Donald Knuth designed MetaFont to describe letters mathematically. Each glyph is a set of geometric equations: pen paths, control points, parameters for serif length and stroke width. Twenty-five of the twenty-six lowercase letters yielded to this approach. Then came S.
The S defeated Knuth's pen-stroke formalism. It is the only common letter that reverses curvature twice while maintaining continuous tangency — curving left, then right, then left again, crossing its own axis through a rotational symmetry rather than a reflective one. O is one curve. C is half of one. B has bumps but they face the same direction. S folds back through itself.
Knuth spent sleepless days producing only ugly results. His formulas — the same ones that handled every other letter — could not express the curvature reversal. Eventually he abandoned the pen-stroke model entirely and described the outline directly, using a construction from classical geometry: an ellipse positioned so its tangent line meets the opposing curve at exactly the right point, creating a bridge between the two halves of the letter. The formula was complex enough that he published it as a mathematical discovery.
Every literate person can draw an S. A child draws it without hesitation — the hand traces the double reversal as naturally as breathing. What Knuth discovered is that this ease is deceptive. The hand has solved a geometric problem that resisted one of the great mathematical minds in computer science. The embodied knowledge of the S is richer than its formal description, not because the description is wrong, but because the description requires machinery (ellipse tangent constructions) that the hand bypasses entirely.
The S is where writing exceeds its own notation.