July 2010 1

Visual and Logical

Beauty in MathematicsLászló Lovász

Eötvös Loránd University, Budapest

lovasz@cs.elte.hu

July 2010 2

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

Quote from a mathematician

July 2010 3

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

Quote from a logician

July 2010 4

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 5

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

1Arctan( ) Arctan( )

2

x

x

Quotes from „everyday life”

July 2010 6

Beautiful formulas

exp( ) 1 0 i

1

3.141592653589793...

2.718281828459045...

i

e1 ie

Euler’s Formula:

1 ( )( )

2

Ñf z

f a dzi z a

Cauchy’s Formula:

geometry

algebra

analysis

July 2010 7

„This is straight from the Book.”

Paul Erdős

The book of the most elegant proof for every theorem

July 2010 8

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 9

A beautiful proof

Theorem: The side and diagonal of a square are not commensurable.

A.k.a. „2 is irrational.”

Hippasus

July 2010 10

Beautiful objects: fractals

July 2010 11

Beautiful objects: internet models

July 2010 12

Beautiful objects: tilings

July 2010 13

Beautiful objects: tilings/Alhambra

All 17 wallpaper groups represented?

Lynn Bodner: 15

July 2010 14

Beautiful objects: tilings/Escher

July 2010 15

Beautiful objects: tilings/Penrose

July 2010 16

Beautiful objects: tilings/squares

Smallest number (21)of „small” squares, all different, tiling a„large” square (demo)

July 2010 17

Self-organization

The Biham-Middleton-Levine traffic model (demo)

Experiments by Raissa d’Souza 256x256

July 2010 18

Experiments by Raissa d’Souza

Self-organization

July 2010 19

Experiments by Raissa d’Souza 233x377

Fibonacci numbers:0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,…

Self-organization

July 2010 20

Experiments by Raissa d’Souza 233x377

3-in-one: - phase transition - self-organization - Fibonacci numbers

Self-organization