De Rham curve

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.

The general form of the curve was first described by Georges de Rham in 1957.

2 with the usual euclidean distance), and a pair of contracting maps on M: By the Banach fixed-point theorem, these have fixed points

, parameterized by a single real parameter x, 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 of Cantor space to distinct points in the plane.

Cantor space is the set of all infinitely-long strings of binary digits.

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:

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.

For any point p in the plane, one has two distinct sequences: and corresponding to the two binary expansions

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.

De Rham curves are by construction self-similar, since 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.

, can be obtained by an Iterated function system using the set of contraction mappings

Because of these constraints, Cesàro curves are uniquely determined by a complex number

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

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.

of the 2-D plane by acting on the vector 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.

, this illustrates the fact that on some occasions, de Rham curves can be smooth.

, one can define a mapping from Cantor 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.

The Mandelbrot set is generated by a period-doubling iterated equation

The corresponding Julia set is obtained by iterating the opposite direction.

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.

If one uses n mappings, then the n-ary decomposition of x has to be used instead of the binary expansion of real numbers.