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Inmathematics, ade Rham curve is a continuousfractal curve obtained as the image of theCantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including theCantor function, Cesàro–Faber curve (Lévy C curve),Minkowski's question mark function,blancmange curve, and theKoch curve are all examples of de Rham curves. The general form of the curve was first described byGeorges de Rham in 1957.^[1]

Consider somecomplete metric space${\displaystyle (M,d)}$ (generally${\displaystyle \mathbb{R}}$² with the usual euclidean distance), and a pair ofcontracting maps on M:

${\displaystyle d_0:M\to M}$

${\displaystyle d_1:M\to M.}$

By theBanach fixed-point theorem, these have fixed points${\displaystyle p_0}$ and${\displaystyle p_1}$ respectively. Letx be areal number in the interval${\displaystyle [0,1]}$, having binary expansion

${\displaystyle x=\sum _{k=1}^{\mathrm{\infty }}\frac{b_k}{2^k},}$

where each${\displaystyle b_k}$ is 0 or 1. Consider the map

${\displaystyle c_x:M\to M}$

defined by

${\displaystyle c_x=d_{b_1}\circ d_{b_2}\circ \cdots \circ d_{b_k}\circ \cdots ,}$

where${\displaystyle \circ }$ denotesfunction composition. It can be shown that each${\displaystyle c_x}$ will map the common basin of attraction of${\displaystyle d_0}$ and${\displaystyle d_1}$ to a single point${\displaystyle p_x}$ in${\displaystyle M}$. The collection of points${\displaystyle p_x}$, parameterized by a single real parameterx, is known as the de Rham curve.

The construction in terms of binary digits can be understood in two distinct ways. One way is as a mapping ofCantor space to distinct points in the plane. Cantor space is the set of all infinitely-long strings of binary digits. It is adiscrete space, and isdisconnected. Cantor space can be mapped onto the unit real interval by treating each string as a binary expansion of a real number. In this map, thedyadic rationals have two distinct representations as strings of binary digits. For example, the real number one-half has two equivalent binary expansions:${\displaystyle h_1=0.1000\cdots }$ and${\displaystyle h_0=0.01111\cdots }$ This is analogous to how one has0.999...=1.000... in decimal expansions. The two points${\displaystyle h_0}$ and${\displaystyle h_1}$ are distinct points in Cantor space, but both are mapped to the real number one-half. In this way, the reals of the unit interval are a continuous image of Cantor space.

The same notion of continuity is applied to the de Rham curve by asking that the fixed points be paired, so that

${\displaystyle d_0(p_1)=d_1(p_0)}$

With this pairing, the binary expansions of the dyadic rationals always map to the same point, thus ensuring continuity at that point. Consider the behavior at one-half. For any pointp in the plane, one has two distinct sequences:

${\displaystyle d_0\circ d_1\circ d_1\circ d_1\circ \cdots (p)}$

and

${\displaystyle d_1\circ d_0\circ d_0\circ d_0\circ \cdots (p)}$

corresponding to the two binary expansions${\displaystyle 1/2=0.01111\cdots }$ and${\displaystyle 1/2=0.1000\cdots }$. Since the two maps are both contracting, the first sequence converges to${\displaystyle d_0(p_1)}$ and the second to${\displaystyle d_1(p_0)}$. If these two are equal, then both binary expansions of 1/2 map to the same point. This argument can be repeated at any dyadic rational, thus ensuring continuity at those points. Real numbers that are not dyadic rationals have only one, unique binary representation, and from this it follows that the curve cannot be discontinuous at such points. The resulting de Rham curve${\displaystyle p_x}$ is a continuous function ofx, at allx.

In general, the de Rham curves are not differentiable.

De Rham curves are by construction self-similar, since

${\displaystyle p(x)=d_0(p(2x))}$ for${\displaystyle x\in [0,1/2]}$ and

${\displaystyle p(x)=d_1(p(2x-1))}$ for${\displaystyle x\in [1/2,1].}$

The self-symmetries of all of the de Rham curves are given by themonoid that describes the symmetries of the infinite binary tree orCantor space. This so-called period-doubling monoid is a subset of themodular group.

Theimage of the curve, i.e. the set of points${\displaystyle \{p(x),x\in [0,1]\}}$, can be obtained by anIterated function system using the set of contraction mappings${\displaystyle \{d_0,d_1\}}$. But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.

Detailed, worked examples of the self-similarities can be found in the articles on theCantor function and onMinkowski's question-mark function. Precisely the samemonoid of self-similarities, thedyadic monoid, apply toevery de Rham curve.

The following systems generate continuous curves.

Cesàro curves, also known asCesàro–Faber curves orLévy C curves, are De Rham curves generated byaffine transformations conservingorientation, with fixed points${\displaystyle p_0=0}$ and${\displaystyle p_1=1}$.

Because of these constraints, Cesàro curves are uniquely determined by acomplex number${\displaystyle a}$ such that and.

The contraction mappings${\displaystyle d_0}$ and${\displaystyle d_1}$ are then defined as complex functions in thecomplex plane by:

${\displaystyle d_0(z)=az}$

${\displaystyle d_1(z)=a+(1-a)z.}$

For the value of${\displaystyle a=(1+i)/2}$, the resulting curve is theLévy C curve.

In a similar way, we can define the Koch–Peano family of curves as the set of De Rham curves generated by affine transformations reversing orientation, with fixed points${\displaystyle p_0=0}$ and${\displaystyle p_1=1}$.

These mappings are expressed in the complex plane as a function of${\displaystyle \overline{z}}$, thecomplex conjugate of${\displaystyle z}$:

${\displaystyle d_0(z)=a\overline{z}}$

${\displaystyle d_1(z)=a+(1-a)\overline{z}.}$

The name of the family comes from its two most famous members. TheKoch curve is obtained by setting:

${\displaystyle a_{\text{Koch}}=\frac{1}{2}+i\frac{\sqrt{3}}{6},}$

while thePeano curve corresponds to:

${\displaystyle a_{\text{Peano}}=\frac{(1+i)}{2}.}$

The de Rham curve for${\displaystyle a=(1+ib)/2}$ for values of${\displaystyle b}$ just less than one visually resembles theOsgood curve. These two curves are closely related, but are not the same. The Osgood curve is obtained by repeated set subtraction, and thus is aperfect set, much like theCantor set itself. The construction of the Osgood set asks that progressively smaller triangles to be subtracted, leaving behind a "fat" set of non-zero measure; the construction is analogous to thefat Cantor set, which has a non-zeromeasure. By contrast, the de Rham curve is not "fat"; the construction does not offer a way to "fatten up" the "line segments" that run "in between" the dyadic rationals.

The Cesàro–Faber and Peano–Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane. By fixing one endpoint of the curve at 0 and the other at 1, the general case is obtained by iterating on the two transforms

and

Beingaffine transforms, these transforms act on a point${\displaystyle (u,v)}$ of the 2-D plane by acting on the vector

${\displaystyle \left(\begin{array}{c}1\\ u\\ v\end{array}\right).}$

The midpoint of the curve can be seen to be located at${\displaystyle (u,v)=(\alpha ,\beta )}$; the other four parameters may be varied to create a large variety of curves.

Theblancmange curve of parameter${\displaystyle w}$ can be obtained by setting${\displaystyle \alpha =\beta =1/2}$,${\displaystyle \delta =\zeta =0}$ and${\displaystyle \epsilon =\eta =w}$. That is:

and

Since the blancmange curve for parameter${\displaystyle w=1/4}$ is a parabola of the equation${\displaystyle f(x)=4x(1-x)}$, this illustrates the fact that on some occasions, de Rham curves can be smooth.

Minkowski's question mark function is generated by the pair of maps

${\displaystyle d_0(z)=\frac{z}{z+1}}$

and

${\displaystyle d_1(z)=\frac{1}{2-z}.}$

Given any two functions${\displaystyle d_0}$ and${\displaystyle d_1}$, one can define a mapping fromCantor space, by repeated iteration of the digits, exactly the same way as for the de Rham curves. In general, the result will not be a de Rham curve, when the terms of the continuity condition are not met. Thus, there are many sets that might be in one-to-one correspondence with Cantor space, whose points can be uniquely labelled by points in the Cantor space; however, these are not de Rham curves, when the dyadic rationals do not map to the same point.

TheMandelbrot set is generated by aperiod-doubling iterated equation${\displaystyle z_{n+1}=z_n^2+c.}$ The correspondingJulia set is obtained by iterating the opposite direction. This is done by writing${\displaystyle z_n=\pm \sqrt{z_{n+1}-c}}$, which gives two distinct roots that the forward iterate${\displaystyle z_{n+1}}$ "came from". These two roots can be distinguished as

${\displaystyle d_0(z)=+\sqrt{z-c}}$

and

${\displaystyle d_1(z)=-\sqrt{z-c}.}$

Fixing the complex number${\displaystyle c}$, the result is the Julia set for that value of${\displaystyle c}$. This curve is continuous when${\displaystyle c}$ is inside the Mandelbrot set; otherwise, it is a disconnected dust of points. However, the reason for continuity is not due to the de Rham condition, as, in general, the points corresponding to the dyadic rationals are far away from one-another. In fact, this property can be used to define a notion of "polar opposites", of conjugate points in the Julia set.

It is easy to generalize the definition by using more than two contraction mappings. If one usesn mappings, then then-ary decomposition ofx has to be used instead of thebinary expansion of real numbers. The continuity condition has to be generalized in:

${\displaystyle d_i(p_{n-1})=d_{i+1}(p_0)}$, for${\displaystyle i=0\dots n-2.}$

This continuity condition can be understood with the following example. Suppose one is working in base-10. Then one has (famously) that0.999...= 1.000... which is a continuity equation that must be enforced at every such gap. That is, given the decimal digits${\displaystyle b_1,b_2,\cdots ,b_k}$ with${\displaystyle b_k\ne 9}$, one has

${\displaystyle b_1,b_2,\cdots ,b_k,9,9,9,\cdots =b_1,b_2,\cdots ,b_k+1,0,0,0,\cdots }$

Such a generalization allows, for example, to produce theSierpiński arrowhead curve (whose image is theSierpiński triangle), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.

Ornstein and others describe amultifractal system, where instead of working in a fixed base, one works in a variable base.

Consider theproduct space of variable base-${\displaystyle m_n}$discrete spaces

${\displaystyle \mathrm{\Omega }=\prod _{n\in \mathbb{N}}{A}_n}$

for${\displaystyle A_n=\mathbb{Z}/m_n\mathbb{Z}=\{0,1,\cdots ,m_n-1\}}$ thecyclic group, for${\displaystyle m_n\ge 2}$ an integer. Any real number in theunit interval can be expanded in a sequence${\displaystyle (a_1,a_2,a_3,\cdots )}$ such that each${\displaystyle a_n\in A_n}$. More precisely, a real number${\displaystyle 0\le x\le 1}$ is written as

${\displaystyle x=\sum _{n=1}^{\mathrm{\infty }}\frac{a_n}{\prod _{k=1}^n{m}_k}}$

This expansion is not unique, if all${\displaystyle a_n=0}$ past some point. In this case, one has that

${\displaystyle a_1,a_2,\cdots ,a_K,0,0,\cdots =a_1,a_2,\cdots ,a_K-1,m_{K+1}-1,m_{K+2}-1,\cdots }$

Such points are analogous to the dyadic rationals in the dyadic expansion, and the continuity equations on the curve must be applied at these points.

For each${\displaystyle A_n}$, one must specify two things: a set of two points${\displaystyle p_0^{(n)}}$ and${\displaystyle p_1^{(n)}}$ and a set of${\displaystyle m_n}$ functions${\displaystyle d_j^{(n)}(z)}$ (with${\displaystyle j\in A_n}$). The continuity condition is then as above,

${\displaystyle d_j^{(n)}(p_1^{(n+1)})=d_{j+1}^{(n)}(p_0^{(n+1)})}$, for${\displaystyle j=0,\cdots ,m_n-2.}$

Ornstein's original example used

${\displaystyle \mathrm{\Omega }=\left(\mathbb{Z}/2\mathbb{Z}\right)\times \left(\mathbb{Z}/3\mathbb{Z}\right)\times \left(\mathbb{Z}/4\mathbb{Z}\right)\times \cdots }$

• Iterated function system
• Refinable function
• Modular group
• Fuchsian group

• Georges de Rham,On Some Curves Defined by Functional Equations (1957), reprinted inClassics on Fractals, ed. Gerald A. Edgar (Addison-Wesley, 1993), pp. 285–298.
• Linas Vepstas,A Gallery of de Rham curves, (2006).
• Linas Vepstas,Symmetries of Period-Doubling Maps, (2006).(A general exploration of the modular group symmetry in fractal curves.)

• De Rham curves

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