DE RHAM CURVE

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.

Contents

Construction

Properties

Example - Césaro curve

Example - Koch curve

Example:General affine maps

Example: Minkowski's question mark function

See also

References

Construction

Consider a pair of contracting maps
:$d\_0:mathbb\{R\}^2\; o\; mathbb\{R\}^2$
and
:$d\_1:mathbb\{R\}^2\; o\; mathbb\{R\}^2$
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 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
:$c\_x:mathbb\{R\}^2\; o\; mathbb\{R\}^2$
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 $p\_xin\; mathbb\{R\}^2$. The collection of points $p\_x$, parameterized by a single real parameter ''x'', is known as the de Rham curve.

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 monoid is a subset of the modular group.

Example - Césaro curve

Let $z=u+ivin\; mathbb\{C\}$ and let $ainmathbb\{C\}$ be a constant such that and . Consider then the maps
:$d\_0(z)\; =\; az$
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'.

Example - Koch curve

The Koch curve and the Peano curve may be obtained by
:$d\_0(z)\; =\; aoverline\{z\}$
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
:
while the Peano curve corresponds to $a=(1+i)/2$

Example:General affine maps

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
:
0 & lpha&delta \
0&eta&epsilon end{matrix}
ight)
and
:
lpha & 1-lpha&zeta \
eta&-eta&eta end{matrix}
ight)
Being affine transforms, these transforms act on a point $(u,v)$ of the 2-D plane by acting on the vector
:
ight)
The midpoint of the curve can be seen to be located at ; the other four parameters may be varied to create a large variety of curves.

Example: Minkowski's question mark function

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

See also

★ Refinable function

★ Modular group

★ Fuchsian group

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)''.