An interactive browser for closed-leg recursions — the family the 91-wave belongs to.
Take a line. Replace it with three legs whose signed heights sum to one. Replace each leg with
a copy of the same figure. Repeat forever. The closure rule — legs summing to one — is
what makes the copies fit without gaps, and a single leg running backwards is what makes the
result fractal. Everything below is that one rule, pushed in a different direction: into the
plane, into terrain, into other numbers of legs, and into a measure you can point at your own
data.
What is claimed here, and what is not. The mathematics is classical throughout —
Barnsley (1986) for the affine fractal interpolation functions, Moran (1946) for the dimension
equation, Mauldin & Williams (1988) for the graph-directed case, Barnsley & Harrington
(1985) for the parameter plane of pairs of complex maps, Allaart (2020) for the Hölder
spectrum. No new theorem is claimed. What is offered is the closure condition treated as
the defining feature rather than an incidental normalisation, and a map of the family you can
drag, where the literature customarily draws one curve at a time.
Every figure on screen is labelled exact or measured. Where a formula gives only
an upper bound, the page says so and declines to print the number as a result.
Deposits ·
The Closed-Leg Atlas ·
Roughness ·
benchmark dataset
Book · The Summer of My Life: Chaos, Fractals, and the Mathematics of Unpredictability
(2026), as M. Y. AI
Runs entirely in your browser. Nothing is uploaded. Save this file and it works offline.