2 \, dx &= \displaystyle \lim_{n\to \infty} \sum_{i=1}^n f(x_i) \Delta x \pause = \lim_{n\to \infty} \frac 1n \sum_{i=1}^n \left(\frac{i}{n}\right)^2 \pause = \lim_{n\to \infty} \frac 1{n^3} \sum_{i=1}^n i^2 \pause \\ &= \displaystyle \lim_{n\to\infty} \frac 1{n^3} \frac{n(n+1)(2n+1)}{6} \pause = \lim_{n\to\infty} \frac{2n^3 + 3n^2 + n}{6n^3} \pause = \frac26 \pause = \frac13 \end{eqnarray*} \pause Not very straightforward. (Actually this is an easy one.) \pause But then we learn the Fundamental Theorem of Calculus. Let's try it that way: \[ \int_0^1 x^2\, dx \pause = \left. \frac{x^3}3 \right|^1_0 \pause = \frac{1^3}3 - \frac03 \pause = \frac13 \] \pause This is much easier than we should ever expect it to be. } \frame{ \frametitle{Green nose: truth} Not every statement is absolutely either ``true'' or ``false''. \fpause Like: ``Fairfield is good.'' (subjective, ambiguous) \fpause ``You never know when I'm hammering, because I'm hammering now'' (nonsensical) \fpause ``This statement is false.''\pause (is not true or false) \fpause You can say some weird things with self-reference. } \frame{ \frametitle{Green nose: truth} You can even get a sentence which can be \emph{both}: \fpause ``This statement is true.'' \fpause It could be both true and false. Certainly it could not be proved either way without more information. \fpause The statement is \emph{undecidable}. } \frame{ \frametitle{Green nose: truth} Surprising fact (G\"odel, 1930s): There are ordinary-looking mathematical statements that are undecidable. \fpause Issues about self-reference are not some silly technicality raised by annoying pedants. They infect all formal logical systems fundamentally. } \end{comment} \begin{comment} \frame{ \frametitle{Green nose: $\infty$} Surprising idea: you can compare sizes of infinite things (Cantor, 1870s). \fpause Here's how: Is there the same number of people as houses? { \bf picture, then draw in arrows} \fpause So there is the same number, even if you can't count that high. } \frame{ \frametitle{Green nose: $\infty$} So there are the same number of even numbers as odd numbers, since: \begin{eqnarray*} 1 & 3 & 5 \dots \\ \updownarrow & \updownarrow & \updownarrow \\ 2 & 4 & 6 \dots \end{eqnarray*} It's a great idea, but a big surprise: \fpause Not all infinite things have the same size. \fpause There are different sizes of infinity. } \frame{ \frametitle{Green nose: $\infty$} Cantor showed the amount of whole numbers is different than the amount of real numbers. The first amount he called $\aleph_0$, the next is $\aleph_1$, and \[ \aleph_0 < \aleph_1, \] though both are infinite. \fpause Are there other sizes of $\infty$? \pause Yes. \fpause There's $\aleph_2$, \pause $\aleph_3$, \dots, \pause also $\aleph_{\aleph_0}$, etc. \fpause Actually there's infinitely many different sizes of infinity! } \frame{ \frametitle{Green nose: $\infty$} Infinitely many different infinities? \fpause How many? \fpause This question breaks the theory developed by Cantor. (The set of infinities is too large to be corresponded with any other set.) \fpause Another big question:\pause are there any infinities in between $\aleph_0$ and $\aleph_1$? } \end{comment} \begin{comment} \frame{ \frametitle{An elegant idea: limits} Calculus is based on \emph{limits}. \fpause For example, \[ \lim_{x \to 3} 2 \times X = 6 \] \fpause This means ``As $X$ approaches $3$, then $2 \times X$ approaches $6$.'' \fpause Do you understand what that means? } \frame{ \frametitle{An elegant idea: limits} ``As $x$ approaches $3$, then $2\times X$ approaches 6.'' \fpause What does ``approach'' mean? \fpause How about ``As $X$ gets really really close to $3$, then $2\times X$ gets really really close to $6$.'' \fpause The phrase ``really really'' should concern you. \fpause Is $2.5$ ``really really'' close to 3? What about $2.99$? } \frame{ \frametitle{An elegant idea: limits} ``As $X$ gets really really close to $3$, then $2\times X$ gets really really close to $6$.'' \fpause Actually the whole thing should concern you. \fpause It implies that numbers move somehow. \fpause Numbers don't actually ``approach'' each other. \fpause So what does this whole thing actually mean? (If anything?) } \frame{ \frametitle{An elegant idea: limits} It takes a true mathematician even to realize that there are issues here. \fpause We fool our students routinely, and nobody notices! \fpause Let's try to formulate this without using any words like ``close'' or ``approach''. \fpause The amazing thing is just how hard it is. } \frame{ \frametitle{An elegant idea: limits} This was first done by Cauchy. Here's what he came up with: \fpause For any $\epsilon > 0$, there is some $\delta>0$ such that if $|X-3|<\delta$, then $|2X-6|<\epsilon$. \fpause In symbols: \[ (\forall \epsilon > 0) (\exists \delta > 0) |X-3|<\delta \Rightarrow |2X-6|<\epsilon \] \fpause (Huh?) } \end{comment} \end{document}