Glyphs from Circles
A single pen is drawing the symbol, held at the tip of a chain of turning circles. Each circle spins at its own fixed speed and carries the next one round on its rim, and together they steer the pen around the glyph. With 60 circles every one of the ten planetary symbols comes out within 1 percent of its height. Slide down to a handful and the glyph melts into a soft loop.
| Points on each path | 512 |
|---|---|
| Symbols | ten, Sun to Pluto |
| Circles by default | 60 |
| Error at 60 circles | under 1% of the height |
What you are looking at
The faint circles are the chain. The first is centred in the middle of the symbol, each next one is centred on the rim of the one before, and every circle turns at a steady whole-number speed, some one way and some the other. The pen rides on the last circle, and the line it leaves fades from its tail as the chain goes round again.
The pen never leaves the paper except once. Where a symbol branches, like the cross under Venus, the pen goes out along the branch and comes back, so the closed path passes some strokes twice. Only the Sun needs a lift, for the jump between its circle and its dot, and the trace is hidden during that jump.
Try the circles slider on Venus. With 1 circle the pen can only draw a circle. With 5 it draws a lumpy Venus that strays 5.9 percent of the symbol's height from the true shape on average, with 20 it is down to 0.85 percent, and with 60 to 0.14 percent.
The math
Joseph Fourier's idea, applied to a drawing: any closed path can be written as a sum of circles, each turning a whole number of times while the pen goes round once, some forwards and some backwards. Sampling the path at 512 points and taking the discrete Fourier transform gives exactly 512 such circles, one of which stands still and marks the centre. Together they pass through every sample exactly.
The circles are sorted by size and only the largest are used. The big slow ones carry the overall shape, and the small fast ones add corners and the sharp turn at the end of a stroke. The average error left over is fixed by the circles that were left out: its square is the sum of their squared radii (Parseval's theorem). So adding a circle can only bring the pen closer, and corners are the last thing to sharpen, because a corner needs many fast circles.
Circles riding on circles are epicycles, the tool Ptolemy used to describe the planets. With enough of them a chain can follow any closed path at all, which is part of why epicycles fitted the sky as well as they did, and a reminder that fitting a path is not the same as explaining it.
How it is computed
Each symbol is drawn by hand as strokes, lines, arcs and curves, in a box two units tall. The strokes form a network whose junctions are where strokes meet. A closed path that uses every stroke exists only when every junction has an even number of stroke ends, which is Euler's rule from 1736. So the piece finds the shortest set of strokes to go over twice that makes every junction even (the Chinese postman problem), then walks the network in one closed loop.
That loop is resampled to 512 points spaced evenly along its length, a discrete Fourier transform turns them into circles, and the largest are chained from the centre. The pen's whole path is computed once when you change a setting, and the chain is redrawn every frame. The small symbol in the corner is the app's own glyph, for comparison.
Questions
How can turning circles draw straight lines and sharp corners?
Many circles of different speeds can cancel each other's curving in some places and add up in others. A straight stroke is where they cancel sideways motion; a corner is where many small fast circles all swing at once. That is why corners are the last thing to appear as you add circles.
Why does the pen go back over some strokes?
The circles can only draw one closed loop. A branch such as the cross on Venus has loose ends, so the pen has to go out along it and come back. The piece picks the shortest set of strokes to retrace so that one loop covers the whole symbol.
What does this have to do with Ptolemy?
Ptolemy described each planet's motion with circles moving on circles, epicycles. The same idea, with as many circles as you like, can draw any closed shape, including a planet's own symbol. See the Ptolemy piece in this gallery for his actual model.