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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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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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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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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]

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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