50 Mathematical Ideas You Really Need to Know

a better theory and the ‘correct’ answer on dimensionality. The smart money seems to be that we’re living in an 11-dimensional u ...

The four-dimensional cube The eight corners of a cube in three dimensions have coordinates of the form (x, y, z) where each of t ...

Topology Dimension theory is part of general topology. Other concepts of dimension can be defined independently in terms of abst ...

25 Fractals In March 1980, the state-of-the-art mainframe computer at the IBM research centre at Yorktown Heights, New York Stat ...

be (0.75)^2 + 0.5 = 1.0625. All these calculations can be done on a handheld calculator. Carrying on we find that the answer get ...

magnification because you will just see more Mandelbrot sets. Before Mandelbrot Like most things in mathematics, discoveries are ...

The Koch snowflake The curious property of the Koch curve is that it has a finite area, because it always stays within a circle, ...

is scaled up by a factor of 3 its area is 9 times its previous value or 3^2 and so the dimension is 2. If a cube is scaled up by ...

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26 Chaos How is it possible to have a theory of chaos? Surely chaos happens in the absence of theory? The story goes back to 181 ...

the second ball. Pinboard box experiment But of course you cannot let the ball go from exactly the same position with exactly th ...

The free pendulum The movement of the bob can be plotted in a phase diagram. On the horizontal axis the (angular) displacement i ...

Phase diagram for the simple pendulum This is not the case for the double pendulum in which the bob is at the end of a jointed p ...

Movement of double pendulum Let’s model the population when a = 2. If we pick a starting value of, say p = 0.3 at time = 0, then ...

Forecasting the weather Even with very powerful computers we all know that we cannot forecast the weather more than a few days i ...

Strange attractors Dynamic systems can be thought of possessing ‘attractors’ in their phase diagrams. In the case of the simple ...

27 The parallel postulate This dramatic story begins with a simple geometric scenario. Imagine a line l and a point P not on the ...

Euclid’s postulates One of the characteristics of mathematics is that a few assumptions can generate extensive theories. Euclid’ ...

Later mathematicians either tried to prove it or replace it by a simpler postulate. In 1795, John Playfair stated it in a form w ...

Bolyai and Lobachevsky constructed a new geometry by allowing there to be more than one line through P that does not meet the li ...