Transcript PowerPoint

Visual and Logical
Beauty in Mathematics
László Lovász
Eötvös Loránd University, Budapest
[email protected]
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
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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
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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
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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
1 
Hey guys … Arctan( x)  Arctan( )  …
x
2
Just wondering what is the most elegant proof of this?
From an internet forum
July 2010
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Beautiful formulas
Euler’s Formula:
i
e  1
exp( i )  1  0
geometry
  3.141592653589793...
algebra
i  1
e  2.718281828459045...
analysis
Cauchy’s Formula:
1
f ( z)
f (a ) 
dz
Ñ

2 i z  a
July 2010
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The book of the most elegant proof for every theorem
„This is straight from the Book.”
Paul Erdős
July 2010
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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
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A beautiful proof
Theorem:
The side and diagonal of a square are not commensurable.
Hippasus
July 2010
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Beautiful objects: fractals
July 2010
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Beautiful objects: internet models
July 2010
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Beautiful objects: tilings
July 2010
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Beautiful objects: tilings/Alhambra
All 17 wallpaper groups represented?
Lynn Bodner: 15
July 2010
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Beautiful objects: tilings/Escher
July 2010
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Beautiful objects: tilings/Penrose
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Beautiful objects: tilings/squares
Smallest number (21)
of „small” squares,
all different, tiling a
„large” square
(demo)
July 2010
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Self-organization
The Biham-Middleton-Levine traffic model (demo)
Experiments by Raissa d’Souza 256x256
July 2010
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Self-organization
Experiments by Raissa d’Souza
July 2010
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Self-organization
Experiments by Raissa d’Souza 233x377
Fibonacci numbers:
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,…
July 2010
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Self-organization
Experiments by Raissa d’Souza 233x377
3-in-one:
- phase transition
- self-organization
- Fibonacci numbers
July 2010
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