matthen

Visualising extinctions over the past million 531 years.  The size of the circle shows how the biodiversity of the earth differs from the long-term trend.  The resulting fluctuations seem to repeat every 62 million years or so, with 5 main extinction events in total.  The most recent was of course the end of the dinosaurs, 65 million years ago.  Does this mean the Earth is due another?! [This follows the analysis of an interesting nature article]  [more] [code]

In maths, very simple rules can create interesting shapes with nice properties. For example to get a cube, we can join all the 3 dimensional points where each coordinate is either 1 or -1.  The shape in the top right, the octahedron is made form all points where only one coordinate is either 1 or -1.  Also you can see how the complicated truncated octahedron shape is made in the bottom left. A more crazy shape made from rectangles, hexagons and octagons is also shown. [more] [code]

In signal processing, there is a very important way of combining two signals called convolution.  [The german word for convolution is Faltung which means folding.]  Here we start by convolving a square signal with another.  One square is slid across from left to right, and we look at how much area there is under the two shapes (coloured in red). The thick black line measures how big this area is, and that ends up being the convolution. Next we convolve the convolution with another square and keep going.  Note how this makes the curve smoother and smoother, and it is actually turning it into a Bell Curve, or Gaussian.  Can anyone explain why this might be, and hence an interesting link between signal processing and probability theory? [more] [code]

It can be hard for computers to find hidden causes for observations.  The above animation shows fictional statistics for the number of people going to the beach, ice cream sales, cinema ticket sales and umbrella sales.  The corresponding icon is bright if one of these statistics is above average on a certain day, otherwise it is dark.  From the data alone a computer might guess the model on the left, because everything is intercorrelated.  A bad statistician might then look at this and suppose for example that umbrella sales are caused by people not going to the beach. A better model is the one on the right, which introduces a ‘hidden variable’ not in the data- the weather.  It’s really the weather that is causing these things, and this gives a simpler model even though it produces exactly the same statistics. [code]

Just a pretty fractal animation! [code]

An animation I made in mathematica roughly showing how the planets will move relative to the Earth throughout 2012. I found it quite hard to make a mapping of the solar system which keeps the Sun at the centre, and puts the planets roughly in a horizontal line- whilst roughly showing what order they appear in the sky. [code]

How would you arrange 100 dots so that no two were too close, but no dot was too far from the centre.  The answer would be useful in nature, where the dots might correspond to growing structures which can’t be overcrowded.  One solution that evolution has found is the Fibonacci spiral, in e.g. a sunflower head.  This animation shows the output of a program which tries to find a good solution. At the start the dots are placed randomly, then each dot is moved in turn to a better position in the image.   [code]

The 100 nearest stars, with the Sun in the middle.  Colours are correct, when mathematica has data on the star’s temperature.  Can you spot any constellations? [code]

Draw a straight line, make a copy of it to its right and make a copy of the copy- including its imperfections.  Repeating this process, can you make it so you end up with a profile of your face?  Here is an animation of a computer program doing just this, ending with a rough profile of me. [more] [code]

Interesting pattern obtained from rolling a small square inside a larger one.  [code]