In mathematics, a de Rham curve is a continuous fractal curve obtained as the image of the Cantor space, or, equivalently, from the base-two expansion of the real numbers in the unit interval. Many well-known fractal curves, including the Cantor function, Cesàro–Faber curve (Lévy C curve), Minkowski's question mark function, blancmange curve, and the Koch curve are all examples of de Rham curves. The general form of the curve was first described by Georges de Rham in 1957.[1]
(M,d)
R
d0: M\toM
d1: M\toM.
By the Banach fixed-point theorem, these have fixed points
p0
p1
[0,1]
x=
infty | |
\sum | |
k=1 |
bk | |
2k |
,
where each
bk
cx: M\toM
defined by
cx=
d | |
b1 |
\circ
d | |
b2 |
\circ … \circ
d | |
bk |
\circ … ,
where
\circ
cx
d0
d1
px
M
px
The construction in terms of binary digits can be understood in two distinct ways. One way is as a mapping of Cantor space to distinct points in the plane. Cantor space is the set of all infinitely-long strings of binary digits. It is a discrete space, and is disconnected. 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, the dyadic rationals have two distinct representations as strings of binary digits. For example, the real number one-half has two equivalent binary expansions:
h1=0.1000 …
h0=0.01111 …
h0
h1
The same notion of continuity is applied to the de Rham curve by asking that the fixed points be paired, so that
d0(p1)=d1(p0)
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 point p in the plane, one has two distinct sequences:
d0\circd1\circd1\circd1\circ … (p)
and
d1\circd0\circd0\circd0\circ … (p)
corresponding to the two binary expansions
1/2=0.01111 …
1/2=0.1000 …
d0(p1)
d1(p0)
px
In general, the de Rham curves are not differentiable.
De Rham curves are by construction self-similar, since
p(x)=d0(p(2x))
x\in[0,1/2]
p(x)=d1(p(2x-1))
x\in[1/2,1].
The self-symmetries of all of the de Rham curves are given by the monoid that describes the symmetries of the infinite binary tree or Cantor space. This so-called period-doubling monoid is a subset of the modular group.
The image of the curve, i.e. the set of points
\{p(x),x\in[0,1]\}
\{d0, d1\}
Detailed, worked examples of the self-similarities can be found in the articles on the Cantor function and on Minkowski's question-mark function. Precisely the same monoid of self-similarities, the dyadic monoid, apply to every de Rham curve.
The following systems generate continuous curves.
Cesàro curves, also known as Cesàro–Faber curves or Lévy C curves, are De Rham curves generated by affine transformations conserving orientation, with fixed points
p0=0
p1=1
a
|a|<1
|1-a|<1
The contraction mappings
d0
d1
d0(z)=az
d1(z)=a+(1-a)z.
For the value of
a=(1+i)/2
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
p0=0
p1=1
These mappings are expressed in the complex plane as a function of
\overline{z}
z
d0(z)=a\overline{z}
d1(z)=a+(1-a)\overline{z}.
The name of the family comes from its two most famous members. The Koch curve is obtained by setting:
a | ||||
|
+i
\sqrt{3 | |
while the Peano curve corresponds to:
a | ||||
|
.
The de Rham curve for
a=(1+ib)/2
b
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
d0=\begin{pmatrix} 1&0&0\ 0&\alpha&\delta\ 0&\beta&\varepsilon \end{pmatrix}
and
d1=\begin{pmatrix} 1&0&0\ \alpha&1-\alpha&\zeta\ \beta&-\beta&η \end{pmatrix}.
Being affine transforms, these transforms act on a point
(u,v)
\begin{pmatrix} 1\\ u\\ v\end{pmatrix}.
The midpoint of the curve can be seen to be located at
(u,v)=(\alpha,\beta)
The blancmange curve of parameter
w
\alpha=\beta=1/2
\delta=\zeta=0
\varepsilon=η=w
d0=\begin{pmatrix} 1&0&0\ 0&1/2&0\ 0&1/2&w \end{pmatrix}
and
d1=\begin{pmatrix} 1&0&0\ 1/2&1/2&0\ 1/2&-1/2&w \end{pmatrix}.
Since the blancmange curve for parameter
w=1/4
f(x)=4x(1-x)
Minkowski's question mark function is generated by the pair of maps
d0(z)=
z | |
z+1 |
and
d1(z)=
1 | |
2-z |
.
Given any two functions
d0
d1
The Mandelbrot set is generated by a period-doubling iterated equation
zn+1
2+c. | |
=z | |
n |
zn=\pm\sqrt{zn+1-c}
zn+1
d0(z)=+\sqrt{z-c}
and
d1(z)=-\sqrt{z-c}.
Fixing the complex number
c
c
c
It is easy to generalize the definition by using more than two contraction mappings. If one uses n mappings, then the n-ary decomposition of x has to be used instead of the binary expansion of real numbers. The continuity condition has to be generalized in:
di(pn-1)=di+1(p0)
i=0\ldotsn-2.
This continuity condition can be understood with the following example. Suppose one is working in base-10. Then one has (famously) that 0.999...= 1.000... which is a continuity equation that must be enforced at every such gap. That is, given the decimal digits
b1,b2, … ,bk
bk\ne9
b1,b2, … ,bk,9,9,9, … =b1,b2, … ,bk+1,0,0,0, …
Such a generalization allows, for example, to produce the Sierpiński arrowhead curve (whose image is the Sierpiński triangle), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.
Ornstein and others describe a multifractal system, where instead of working in a fixed base, one works in a variable base.
Consider the product space of variable base-
mn
\Omega=\prodn\inNAn
An=Z/mnZ=\{0,1, … ,mn-1\}
mn\ge2
(a1,a2,a3, … )
an\inAn
0\lex\le1
infty | |
x=\sum | |
n=1 |
an | |||||||||
|
an=0
K<n
a1,a2, … ,aK,0,0, … =a1,a2, … ,aK-1,mK+1-1,mK+2-1, …
For each
An
(n) | |
p | |
0 |
(n) | |
p | |
1 |
mn
(n) | |
d | |
j |
(z)
j\inAn
(n) | |
d | |
j |
(n+1) | |
(p | |
1)=d |
(n) | |
j+1 |
(n+1) | |
(p | |
0) |
j=0, … ,mn-2.
Ornstein's original example used
\Omega=\left(Z/2Z\right) x \left(Z/3Z\right) x \left(Z/4Z\right) x …