The 91-Wave: A closed three-leg recursion, its parameter atlas, and the geometry of the fold

A closed three-leg recursion replaces a line segment with three legs whose signed heights sum to one, then replaces each leg with a scaled copy of the same figure, without limit. The closure condition is what makes the construction possible: it is exactly the condition under which three legs land where the parent lands, so that copies nest without gaps at every scale. This work specifies the family, derives its well-posedness, variation, box dimension and Hölder spectrum, classifies its parameter plane, extends it to unequal leg durations, and names one member — the 91-wave, with coefficients 50/91, −24/91, 65/91, of box dimension 1.385590468069 and typical Hölder exponent 0.688175569997. Two results are worth singling out. First, the tame region of the parameter plane is exactly the region in which all three legs run the same direction, so every fractal member of the family owes its roughness to a leg running backwards. Second, freeing the leg durations and maximising the box dimension assigns the shortest duration to the leg of smallest amplitude, lifting the dimension from 1.385590 to 1.430240. The deposit includes a reference implementation in Python and two interactive atlases that render any member of the family on demand. This family is Barnsley's affine fractal interpolation function on a uniform partition (1986); the dimension formulae are classical and the pointwise Hölder spectrum of the class is treated by Allaart (2020). No new theorem is claimed, and the prior art is set out in full in section 11. What is offered is the closure condition treated as the defining feature rather than an incidental normalisation, a named worked address, and an atlas of the whole two-parameter family, where the literature customarily presents one curve at a time. Prepared with AI assistance for derivation checking, numerical verification, and code. All numerical claims were independently recomputed and every derivation checked against the closed forms given.