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Things of interest in Maths & Science

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Any three points uniquely define one circle, their circumcircle, running through each of them. Here 30 points are randomly added to the picture, so on the final frame there are 4,060 circles generated from all possible triples. [more] [code] 

Any three points uniquely define one circle, their circumcircle, running through each of them. Here 30 points are randomly added to the picture, so on the final frame there are 4,060 circles generated from all possible triples. [more] [code

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A Julia set fractal is a collection of points which don’t escape to infinity after repeated application of certain geometric transformations. Here we work the transformations backwards- a shift up and to the right, a squeezing so everything is in the top half, then a reflection to fill up the bottom half too. After a few iterations, the pattern of circles starts to resemble a Julia set.  [more] [code]

A Julia set fractal is a collection of points which don’t escape to infinity after repeated application of certain geometric transformations. Here we work the transformations backwards- a shift up and to the right, a squeezing so everything is in the top half, then a reflection to fill up the bottom half too. After a few iterations, the pattern of circles starts to resemble a Julia set.  [more] [code]

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Here a line of fixed length is moved along the edge of an ellipse, tracing out a collection of new shapes. Consider the area of the shape traced by a point p units from one end of the line and q from the other. Holditch’s theorem, regarded as a milestone in the history of maths, tells us this area is less than the area of the ellipse by at least π×p×q. Curiously, this formula holds not just for an ellipse, but any closed curve. [more] [code]

Here a line of fixed length is moved along the edge of an ellipse, tracing out a collection of new shapes. Consider the area of the shape traced by a point p units from one end of the line and q from the other. Holditch’s theorem, regarded as a milestone in the history of maths, tells us this area is less than the area of the ellipse by at least π×p×q. Curiously, this formula holds not just for an ellipse, but any closed curve. [more] [code]

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Another interesting property of the logarithmic spiral is revealed if you roll it along a horizontal line. This animation shows the curves traced by points on the spiral, and note that the very centre follows the path of a straight line. The angle between this line and the horizontal is called the pitch of the spiral, and for our spiral galaxy the pitch is around 12 degrees. [more] [code] 

Another interesting property of the logarithmic spiral is revealed if you roll it along a horizontal line. This animation shows the curves traced by points on the spiral, and note that the very centre follows the path of a straight line. The angle between this line and the horizontal is called the pitch of the spiral, and for our spiral galaxy the pitch is around 12 degrees. [more] [code

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We are only able to see at most three faces of a cube at any one time. This animation shows how a shape built from cubes can appear to change into several different forms by simply rotating the three originally visible faces. [code]

We are only able to see at most three faces of a cube at any one time. This animation shows how a shape built from cubes can appear to change into several different forms by simply rotating the three originally visible faces. [code]

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The arms of spiral galaxies are roughly logarithmic, as are the bands of tropical cyclones and many biological structures such as shells. These are spirals which make a constant angle with the line connecting any point to the centre. Rays of light emanating from the centre reflect off the spiral and burn the image of a second identical spiral, slightly rotated. [more] [code]

The arms of spiral galaxies are roughly logarithmic, as are the bands of tropical cyclones and many biological structures such as shells. These are spirals which make a constant angle with the line connecting any point to the centre. Rays of light emanating from the centre reflect off the spiral and burn the image of a second identical spiral, slightly rotated. [more] [code]

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If you were to start on the equator, pick a compass bearing between North and East, and then keep moving in exactly that bearing, what path would you take? You would spiral in towards the North pole, winding tighter and tighter, following a course called a loxodrome. On many maps of the world, this would look like a straight line. This animation shows a loxodrome spiral on a rotating globe. The shadow it casts from a light source at the top of the globe creates a logarithmic spiral which is inverted on itself as the globe spins. The logarithmic spiral is a self-similar curve which often appears in nature. [more] [code]

If you were to start on the equator, pick a compass bearing between North and East, and then keep moving in exactly that bearing, what path would you take? You would spiral in towards the North pole, winding tighter and tighter, following a course called a loxodrome. On many maps of the world, this would look like a straight line. This animation shows a loxodrome spiral on a rotating globe. The shadow it casts from a light source at the top of the globe creates a logarithmic spiral which is inverted on itself as the globe spins. The logarithmic spiral is a self-similar curve which often appears in nature. [more] [code]

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You can get interesting star patterns by joining up pairs of points along a circle, separated by a fixed angle. But if you imagine you are actually joining up one circle to another sitting below in the third dimension, you are actually creating a wireframe of a hyperboloid. [code] [more]

You can get interesting star patterns by joining up pairs of points along a circle, separated by a fixed angle. But if you imagine you are actually joining up one circle to another sitting below in the third dimension, you are actually creating a wireframe of a hyperboloid. [code] [more]

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Draw a straight line, and then continue it for the same length but deflected by an angle. If you continue doing this you will eventually return to roughly where you started, having drawn out an approximation to a circle. But what happens if you increase the angle of deflection by a fixed amount at each step? The curve will spiral in on itself as the deflection increases, and then spiral out when the deflection exceeds a half-turn. These spiral flourishes are called Euler spirals. [code] 

Draw a straight line, and then continue it for the same length but deflected by an angle. If you continue doing this you will eventually return to roughly where you started, having drawn out an approximation to a circle. But what happens if you increase the angle of deflection by a fixed amount at each step? The curve will spiral in on itself as the deflection increases, and then spiral out when the deflection exceeds a half-turn. These spiral flourishes are called Euler spirals. [code

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Though 3753 Cruithne is sometimes called our second moon, it does not actually orbit the Earth. Its eccentric orbit just happens to be in 1:1 resonance with ours, taking a year to go once around the Sun.  From our frame of reference, Cruithne appears to be moving in a bean-shaped orbit about the Earth. This animation shows a similar orbit, and how the two motions can combine to give the illusion of the oddly shaped orbit. [more] [code]

Though 3753 Cruithne is sometimes called our second moon, it does not actually orbit the Earth. Its eccentric orbit just happens to be in 1:1 resonance with ours, taking a year to go once around the Sun.  From our frame of reference, Cruithne appears to be moving in a bean-shaped orbit about the Earth. This animation shows a similar orbit, and how the two motions can combine to give the illusion of the oddly shaped orbit. [more] [code]

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What image is traced by a line moving according to the digits of π? The 3 corresponds to 3 tenths of a full turn, the 1 is one tenth, etc… An infinitely complex random walk is generated, changing in shape the longer you allow it to generate. More than 100 thousand digits are drawn here. [code]

What image is traced by a line moving according to the digits of π? The 3 corresponds to 3 tenths of a full turn, the 1 is one tenth, etc… An infinitely complex random walk is generated, changing in shape the longer you allow it to generate. More than 100 thousand digits are drawn here. [code]

Matt Henderson

I post original stuff about maths, space, computational linguistics and other things that I like. This blog is meant to be accessible and interesting to people of all backgrounds. My undergrad was maths in Cambridge, and I'm now starting research in Speech and Language technology. Email me at If you're new, check out this overview of my posts. All code posted is in Mathematica.

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