Beauty by the Numbers
Aesthetics from a mathematician's perspective
By Percy Wong
If you ever chance upon a mathematician and strike up a conversation, you might be surprised to find that a lot of us have a penchant for describing certain work in mathematics, be it a concept, a theorem, or a proof, as beautiful or aesthetically pleasing. What do we mean by that? Most of the time, when people talk about beauty, they are associating that notion with arts: painting, sculpture, music, poetry, photography, etc. So have we mathematicians as a group hijacked these words, “beauty,” “aesthetics,” and turned them into homonyms that have entirely different meanings from their traditional usages? I would like to make a humble effort to answer this question in the negative. More ambitiously, I want to show that what we call “beauty” can be appreciated by someone with very little formal training in mathematics, similar to how one can enjoy a painting by Monet or a mazurka by Chopin without ever having lifted a paintbrush or laid a finger on the ivory keys of a piano. While very few mathematicians, other than the most vainglorious, would ever consider themselves “artists,” our perception of what is considered beautiful is perhaps less alien than what it appears to be prima facie.
Not every single piece of artwork that displays symmetry is considered a masterpiece; conversely, not every single masterpiece needs to display symmetry. However, it is undeniable that there are artworks that derive their beauty, at least partially, from the symmetry they display. Similarly, not all theorems in mathematics that show symmetry are considered beautiful, but the history of progress in mathematics is rife with examples of mathematical work that possess symmetry and also turn out to be both beautiful and influential. Let us look at one such example: the Fourier transform.
Imagine a sound, say the concert pitch of A, which is a pure waveform of 440 Hz. This can be represented graphically as a sound wave:
The reason it is called A440 is because in one second there are 440 periods (cycles) of this wave. Analogously, a different pitch, say middle C (261.6 Hz), will have a different waveform, and will be represented graphically as:
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