The Approximate Present

Suppose you live in a small village just outside a city, and every morning you commute to work (along with 9 other residents) by one of two routes linking the village and the city, one via Town A and one via Town B.

On route A, the first portion of the road is relatively short, but narrow, and so takes N minutes to complete, where N is the number of cars taking the road. The second portion of the drive is longer, but with a much better carrying capacity, so takes a flat 12 minutes.

Route B is the opposite, with the longer, wider section first, and the shorter, narrower section second. Assuming that every driver wants to make their drive as short as possible, 5 people will take route A, and 5 will take route B, giving a journey time of N+12 = 5+12 = 17 minutes.

The council thinks this isn't ideal, too many traffic jams on the narrow portions they say. So they add a road connecting Town A to Town B directly that takes 0 minutes (they're really very close together). People have the option to change routes halfway through, however something very strange happens.

Now, when leaving the village, people deduce that the first portion of A is going to take 10 minutes tops (with all 10 cars on it) whereas the first portion of B will take 12! Of course they choose route A. Then they see that the second half of route B is again, at least 2 minutes quicker than the second part of A, a no-brainer. So everybody takes the first part of A, followed by the second part of B, however, the average travel time has now gone up to 20 minutes, 3 minutes longer than before.

Increasing the capacity of the road network has made the average commute slower! This is not a purely mathematical issue, and has been observed in cities around the world.

I don't know why I made this.

Suppose you enter a casino offering a very simple game. The croupier tosses a coin, if it shows a tail, you win $1, if it's heads, he flips again and the pot doubles. If the second flip is a tail, you win $2, if it's heads, he flips again and the pot doubles. If the first tail arrives on the nth flip, you win $2^(n-1).

Here's the question, how much should you pay to enter?

Naive probability suggests that the expected winnings is infinite, but that clearly doesn't make much sense...

Many solutions have been proposed for this, in the 300 years since it's proposal, some as recent as 2011 employing techniques from ergodic theory and statistical mechanics. Exactly when does taking an expected value make sense, and what else needs to be taken in to account?

Julia Sets.

Consider the function f(z) = z^2 + c where z is a complex variable and c a complex constant. The Julia set is the set of points on which f acts chaotically. 

Patterns in 2 dimensional billiards.

Suppose we place a particle in a 2D bounded area. We then give it an initial velocity, and trace its path. The behaviour is incredibly varied, depending not only on the shape of the billiard, but on the initial condition.

Top left: Circular billiard. All orbits (except for one that traces the diameter) leave an uncovered "hole" in the center. If the initial angle is a rational multiple of pi, the orbit is periodic. Otherwise (as in this case) it will fill the outer ring densely.

Top right: Elliptical billiard. The paths avoid the narrow ends of the ellipse, and fill the center densely (except again if it is a periodic orbit).

The triangle with side lengths 1, 2 and root 5 can be split in to 5 similar copies of itself. Repeating this procedure and removing the central triangle creates the pinwheel fractal. This illustration shows the triangle after 3 iterations.

A mathematical approximation of Brownian Motion.

Brownian motion is the name used for the seemingly random movement of particles suspended in fluid.

In this model, at each time step the particle moves in a random direction in 3 dimensional space according to the Wiener process.

A spiral of Fibonacci spirals.

The Fibonacci spiral is a common approximation for the golden spiral. The golden spiral grows by a factor equal to the golden ratio for each quarter turn.

[http://en.wikipedia.org/wiki/Golden_spiral]

The Weierstrass function is a function that is continuous everywhere (there are no discontinuous jumps in value), but differentiable nowhere (it is smooth nowhere).

If you zoom in far enough, most continuous functions will look smooth, or at least be made up of separate smooth parts. The Weierstrass function is self-similar like a fractal and this does not happen.

The double pendulum is chaotic, meaning small changes in initial condition have large changes on the behaviour. These two double pendulums begin very close to each other but quickly diverge.

"In this installation two purple dots travel over a plane curve that's a simple modification of Bernoulli's Lemniscate.