Visual and Logical Beauty in Mathematics Lszl Lovsz

Visual and Logical Beauty in Mathematics László Lovász Eötvös Loránd University, Budapest lovasz@cs. elte. hu July 2010 1

Quote from a mathematician The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G. H. Hardy July 2010 2

Quote from a logician Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. Bertrand Russel July 2010 3

Quotes from physicists To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. . . Richard Feynman Elegance should be left to shoemakers and tailors. Ludwig Boltzmann July 2010 4

Quotes from „everyday life” Tao and co-author Allen Knutson produced beautiful work on a problem known as Horn's conjecture… Report on the work of Fields medalist Terence Tao Zermelo gave a beautiful proof that every set can be well ordered… Daniel Grayson lecture notes on the internet Hey guys … … Just wondering what is the most elegant proof of this? From an internet forum July 2010 5

Beautiful formulas Euler’s Formula: geometry algebra analysis Cauchy’s Formula: July 2010 6

The book of the most elegant proof for every theorem „This is straight from the Book. ” Paul Erdős July 2010 7

What is a beautiful proof? An elegant proof is a proof which would not normally come to mind, like an elegant chess problem: the first move should be paradoxical. Claude Berge July 2010 8

A beautiful proof. ” l a n o ti Theorem: a r r i s i 2 „ The side and diagonal. of a. square not commensurable. A k. a Hippasus July 2010 9

Beautiful objects: fractals July 2010 10

Beautiful objects: internet models July 2010 11

Beautiful objects: tilings July 2010 12

Beautiful objects: tilings/Alhambra All 17 wallpaper groups represented? Lynn Bodner: 15 July 2010 13

Beautiful objects: tilings/Escher July 2010 14

Beautiful objects: tilings/Penrose July 2010 15

Beautiful objects: tilings/squares Smallest number (21) of „small” squares, all different, tiling a „large” square (demo) July 2010 16

Self-organization The Biham-Middleton-Levine traffic model (demo) Experiments by Raissa d’Souza 256 x 256 July 2010 17

Self-organization Experiments by Raissa d’Souza July 2010 18

Self-organization Experiments by Raissa d’Souza 233 x 377 Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, … July 2010 19

Self-organization Experiments by Raissa d’Souza 233 x 377 3 -in-one: - phase transition - self-organization - Fibonacci numbers July 2010 20