Circles That Draw

1. Circles that draw

Each circle below turns at a steady speed, and each one rides on the rim of the circle before it. Every circle turns a whole number of times while the drawing is made once: 1 turn, 2 turns, 3 turns, and so on, one way or the other. The pen at the end of the chain draws the red line. Pick a shape, then use the slider to add circles one at a time.

How this drawing stays honest. Nothing here is traced over a picture. The page samples the shape at 256 evenly spaced points along its outline and works out the 256 circles with a discrete Fourier transform (step 2). The red line is then drawn only by adding up the turning arms of the circles in use: the pen is always exactly at the end of the chain. Turn on all 256 circles and the line passes through every sample point. With fewer circles you see the honest approximation, and the readout measures how far off it is.

2. Why circles work

Any repeating motion is a sum of steady circular motions. If a point goes round a closed path and comes back to the start, its motion can be split into circles turning 0, 1, 2, 3, … times per trip around the path, in either direction. Each circle needs only two numbers: how big it is and where it starts. Small, fast circles add the fine detail; big, slow circles give the overall shape.

How the page finds the circles. Write each sample point as a complex number z = x + i y. To find the circle that turns k times per drawing, the page turns every sample point backwards by k turns' worth and takes the average. Every other circle still goes round and averages out to zero. Only the speed-k circle is now standing still, so it is what is left:

c_k = (1/N) Σ z_j e^{−2πi k j/N},   and then   z(t) = Σ c_k e^{2πi k t}

The size of c_k is the circle's radius and its angle is where the circle starts. The second formula is the chain of circles itself: each term c_k e^{2πi k t} is one arm turning k times while t runs from 0 to 1.

Here only the height of the pen matters. The circles turn 1, 3, 5, 7, … times per cycle, with radii 1, ⅓, ⅕, 1/7, … (times 4/π). Their heights add up to a wave with flatter tops and steeper sides the more circles you use. This is the Fourier series of a square wave:

y = (4/π) [ sin x + ⅓ sin 3x + ⅕ sin 5x + … ]

Each term is a sine wave with a smaller amplitude and a shorter period, just like the graphs in a trigonometry course. In sound, the same terms are a note's harmonics or overtones, which is why a violin and a flute playing the same note sound different. Near each jump the sum always overshoots by about 9% of the jump, however many circles you add. This is called the Gibbs phenomenon.

Fourier. In 1807 Joseph Fourier told the French Academy of Sciences that the way heat spreads through a solid body could be described with sums of sines and cosines. He published the full theory in The Analytical Theory of Heat (1822). Leading mathematicians, Lagrange among them, doubted that a sum of smooth waves could make a shape with corners. It took later mathematicians, beginning with Dirichlet in 1829, to prove when Fourier's series really does add up to the function. Today the same idea is at work in music recordings, radio, medical scanners and the compression of photos.

Circles in the sky. Long before Fourier, Greek astronomers described the planets with circles on circles. Apollonius and Hipparchus used them, and Claudius Ptolemy brought the method together in the Almagest (about A.D. 150). Each planet rode a small circle, the epicycle, whose centre moved around a larger circle, the deferent. This explained why a planet sometimes seems to stop and move backwards against the stars. Ptolemy's tables were used to predict planetary positions for well over a thousand years, and Copernicus still used epicycles in his Sun-centred system of 1543. In 1609 Kepler showed that Mars moves on an ellipse with the Sun at a focus. He went on to describe every planet the same way, and the ellipses fit the observations better than any arrangement of circles had.

The psalmist wrote, “The heavens declare the glory of God” (Psalm 19:1). Astronomers from Ptolemy to Kepler kept looking for the simple order beneath what they saw, and the steady turning circle turns out to sit underneath tides, sound, light and the swing of a pendulum.

3. Quick check