## Math

Disk point picking is an elementary problem in geometry. To evenly distribute points on a disk, an intuitive idea is to sample pairs of numbers in polar coordinate (radius and angle) randomly from uniform distributions. After all, “uniformly” is just another word for “evenly”, isn’t it? But it doesn’t work. When the points are sampled this way, the density of the dots fall off linearly with the distance to the origin.

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Is the salesman travelling on foot or on an airplane? This article describes an experiment to develop a version of the elastic net algorithm that works on spherical surfaces. I needed it for a computational neuroscience problem, but for those who are mainly interested in machine learning, it also serves a simple and intuitive demonstration of using gradient descent on non-Euclidian surfaces. The Tensorflow source code is available on GitHub. I also made a youtube video showing the algorithm in action.

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Installed on the northern wall of the Monash University student centre (Melbourne) is a curious geometrical object. If you don’t pay close attention, you might think it’s just a decorative sculpture. But it is actually a functional sundial — conceived and constructed by the mathematician Carl Moppert in the 80’s. This design appears to be unique. I browsed through many photos on Instagram tagged with #sundial, and I couldn’t find another one that looks quite like it.

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Fourier analysis says that complex patterns can be created by adding up a large number of patterns as simple as sinusoidal waves. To make the idea more concrete, I like to use the following analogy in teaching: Imagine that you lived in the early 19th century. If you wanted to listen to a symphony, the only way to make it happen was to hire a few dozen highly-trained musicians to perform it for you.

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