Given a real matrix and a vector , produces an dimensional vector . This is a stretched, rotated version of . Two-dimensional matrix multiplication is easily visualized on a plane. Two dimensional case: Given matrix , multiplying by vector produces . Say we wanted to find a matrix to rotate a vector 90 degrees clockwise. The desired result of multiplication will be . Studying the formula above, this is accomplished

## What do functions sound like?

I was a band geek in middle and high school, the euphonium being my instrument of choice. When the band director handed out a new piece of sheet music for the band to study, the first thing we had to do was sight read it, and use the ink on the page to imagine what the song sounds like. To do this, I’d hum some tone, and if the notes

## Guessing Fourier Coefficients

Given a ‘nice enough’ periodic function, it can be expressed as an infinite sum of sine and cosine functions, known as its Fourier Series. In some cases, formulas specifying and can be computed by hand using the Euler-Fourier formulas. When these these methods are inconvenient, approximate coefficients can be found using a random search in the space of all possible coefficients. The Coefficient Arrays The random search works like this:

## Accidental Sierpinski Gasket

I was trying to visualize the 3-adic numbers as dots on a plane. The idea was to use the coefficients in each number’s representation as the sum of powers of p to uniquely define a pair of coordinates. Since the coefficients can only be 0, 1, or 2, the plan was to create a tree (inspired by Bruhat-Tits trees) with three branches at each level. To stop the points from