Abstract
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.
The most discussed is the one shown in Figure 4.
For example, Moore in [6] and Milne in [7] conducted a very thorough study of it.
More discussions of constructions of such curves can be found in Sagan [8] and Jaffard and Nicolay [9, 10]. Recently, Makarov and Podkorytov [11] constructed a nonsymmetric plane Peano curves whose coordinate functions satisfy the Lipschitz conditions. However, there has not been any analytic expression of the Peano curve with which we can study them analytically and more thoroughly.
The goal of this work is to use fractal iteration method to establish an analytic expression of the iterated function system (IFS), then a series representation of the Peano curve; thus, we are able to discuss various properties of the curve. Specifically, we will prove the coordinate functions of the curve satisfy a Hölder inequality with index , which shows that the Peano curve is everywhere continuous and nowhere differentiable. We will also show the analytically constructed curve passes each point in a unit square.
2. Peano Curve Generated by Iteration
Sagan (cf. [8]) studied the Peano curve with an IFS expressed in matrix form. Matrix multiplication makes it very complicated to work with the IFS. In this section, we will obtain the IFS for the Peano curve in an analytic (series) form, which allows us not only to obtain more convenient coordinate functions for the curve but also to prove the nondifferentiability of the functions.
Divide into 9 equal subintervals, each of which is divided into 9 equal subintervals. We have, k-steps later, subintervals , each with length . Similarly keep dividing into 9 smaller equal squares, k-steps later, we have squares each with side length . The problem is to find for mapping to .
The initiator of the Peano curve is a diagonal of a unit square and can be obtained by stretching line segment times and rotating counterclockwise, represented in complex coordinate system by
The generator of the Peano curve can be expressed aswhich represent the nine diagonals in Figure 4 as functions of . Divide into nine equal sections:
Eliminating in (3) and (4), we get
Denote . Separating the real and imaginary parts in (5), we have
Introduce in ternary form. Equations (6) and (7) can be simplified as, for ,
Applying (8) and (9) again for , and ,
Substituting (10) back into (8) and (9), we have
Continuing this process, withwe obtain
Since , , sending , we arrive at the series expression of the Peano curve:
Both series in (14) converge because .
3. Properties of the Peano Curve
in (14) are functions of with
Theorem 3.1. Curve (14) fills a unit square up.
Proof. Let us first check the expression of :Denoting or ,We havethat is,Similarly,Therefore,Because the arbitrarity of ,fills the square up.
Theorem 3.2. Curve (14) is everywhere continuous and nowhere differentiable.
Proof. We will first show forwhich means x and y satisfy the Hölder inequality with index .
In fact, from (15), i.e.,Thus,Similarly,Inequalities (24) and (25) imply that and are continuous everywhere.
Next, we prove that and are nowhere differentiable.
TakeThen,Therefore,For any and any positive integer l, there exists , such thatIf , it follows from (31) thatIf , then eithermust hold. Otherwise, we would have bothwhich implycontradicting the triangle inequality. Therefore, we can always take or , such that andTherefore,which shows is not differentiable.
Similarly, we can also show that is not differentiable.
Theorems 3.1 and 3.2 imply that the Peano curve is a continuous curve that fills a unit square and it is nowhere differentiable.
Data Availability
The iteration data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.