matthen

Things of interest in Maths & Science

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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]

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An intricately patterned image made entirely from drawing lines straight from one side to the other. [code]

An intricately patterned image made entirely from drawing lines straight from one side to the other. [code]

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An interesting construction which results in a golden ratio rectangle (one with sides a and b such that (a+b)/a = a/b).  [more] [code] [interactive]

An interesting construction which results in a golden ratio rectangle (one with sides a and b such that (a+b)/a = a/b).  [more] [code] [interactive]

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Today is the Solstice, the longest or shortest day depending on which hemisphere of the Earth you’re in.  Shown here is the movement of the Sun on a projected celestial sphere, as observed from my hometown of Edinburgh. The Sun’s path over 8 equally spaced dates in the year is shown, and its position at noon is marked, revealing a figure of eight pattern. The two Solstices are at the top and bottom of the figure of the eight, where the Sun (sol) stands still (sistit). [more] [code] [interactive]

Today is the Solstice, the longest or shortest day depending on which hemisphere of the Earth you’re in.  Shown here is the movement of the Sun on a projected celestial sphere, as observed from my hometown of Edinburgh. The Sun’s path over 8 equally spaced dates in the year is shown, and its position at noon is marked, revealing a figure of eight pattern. The two Solstices are at the top and bottom of the figure of the eight, where the Sun (sol) stands still (sistit). [more] [code] [interactive]

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A simple animation showing how connecting points rotating on circles at different phases can create the illusion of a 3D figure moving, rotating and skewing. [inspired by] [code]

A simple animation showing how connecting points rotating on circles at different phases can create the illusion of a 3D figure moving, rotating and skewing. [inspired by] [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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