## On Fourier Series

Posted in mathematics, Maynooth, The Universe and Stuff with tags , , , , , , on November 30, 2021 by telescoper

So here we are, in the antepenultimate week of the Autumn Semester, and once again I find myself limbering up for the “and” bit of my second-year module on Vector Calculus and Fourier Series, i.e. Fourier Series.

As I have observed periodically, I don’t like to present the two topics mentioned in the title of this module as completely disconnected, so I linked them in a lecture in which I used the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding.

## Fourier Series, Epicycles and Haemorrhoids

Posted in The Universe and Stuff with tags , , , , on August 13, 2015 by telescoper

My attention was drawn to this little video some time ago by esteemed Professor George Ellis. I don’t know why it has taken me so long to share it here. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about Astronomy. It’s a common view that Kepler’s laws of planetary motion  according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as parts the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed form expression for such an orbit is considerably more compact and elegant than a Fourier representation and also encapsulates a deeper level of physical understanding.

It’s nore entirely relevant to the rest of this post but I discovered last week – by reading a book – that Johannes Kepler suffered so badly from haemorrhoids (piles) that he did all his calculations standing up. I just thought I’d share that with you.