Transcript The Beauty of Mathematics

The Beauty
of
Mathematics
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of
mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic
aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple
and elegant proof and in an elegant numerical method that speeds calculation.
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The Most Beautiful Equation In Mathematics
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Euler's identity is considered by many to be remarkable for its
mathematical beauty. Three basic arithmetic operations occur
exactly once each: addition, multiplication, and exponentiation.
The identity also links five fundamental mathematical
constants:
The number 0.
The number 1.
The number p, which is ubiquitous in trigonometry, geometry
of Euclidean space, and mathematical analysis.
The number e, the base of natural logarithms, which also
occurs widely in mathematical analysis (e ≈ 2.71828).
The number i, imaginary unit of the complex numbers, which
contain the roots of all non-constant polynomials and lead to
deeper insight into many operators, such as integration.
Furthermore, in mathematical analysis, equations are
commonly written with zero on one side.
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An Example of a Fractal
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A revolutionary step in the description of many natural shapes and phenomena was taken by
Mandelbrot, when he discovered the meaning of fractality and fractal objects. "Fractal" came from
the latin word fractus, meaning broken.
While a formal definition of a fractal set is possible, the more intuitive notion is usually offered,
that in a fractal, the part is reminiscent of the whole. Fractals have two important properties:
Self similarity, and Self affinity.
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An Example of Koch Snowflake
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An Example of a Julia Set
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The Mandelbrot Set
• The Mandelbrot set, named after Benoit
Mandelbrot, is a fractal. Fractals are
objects that display self-similarity at
various scales. Magnifying a fractal
reveals small-scale details similar to the
large-scale characteristics.
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The Golden Ratio: Φ = 1.618 033 ...
Leonardo da
Vinci's drawings
of the human
body emphasised
its proportion.
The ratio of the
following
distances is the
Golden Ratio:
(foot to navel) :
(navel to head)
Similarly, buildings are more attractive if the
proportions used follow the Golden Ratio.
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The ABC logo is a Lissajous figure. The
parametric equations that describe the
logo are:
x = sin t
y = cos 3t
The graph is as follows:
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Equiangular Spiral
• The equation for the
equiangular spiral was
developed by Rene
Descartes (1596-1650)
in 1638.
• This spiral occurs
naturally in many places
like sea-shells where the
growth of an organism is
proportional to the size of
the organism.
• The general polar
equation for the
equiangular spiral curve
is
r = ae
θ cot b
Nautilus Shell
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References:
http://en.wikipedia.org/wiki/Mathematics
http://users.forthnet.gr/ath/kimon/Euler/Euler.htm
http://users.forthnet.gr/ath/kimon/Fractals/Fractal.html
http://www.olympus.net/personal/dewey/mandelbrot.html
http://www.lboro.ac.uk/departments/ma/gallery/mandel/index.html
http://www.intmath.com/Trigonometric-graphs/7_Lissajous-figures.php
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