[Mathematics] Most fascinating Topics

From the Book '50 mathematical ideas you really need to know' by Tony Crilly.

1. Topology
   
2. Fractals
   
3. Game theory
   
4. Euclid's Algorithm
   
5. Discrete Geometry
   
6. Dimension
   
7. Conventional(?) Number Theory
   

Topology:
Topology is the branch of geometry that deals with the properties of surfaces and general shapes but is unconcerned with the measurement of lengths or angle. High on the agenda are qualities which do not change when shapes are transformed into other shapes. We are allowed to push and pull the shape in any direction and for this reason topology is sometimes described as 'Rubber sheet geometry'. Funny thing is topologists are people who cannot tell the difference between a donut and a coffee cup!

Fractals:
In March 1980, the state-of-the-art mainframe computer at the IBM research centre at Yorktown Heights, New York State, was issuing its instructions to an ancient Tektronix printing device, It dutifully struck dots in curious places on a white page, and when it had stopped its clatter the result looked like a handful of dust smudged across the sheet. Benoit Mandelbrot rubbed his eyes in disbelief. He saw it was important, but what was it? The image that slowly appeared before him was like the black and white print emerging from a photographic developing bath. It was a first glimpse of that icon in the world of fractals. - the Mandelbrot set.

Game Theory:

Some said Johnny was the smartest person alive. John von Neumann was a child prodigy who became a legend in the mathematical world. When people heard that he arrived at a meeting in a taxi having just scribbled out his 'Minimax Theorem' in game theory, they just nodded. It was exactly the sort of thing von Neumann did. He made contributions to quantum mechanics, logic, algebra, so why should game theory escape his eye? It didn't - with Oskar Morgenstern he coauthored the influential Theory of Games and Economic Behavior: In its widest sense game theory is an ancient subject, but von Neumann was key to sharpening the theory of the 'two-person zero-sum game'.

Euclid's Algorithm:

Al-Khwarizmi gave us the word 'algebra', but it was his ninth-century book on arithmetic that gave us the word 'algorithm'. Pronounced 'Al Gore rhythm' it is a concept useful to mathematicians and computer scientists alike. But what is one? If we can answer this we are on the way to understanding Euclid's division algorithm.

Discrete Geometry:

Geometry is one of the oldest crafts - it literally means earth [geo] measuring [metry]. In ordinary geometry there are continuous lines and solid shapes to investigate, both of which can be thought of as being composed of points 'next to' each other. Discrete mathematics deals with whole numbers as opposed to the continuous real numbers. Discrete geometry can involve a finite number of points and lines or lattices of points - the continuous is replaced by the isolated.

Dimension:

Leonardo da Vinci wrote in his notebook: 'The science of painting begins with the point, then comes the line, the plane comes third, and the fourth the body in its vesture of planes.' In da Vinci's hierarchy, the point has dimension zero, the line is one-dimensional, the plane is two-dimensional and space is three-dimensional. What could be more obvious? It is the way the point, line, plane and solid geometry had been propagated by the Greek geometer Euclid, and Leonardo was following Euclid's presentation.