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Things of interest in Maths & Science

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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]

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How to cut an equilateral triangle into only four pieces so they can be rearranged into a square? Henry Dudeney's solution to this (the Habberdasher's problem) is particularly neat as it can work using hinged pieces. [more] [thanks to] [code]

How to cut an equilateral triangle into only four pieces so they can be rearranged into a square? Henry Dudeney's solution to this (the Habberdasher's problem) is particularly neat as it can work using hinged pieces. [more] [thanks to] [code]

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Just finished my new gravity game, where you explore a universe and test your orbital mechanics skills. If you can get into orbit around a planet, you will start mining it for fuel. You can hop from system to system in an infinite universe using wormholes, but they get more and more hostile the further you travel. Just click to fire your engines.  Try it out!

Just finished my new gravity game, where you explore a universe and test your orbital mechanics skills. If you can get into orbit around a planet, you will start mining it for fuel. You can hop from system to system in an infinite universe using wormholes, but they get more and more hostile the further you travel. Just click to fire your engines.  Try it out!

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Unlike with film, most digital cameras don’t record every part of the image at the same time. Often the image is scanned from left to right and top to bottom, which can create some interesting effects when recording moving objects like the blades of a fan. Here we simulate a digital movie of a rotating chess-board. The scan is moving from left to right, catching up with the rotation in the bottom of the image, and moving against it at the top. [inspiration from danielwalsh] [more] [code]

Unlike with film, most digital cameras don’t record every part of the image at the same time. Often the image is scanned from left to right and top to bottom, which can create some interesting effects when recording moving objects like the blades of a fan. Here we simulate a digital movie of a rotating chess-board. The scan is moving from left to right, catching up with the rotation in the bottom of the image, and moving against it at the top. [inspiration from danielwalsh] [more] [code]

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Two touching identical circles have the same area as the negative space they create in a circumscribing larger circle. That allows us to create this gif, where the circles transform without changing area. [can you prove the first sentence?] [code]

Two touching identical circles have the same area as the negative space they create in a circumscribing larger circle. That allows us to create this gif, where the circles transform without changing area. [can you prove the first sentence?] [code]

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If liquid starts at the top of some porous material, will it be able to filter to the bottom? This is the kind of question analysed in Percolation theory, applicable to coffee making and material science. Consider a lattice of points, with edges deleted with a fixed probability. The threshold probability for us to expect an arbitrarily large air pocket is known in two dimensions, in dimensions larger than 18, but not known inbetween.  [more] [graph] [code]

If liquid starts at the top of some porous material, will it be able to filter to the bottom? This is the kind of question analysed in Percolation theory, applicable to coffee making and material science. Consider a lattice of points, with edges deleted with a fixed probability. The threshold probability for us to expect an arbitrarily large air pocket is known in two dimensions, in dimensions larger than 18, but not known inbetween.  [more] [graph] [code]

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The aesthetics of pixel art can be tricky to get right, but this illustrates one simple method which makes curves look natural. The initial pixelation is done by Photoshop, which creates curves which look bumpy and jaggy. Sorting the blocks by their slope causes the diagonals to form smoother curves, and gives a better pixelation. [Tiffany Inglis] [code] [more]

The aesthetics of pixel art can be tricky to get right, but this illustrates one simple method which makes curves look natural. The initial pixelation is done by Photoshop, which creates curves which look bumpy and jaggy. Sorting the blocks by their slope causes the diagonals to form smoother curves, and gives a better pixelation. [Tiffany Inglis] [code] [more]

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A visualisation of some recorded speech, based on triangulating a mesh of all the recorded samples from the microphone. [what is being said in the recording?] [code] [full res]

A visualisation of some recorded speech, based on triangulating a mesh of all the recorded samples from the microphone. [what is being said in the recording?] [code] [full res]

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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