Minimal Peano curve

Abstract

A Peano curve p(x) with maximum square-to-linear ratio |p(x)−p(y)|2/|x−y| equal to 5 2/3 is constructed; this ratio is smaller than that of the classical Peano-Hilbert curve, whose maximum square-to-linear ratio is 6. The curve constructed is of fractal genus 9 (i.e., it is decomposed into nine fragments that are similar to the whole curve) and of diagonal type (i.e., it intersects a square starting from one corner and ending at the opposite corner). It is proved that this curve is a unique (up to isometry) regular diagonal Peano curve of fractal genus 9 whose maximum square-to-linear ratio is less than 6. A theory is developed that allows one to find the maximum square-to-linear ratio of a regular Peano curve on the basis of computer calculations.