Basic Explanation
A whole number is prime when it isn't equal to any product of whole numbers other
than itself and 1. However, if we allow some "new" whole numbers, something that was
previously prime may now be equal to a product of others. Each diagram and chime
represents how a prime number factors in a bigger number system.
Advanced Explanation
More technically, if \(p\) is our prime number, each blue circle represents one of the
prime ideals dividing \(p\mathcal{O}_F\), with the circle's area representing inertial degree,
and the shade of blue representing ramification index. Thus the amount of blue above each
prime is always the same, \([F:\mathbf{Q}]=8\) times the amount of green.
For each prime ideal dividing \(p\mathcal{O}_F\) a chime is played, whose pitch is
determined by its inertial degree. Celesta or clavichord is used depending on whether that prime
ideal is unramified or ramified, respectively. However, when \(p\mathcal{O}_F\) is itself prime,
a "swell" is played. For a given \(F\), all but finitely many primes will be unramified.
Implementation
This page is built with PHP and JavaScript.
The chimes are implemented with the howler.js
library.
The actual factorizations are computed by Sage. To factor the ideals \(p\mathcal{O}_F\), Sage must
first compute \(\mathcal{O}_F\), which can be quite costly. Therefore my Sage script memoizes its
computation of \(\mathcal{O}_F\) using the pickle module.
The source code is available
on Github.
Acknowledgements
This page uses the audio files from Listen
to Wikipedia, a project created by Stephen LaPorte and Mahmoud Hashemi. Here is
a copy of
their license.
The idea to make this "audiation" was somewhat inspired by Kazuya Kato's poetry
about prime numbers.
