De Rham curve

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In mathematics, a de Rham curve is a certain type of fractal curve. The Cantor function, Minkowski's question mark function, the Lévy C curve, the blancmange curve and the Koch curve are all special cases of the general de Rham curve. The curve is named in honor of Georges de Rham.

1 Construction
2 Properties
3 Example - Césaro curve
4 Example - Koch curve
5 Example:General affine maps
6 Example: Minkowski's question mark function
7 See also
8 References

[edit] Construction

Consider a pair of contracting maps

$d_0:\mathbb{R}^2 \to \mathbb{R}^2$

and

$d_1:\mathbb{R}^2 \to \mathbb{R}^2$

By the Banach fixed point theorem, these have fixed points $p 0$ and $p 1$ respectively. Let x be a real number in the interval $x \in [0,1]$ , having binary (2-adic) expansion

$x = \sum_{k=1}^\infty b_k 2^{-k}$

Here, each $b k$ is understood to be an integer, 0 or 1. Consider the map

$c_x:\mathbb{R}^2 \to \mathbb{R}^2$

given by

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

where $\circ$ denotes function composition. It can be shown that each $c x$ will map the common basin of attraction of $d 0$ and $d 1$ to a single point $p_x\in \mathbb{R}^2$ . The collection of points $p x$ , parameterized by a single real parameter x, is known as the de Rham curve.

[edit] Properties

When the fixed points are paired such that

d 0 (p 1) = d 1 (p 0)

then it may be shown that the resulting curve $p x$ is a continuous function of x. When the curve is continuous, it is not in general differentiable. 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 set. This so-called period-doubling moniod is a subset of the modular group.

[edit] Example - Césaro curve

Cesaro curve for a=0.3+i0.3

Cesaro curve for a=0.5+i0.5

Let $z=u+iv\in \mathbb{C}$ and let $a\in\mathbb{C}$ be a constant such that $| a | < 1$ and $| 1 - a | < 1$ . Consider then the maps

d 0 (z) = a z

and

d 1 (z) = a + (1 - a) z

For the value of $a = (1 + i) / 2$ , the resulting curve is the Lévy C curve. For general values of a, the curve is often known as the Césaro curve or the Césaro-Faber curve.

[edit] Example - Koch curve

Koch-Peano curve for a=0.6+i0.37

Koch-Peano curve for a=0.6+i0.45

The Koch curve and the Peano curve may be obtained by

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

and

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

where $\overline{z}$ denotes the complex conjugate of $z$ . The classic Koch curve is obtained by setting

$a=\frac{1}{2} + i\frac{\sqrt{3}}{6}$

while the Peano curve corresponds to $a = (1 + i) / 2$

[edit] Example:General affine maps

Generic affine de Rham curve

The Cesaro-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 one, the general case is obtained by iterating on the two transforms

$d_0=\left( \begin{matrix} 1&0&0 \\ 0 & \alpha&\delta \\ 0&\beta&\epsilon \end{matrix}\right)$

and

$d_1=\left( \begin{matrix} 1&0&0 \\ \alpha & 1-\alpha&\zeta \\ \beta&-\beta&\eta \end{matrix}\right)$

Being affine transforms, these transforms act on a point $(u, v)$ of the 2-D plane by acting on the vector

$\left( \begin{matrix} 1 \\ u \\ v \end{matrix}\right)$

The midpoint of the curve can be seen to be located at $(u, v) = (α,β)$ ; the other four parameters may be varied to create a large variety of curves.

[edit] Example: Minkowski's question mark function

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

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

and

$d_1(z)= \frac{1}{z+1}$

[edit] See also

[edit] References

Georges de Rham, On Some Curves Defined by Functional Equations (1957), reprinted in Classics on Fractals, ed. Gerald A. Edgar, (Addison-Wesley, 1993) p285-298
Linas Vepstas, Symmetries of Period-Doubling Maps, (2004). (A general exploration of the modular group symmetry in fractal curves).

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Category: Fractal curves