Can circles draw anything?

Watch Mars against the stars for a few months and it drifts slowly eastward, night after night. But roughly every two years and two months it slows, stops, runs backward for about ten weeks, stops again and carries on. Astronomers call the backward stretch retrograde motion. Any theory of the sky has to explain it.

For fourteen centuries the standard explanation was the one written down by Claudius Ptolemy in Alexandria around 150 CE, in the book we know as the Almagest. The planet rides on a small circle, the epicycle, whose centre is carried around the Earth on a big circle, the deferent. It is often retold as a joke: whenever the theory went wrong, add another circle. The joke hides something true and surprising. Circles on circles are not a weak tool. They are, in a precise sense, a tool that can fit anything — and that turns out to be the problem.

An arrow on an arrow

Think of each circle as an arrow spinning about its tail at a steady rate. One arrow, alone, sweeps its tip around a circle. Now fix the tail of a second, shorter arrow to the tip of the first, and let both spin. The tip of the second arrow is our planet.

In the figure below, Earth sits at the centre. The big arrow turns once every 687 days; the small one is 0.66 times as long and turns 1.88 times as fast. These are not invented numbers, as we will see. The outer ring stands for the sky: the dashed line is our line of sight, and the marks on the ring show where the planet has appeared over the last 300 days. When the planet’s place in the sky slides backward, both its path and its marks turn red.

Why does the planet go backward? Its motion is the sum of two turnings. Along the outer part of a loop the two arrows carry it the same way. On the inner part, nearest Earth, the small arrow sweeps the planet backward faster than the big arrow carries it forward, so for a while the net motion across our sky is reversed. If the small arrow is too short or too slow, the forward motion always wins and there is no retrograde stretch at all.

With the default settings the model makes a backward stretch of 73 days, covering 15.9° of sky, once every 780 days. Mars really does this every 780 days — NASA lists its synodic period, the time between one lining-up of Sun, Earth and Mars and the next, as 779.94 days — and its backward stretches last around ten weeks.

The match is no accident. The big arrow is Mars’s own orbit around the Sun (687 days). The small arrow is Earth’s orbit, seen in reverse: it turns once a year, which is 1.88 times as fast, and its length is Earth’s distance from the Sun divided by Mars’s, 1 ÷ 1.524 ≈ 0.66. Adding the two arrows gives the line from Earth to Mars exactly. Ptolemy’s epicycle for Mars was, in effect, a picture of Earth’s motion that he did not know he was drawing.

Nicolaus Copernicus, in 1543, put the Sun near the centre and showed that retrograde motion is only apparent: it is what you see from a moving Earth as it overtakes a slower outer planet. But he did not get rid of circles on circles. To match the observations with uniform circular motion he still needed small epicycles, which he preferred to Ptolemy’s equant — a point off the centre from which the motion looked uniform. The circles only went away in 1609, when Johannes Kepler showed that Mars moves on an ellipse, faster when nearer the Sun.

More circles, any path

Ptolemy’s model of Mars used two circles, plus an offset centre and the equant. What happens if you keep going — a third arrow on the tip of the second, a fourth on the third? In 1960 the philosopher Norwood Russell Hanson showed that with enough of them, circles can trace even a path with sharp corners, such as a square. Each new arrow adds one term to a sum, and that sum is what mathematicians now call a Fourier series: a way of writing a repeating motion as a combination of steady rotations, each turning a whole number of times per cycle.

Here is the hardest kind of case: a value that jumps. Below, each circle turns an odd number of times per cycle (1, 3, 5, …), and circle number n has radius 4/(πn). We track only the height of the last arrow’s tip, and draw it out to the right over time. The target, in grey, is a square wave: +1 for half of each cycle, −1 for the other half.

This stubborn overshoot is the Gibbs phenomenon, first noticed by Henry Wilbraham in 1848 and rediscovered by J. Willard Gibbs in 1899. Wherever the target jumps, the sum of circles overshoots by about 8.95% of the size of the jump, however many circles you add. Its width shrinks as circles are added, so on average the fit keeps improving, but its height does not. Circles, which are smooth, cannot make a clean jump.

Draw your own

Now for the reverse question: given a shape, which circles draw it? Walk around the shape at an even pace and write down 256 equally spaced points. Treat each point as a single number, a complex number (its sideways position plus i times its height), which is the natural language for arrows. A calculation called the discrete Fourier transform then gives, for each whole-number speed k from −128 to 127 turns per trip around the shape, the one arrow of that speed that belongs in the recipe — its length and its starting angle:

c_k = average over all points of (point × a rotation of −k turns per trip)

In words: spin the whole shape backwards at speed k and find its average position. Parts of the shape moving at speed k stand still in this spinning view and survive the averaging; everything else goes round and cancels out. Chain all 256 arrows together and their tip passes exactly through all 256 points. Use only the biggest few and you get the best rough sketch that those few can make.

Try the square outline with 10 or 20 circles. The corners come out rounded, but unlike the square wave, the green path does not shoot past them: here the pen never jumps, it only turns sharply, and for a continuous path like that the gap shrinks everywhere as circles are added. For any closed path made of smooth pieces — anything you can draw without lifting the pen — the circles’ path closes in on it at every point. That is the precise sense in which circles can draw anything.

Notice too how fast the energy share climbs. For the heart, the 12 biggest circles already hold about 99.9% of it, and their path is on average less than 1% of the heart’s width away from it. Most shapes people draw are like this: a few slow, large circles carry the outline, and many fast, tiny ones add the details. Related frequency-based ideas — though not epicycles as such — let formats like JPEG and MP3 spend few bits on the fine, fast-changing detail that we barely notice.

What the circles cannot tell you

Here is the twist that follows from Hanson’s result. Because circles on circles can fit any closed path, fitting the planets with them proves nothing about why the planets move. Ptolemy’s system was not wrong because its mathematics was weak; its mathematics was, if anything, too strong. It could describe whatever the sky did, so it could not explain it. Kepler’s ellipses made a much riskier claim — one simple curve per planet, with the Sun at a fixed point inside it — and Newton’s law of gravity then said why it had to be so. A good theory rules things out.