In this work, we obtained a nonmatrix analytic expression for the generator of the Peano curve. Applying the iteration method of fractal, we established a simple arithmetic-analytic representation of the Peano curve as a function of ternary numbers. We proved that the curve passes each point in a unit square and that the coordinate functions satisfy a Hölder inequality with index , which implies that the curve is everywhere continuous and nowhere differentiable.

1. Introduction

Space-filing curves, such as the Peano curves, are geometrically interesting curves and have important applications, particularly in parallel computing. Bagga et al. [1] developed a matrix multiplication utilizing the Peano curves in designing a cache oblivious algorithm. Platos et al. [2] created a model of signal coverage based on optimized representation by space-filling curves to reduce memory consuming in computation. Sasidharan and Snir [3] showed how to reduce communication and improve the quality of partitions using a space-filling curve. Much more applications of space-filing curves can be found in Bader’s book titled Space-Filling Curves: An Introduction with Applications in Scientific Computing [4].

Peano in 1890 [5] geometrically constructed a continuous curve, later called the Peano curve, that fills the unit square . The idea of the construction is to divide each square into 9 smaller equal squares continuously and to determine a path, or curve, so it goes through each square. The limit of this path is the Peano curve. Many different such paths can be designed as shown in Figures 1–3.