matthen

If a mathematician wants to cross a road, they will think carefully about their optimal path. The total distance of the path should be minimised, but they prefer walking on the sidewalk to the road. If there is no extra discomfort from being on the road, the best path is a straight line, but as it increases it is better to cross the road more directly.  The resulting path is exactly the same as a ray of light refracting through a block of glass [with relative refractive index equal to the ratio of these ‘discomfort levels’]. Fermat’s principle says that light will want to spend less time in the glass (on the road), as it actually travels more slowly in the glass. [video] [code] [more]

In 1952 Alan Turing, a british mathematician, logician, cryptanalyst, and computer scientist, wrote a paper which remains influential in computational biology today. He explained how stripes might form on a snake’s skin [and other patterns on animals], using the dispersion of two chemicals; an activator [red] and an inhibitor [yellow]. The activator causes the colouration, and the inhibitor inhibits it. Turing wrote a pair of equations which say that concentrations of the activator cause creation of more inhibitor, but that the inhibitor diffuses and spreads out more quickly than the activator. As shown in the animation, this causes the activator to form peaks with surrounding basins of inhibitor. The concentrations of the two chemicals quickly converge to a stripey pattern where the red activator is periodically in higher concentration than the yellow inhibitor. [video] [more] [code]

The bending of a light ray depends only on the angle it hits the lens, and not on the thickness of the glass. This means thick lenses can be made thinner, as shown here.  This is called a Fresnel lens, and is used to make magnifying sheets as well as in lighthouses. [more] [video] [code]

How do Centrifugal and Coriolis forces affect the motion of particles as the speed of rotation is varied?  This animation shows how the underlying grid of my previous post is warped at differing speeds of rotation. Note the square grid when there is no rotation. [code] [video]  

Disks moving in a square grid trace out an interesting pattern as they pass over a rotating wheel.  Some traces appear to go towards the centre only to be flung back out, in some cases forming a sharp corner. A few of the traces form loops, as if there was also a force acting sideways rather than outwards.  This demonstrates how the Centrifugal and Coriolis ‘forces’ appear to affect the motion of objects in a rotating frame. [making of video (new)] [more] [code] 

Though it seems as though the top row of diamonds is darker than the bottom, all the diamonds are actually identical. The sliding diamond seems to get darker as it ascends, but is in fact unchanging.  Illusions like this demonstrate that our perception of the darkness of an object is highly dependent on context. [based on] [code]

A gingerbreadman drawn using chaos. Points are chosen at random, then repeatedly moved to new locations according to a simple rule [the new y coordinate is the old x one, and the new x is 1 - the old y + |the old x|]. This rule is called the gingerbreadman map, of course. The trajectories shown are chaotic, showing complex behaviour from such a simple rule. Hexagonal areas seem to build up the picture. [detailed version] [more] [code]

Streamlines exposing the structure of a gravitational field surrounding multiple masses. Streamlines run in the direction that a mass would be pulled within the field. Between masses, the streamlines form crosses at what are called Lagrangian points where the pulls cancel each other out. [code]

Two bubbles merge in an arrangement which minimises the total surface area, given the air stored in each compartment is equal to the original air in each bubble. [A single bubble is a sphere, which is the minimum surface are for a given volume.] The two compartments are parts of spheres, and the boundary between them is part of another sphere meeting the other walls at 120 degree angles. If the two bubbles were originally the same size, the boundary sphere has an infinite radius, giving a flat wall. Though it is a familiar picture to anyone who has blown bubbles, it was only proven that this was how double bubbles are made in 2002. [more₁] [more₂] [code]

Organisation can emerge from chaos in complex systems made of relatively simple parts. This mathematics of emergence helps understand networks in the brain, structures in societies, and how groups of fireflies can flash in sync without requiring a leader to keep the beat.  These animations show one mathematical model where runners running round a track can synchronise to the same pace with no leader. Each runner runs at their own pace, but adjusts it slightly if they feel they are slower or faster than their perceived average. [This can be succinctly written as a set of simple differential equations]. In the top row, each runner is more strongly coupled to the group than the bottom row. [The two animations on the right are the same as the ones on the left, except the camera is rotating with the group.] [more] [code]

3D Valentines card-ioid [code]