Typography and the Letter S
Every "S" you've ever read was built from circles and tangent lines long before computers existed. In 1517, printer Francesco Torniello worked out how to draw a perfect S using nothing but a compass — find where a straight line just grazes a circle, and the curve reveals itself.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Every "S" on this page was ultimately built from circles and straight lines. In 1517, the Italian printer Francesco Torniello worked out a ruler-and-compass method for constructing the letter, and centuries later the computer scientist Donald Knuth showed exactly what mathematics that method relies on: a classic tangent-line-to-a-circle problem, solvable with nothing more than simultaneous equations. This idea walks through Torniello's original construction and the analytic geometry behind it.
Guiding Questions
- Draw a circle and a point outside it. Where exactly does a straight line from that point just touch the circle without crossing it?
- Torniello needed fourteen points and several circles to build one S by hand. Try building a smooth S shape from your own circles and straight lines — how few pieces can you get away with?
- Change the radius or position of your circles. What happens to the shape of the S?
- (stretch) Donald Knuth later found a formula for exactly this tangent problem — pin down your own version of it, and check it against a circle and line you draw yourself.