Ptolemy’s discoveries, the beauty of circles, and Homer Simpson? Here’s how Ptolemy’s theory of epicycles can be used to create any closed orbit.
From the long-running television phenomenon The Simpsons, Homer Simpson is often portrayed as an incompetent, slothful, and indolent plebeian, but yet, he continues to baffle all of us with his bizarre and almost clairvoyant predictions that prove to be true time and time again (take, for example, his prediction of the Higgs boson’s mass1). A true front of innovative scientific discovery, Homer Simpson has done it again, this time not discovering, but embodying, the reconstruction of a mysterious orbit found in 2008. Aptly dubbed the “Homer Simpson orbit,” this discovery sparked interest in a field that now permeates our technological world today…
In all seriousness, despite Homer Simpson’s seemingly prophetic powers, which should actually be largely attributed to the Harvard and UC Berkeley educated writers2 behind the works, the reconstruction of the “Homer Simpson orbit” utilizes a concept that is fundamental in many scientific fields: epicycles. Epicycles are essentially orbits around currently orbiting objects, or mini orbits. They have long been a part of our scientific literature, dating all the way back to early 200 BCE when Greek geometer and astronomer Apollonius of Perga3 used epicycles to explain the eccentric orbits of bodies around the earth in the geocentric model of our solar system. Although the Greeks later discovered that the Earth was indeed not the center of the solar system, the use of epicycles turned out to have a plethora of various other interesting uses.
Claudius Ptolemy first refined the theory of epicycles developed by Apollonius in the 2nd century AD. He introduced the proposal that epicycles were able to explain all of the strange orbits that occurred from the viewpoint of the Earth. Using a few tricks by placing the center of motion at an equant4 and shifting our observer’s view, he was able to sufficiently create a Ptolemaic model that possibly explained the movement of celestial bodies in our close vicinity. As it turns out, his ideas could not only explain a geocentric model of our solar system, but they could explain any possible orbit, no matter how erratic the curve.
Fourier transformations are quite complex to understand from a purely mathematical point of view. Instead, the image of a smoothie5 can help to intuitively see what a Fourier transformation does. The metaphor states that a Fourier Transform takes a given “smoothie” and performs an operation on it in order to find its ingredients. How exactly does this happen? Essentially, many filters are run through the smoothie in order to extract the specific ingredients that comprise of the smoothie. Recipes are often easier to work with than the whole of the smoothie itself, and modification often happens not with the final product, but with the original ingredients used. Once any operations are performed on the ingredients of the smoothie are complete, all that needs to be done is to mix the “smoothie” back together!
So what does this have to do with epicycles? When using Fourier Transformations on signals, the signals are typically processed as waves, or in the case of images, spatial frequencies. Take, for example, audio waves. Audio waves are one of the classic examples of a time-based signal which can be broken down into fundamental frequencies5 (the ingredients of the “smoothie”) in order to find the most prevalent signals in the audio wave. As it turns out, these fundamental frequencies are all simply sinusoidal waves that can be expressed through a time-based circular path with a particular amplitude, frequency, and phase shift. Once these three characteristics of the circular path are designated, the circular paths can all simply be added on one another as circles in an epicycle to produce the final, desired signal. With Ptolemy’s theory of epicycles, we can even create an image of Homer Simpson using a series of hundreds of epicycles on one another.
Sometimes, background noise or undesired signals are present in signals, and this is where the Fourier Transformation truly shines. With the Fourier Transformation, during the decomposition phase when we have the fundamental frequencies, we have the ability to easily remove the undesirable signals before synthesizing the final product once again. This is precisely the process that audio and JPEG compression use to send large data files over the internet, but the applications beyond signal processing are plentiful.
Fourier Transformations have helped to shape science from quantum physics to voice recognition to even MRI scans. With its vast applications in our modern world, this beautiful mathematical tool has been a keystone in advancing scientific discoveries throughout history. It was even famously used by James Watson and Francis Crick6 when decoding the double-helix structure of DNA based on X-ray patterns from Rosalind Franklin’s research! Though we may be oblivious to the pervasive nature of the Fourier Transformation, it has revolutionized countless numbers of fields and continues to be used every single day whether we are cognizant of its beauty or not.
1 Schwartz, R. (2015, March 5). Homer Simpson (Almost) Discovered The Elusive ‘God Particle? Fourteen Years Before Science Did. Retrieved from https://www.good.is/articles/homer-simpson-discovers-god-particle.
2 Salam, M. (2018, February 2). ‘The Simpsons? Has Predicted a Lot. Most of It Can Be Explained. Retrieved from https://www.nytimes.com/2018/02/02/arts/television/simpsons-prediction-future.html
3 Khan Academy. (n.d.). Planets & epicycles. Retrieved from https://www.khanacademy.org/partner-content/nasa/measuringuniverse/spacemath1/a/planets-epicycles
4 Ptolemaic system. (2008, February 9). Retrieved from https://brunelleschi.imss.fi.it/itineraries/multimedia/PtolemaicSystem.html
5 BetterExplained. (n.d.). An Interactive Guide To The Fourier Transform. Retrieved from https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/
6 Bhatia, A. (2013, November 6). The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face. Retrieved from http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face