De Rham curve

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In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.

The Cantor function, Cesaro curve, 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.

Contents 1 Construction 2 Properties 3 Classification and examples 3.1 Césaro curves 3.2 Koch-Peano curves 3.3 General affine maps 3.4 Minkowski's question mark function 4 See also 5 References

[edit] Construction

Consider some metric space (M,d) (generally ² with the usual euclidean distance), and a pair of contracting maps on M:

By the Banach fixed point theorem, these have fixed points p₀ and p₁ respectively. Let x be a real number in the interval [0,1], having binary expansion

where each b_k is 0 or 1. Consider the map

defined by

where denotes function composition. It can be shown that each c_x will map the common basin of attraction of d₀ and d₁ to a single point p_x in M. 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

d₀(p₁) = d₁(p₀)

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.

[edit] Classification and examples

[edit] Césaro curves

Césaro curves (or Césaro-Faber curves) are De Rham curves generated by affine transformations conserving orientation, with fixed points p₀ = 0 and p₁ = 1.

Because of these constraints, Césaro curves are uniquely determined by a complex number a such that | a | < 1 and | 1 − a | < 1.

The contraction mappings d₀ and d₁ are then defined as complex functions in the complex plane by:

d₀(z) = az

d₁(z) = a + (1 − a)z

For the value of a = (1 + i) / 2, the resulting curve is the Lévy C curve.

[edit] Koch-Peano curves

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 and p₁ = 1.

These mappings are expressed in the complex plane as a function of , the complex conjugate of z:

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

while the Peano curve corresponds to:

[edit] 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

and

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

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.

The blancmange curve of parameter w can be obtained by setting α = β = ε = 1 / 2, δ = ζ = 0 and η = w. That is:

and

Since the blancmange curve of parameter w = 1 / 4 is the parabola of equation f(x) = 4x(1 − x), this illustrate the fact that in some occasion, de Rham curves can be smooth.

[edit] Minkowski's question mark function

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

and

[edit] See also

• Refinable function
• Modular group
• Fuchsian group

[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), 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.)