Celebrating 𝜋 Day

March 14th marks the day to celebrate the iconic irrational mathematical constant of 𝜋, which serves as a mysterious fundamental constant of our universe. Chosen because the date reflects the first digits of 𝜋 in 1988 by American physicist Larry Shaw, 𝜋 day has since become a global phenomena celebrated each year by maths aficionados where they participate in events such as competitions, games and eating pies! Interestingly, it also happens to be the birthdate of Albert Einstein.

This year, our school will commemorate the upcoming 𝜋 day with an event to show the wide range of applications of mathematical concepts in various disciplines as well as the many projects students have worked on throughout the year. As a member of Project 0, and an avid mathematician and computer scientist, I took this opportunity to help arrange the event structure and showcase my projects. In particular, I want to highlight the connection between maths and the arts and how we can model various complex environments with simulations.

A prominent example that showcases the direct relationship between mathematics and art is through computer graphics and shaders. The signed distance function (SDF) shader allows us to mathematically represent geometry based on simple functions in an efficient manner. Additionally by interpolating in different ways between different functions we can blend these geometries together in interesting ways. This simple technique can be used to quickly create stunningly complex and appealing designs from the equations that describe them.

A time-lapse of the creation of a snail using the signed distance functions

Source: https://www.alanzucconi.com/2016/07/01/signed-distance-functions/#part1

Another relevant example comes in the form of Google's deep dreams and appearance transformation networks. By inverting an image recognition neural network to mutate it's input from it's output we can get it to "interpret" what the images it's seeing and mutate the initial image with it's interpretation. This leads to bizarre hallucinations that give closer insight into how these commonly used image networks see the world. Neural networks act almost like a model of the key features of the dataset they encode. From the features of a certain animal to the art style of a particular artist, we can encode their essence in a neural network which captures their unique expression. Using mathematics, we are able to express and capture a snapshot of a subject. This is an idea that is deeply artistic in it's nature, showing a clear link between the two areas. 

Lastly we'll explore how the field of signal processing can allow us to instantly stream and break down audio, and in particular one innovation which has enabled endless possibilities in the realms of signal processing and audio analysis and has revolutionised compression. The Fourier transform and it's respective inverse function allows us to capture, combine, distort, and filter audio in interesting and unique ways. We'll go over how timbre and various different qualities of the audio can be expressed within it's Fourier decomposition and how drastically different sounds can still embody the same pitch. The Fourier transform can also be used to encode pretty much any information including image data (like the PNG - a popular image format for it's efficacy and compressive power) and has wide reaching cababilities beyond the obvious signal processing applications.