1. 1 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.

2. 2 The Most Beautiful Equation In Mathematics 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 , 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.

3. 3 An Example of a Fractal 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.

4. 4 An Example of Koch Snowflake

5. 5 An Example of a Julia Set

6. 6 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.

7. 7 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.

8. 8 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:

9. 9 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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15. 15 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 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