Explore a Value
How It Works
The ?-function encodes continued fractions as binary strings via run-length encoding. For x = [0; a₁, a₂, a₃, ...], write a₁ zeros, then a₂ ones, then a₃ zeros, alternating. Read the result as a binary fraction: that's ?(x).
?(x) = Σ (-1)^(k+1) · 2^(-(a₁+...+aₖ))
The Devil's Staircase
Why "Devil's Staircase"?
?(x) is strictly increasing and continuous, yet its derivative equals 0 at almost every point. The function gains all its height on a set of measure zero—a fractal dust of quadratic irrationals and their limits.
Quadratic Irrationals ↔ Rationals
The ?-function's deepest property: it maps quadratic irrationals (solutions to ax² + bx + c = 0) bijectively to rational numbers. This works because quadratic irrationals have periodic continued fractions, which become periodic binary expansions—i.e., rationals.
Self-Similarity & PSL(2,ℤ)
The ?-function exhibits fractal self-similarity generated by two operations:
S: Shrink to bottom-left
S(x, y) = (x/(1+x), y/2)
?(x/(1+x)) = ?(x)/2
R: Point reflection
R(x, y) = (1-x, 1-y)
?(1-x) = 1 - ?(x)
Combining S and R generates a monoid isomorphic to a subset of PSL(2,ℤ), the modular group. Each element corresponds to a rational via continued fractions, and describes a self-similarity of the devil's staircase.