matthen

The smooth motion of rotating circles can be used to build up any repeating curve even one as angular as a digital square wave. Each circle spins at a multiple of a fundamental frequency, and a method called Fourier analysis shows how to pick the radiuses of the circles to make the picture work. Decomposing signals like this lies at the heart of a lot of signal processing. [more] [code]

A cloud created indoors by Berndnaut Smilde looks just like the real thing.  If it were on a blue background then it would have been almost impossible to distinguish it from a photo of a cloud in the sky. And it seems hard to say which of the two real clouds shown is bigger [can you tell?]. This property of clouds to look similar at many scales has lead mathematicians to model them as fractals, shapes with a fixed volume but infinite surface area. Benoît Mandelbrot starts The Fractal Geometry of Nature, ”Clouds are not spheres”, but rather they exhibit much more complex geometry.  [more]

The distinct forms of five vowel sounds; a, e, i o and u. The bright bands in the visualisations show which frequencies are active in the voice, the further from the centre the higher the pitch. Time is wrapped around anti-clockwise to create a loop. Computers convert speech into patterns like this, so speech recognition can become pattern matching.  [inspiration] [code]

Draw a circle on a piece of paper, and a random point inside. If you continually fold points on the edge of the circle on top of the point inside, then the fold marks will combine to form the shape of an ellipse. [code] [more] [inspiration] [bonus]

Creating some fractal patterns by pressing paint between two sheets of glass, and separating them. This leaves the paint in branching tree-like channels which can be transferred to paper.  

The Cycloid [red] is generated by tracing a point on a rolling circle. This animation shows a curious symmetry between the Cycloid and itself shifted. The traced lines are always crossing at right angles to the top cycloid and are tangent to the bottom. [The Cycloid is its own evolute]  [code] [more]

The Cardioid [lovely mathsy heart shape] has a secret reflected heart inside itself.  It can be found by taking the evolute, which is shown above. The evolute of a curve is enveloped by all the lines which cross the curve at right angles. [more] [code]

Draw some random points on a piece of paper and join them up to make a random polygon. Find all the midpoints and connecting them up to give a new shape, and repeat. The resulting shape will get smaller and smaller, and will tend towards an ellipse!  [code] [more] [bigger version]

Recently I described saccades and fixations, the jittery movement of the eye as you scan a horizon. The other way in which eyes move, called smooth pursuit, happens when you follow a moving object. Most people find it hard to do this without the help of a moving object [I was watching my finger], and are better at horizontal than vertical movement. [more]

A Cardioid, a lovely mathsy heart shape, can be constructed as shown in the animation as the combination of many circles generated from a single underlying circle. This shape describes the sensitivity regions of many directional microphones. [more] [code]  

This animation of falling sand was created by defining 16 simple rules [shown below] about how the sand ought to behave in 2×2 grids. Breaking up the problem makes it easy to simulate the complicated patterns of falling sand.  Systems which define rules on a grid like this are called cellular automata. Other famous examples include Conway’s Game of Life.

[more] [code]