Mathematical images and sounds
I’ve written two blog posts since my last Substack article. One produces and interesting image and the other produces an interesting sound.
Suppose you draw a triangle, then draw a circle around that. Then draw a square and a circle around that. And then a pentagon, etc. It’s not obvious at first whether this process will converge to a figure with finite radius or grow without bound.
The process does converge, and it produces an interesting image.
The image above shows 100 steps of the process, which is effectively infinite due to the limitation of finite pixels. For more information see the blog post Limit of a doodle.
This morning I wrote a post about the Clausen function. The function looks sorta like a sawtooth wave, so I wondered whether it sounds like a sawtooth wave, and it kinda does.
The Fourier coefficients of the Clausen function decay slowly. This accounts for the harsh sound of a sound wave with this shape, and for the vertical tangent of the function at 0. It’s not clear from the plot above whether the derivative is finite or infinite at 0, but the blog post shows its infinite.