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Concept
De Rham curve
Summary
In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all special cases of the general de Rham curve.
Consider some complete metric space (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 and respectively. Let x be a real number in the interval , having binary expansion
where each is 0 or 1. Consider the map
defined by
where denotes function composition. It can be shown that each will map the common basin of attraction of and to a single point in . The collection of points , parameterized by a single real parameter x, is known as the de Rham curve.
When the fixed points are paired such that
then it may be shown that the resulting curve is a continuous function of x. When the curve is continuous, it is not in general differentiable.
In the remaining of this page, we will assume the curves are continuous.
De Rham curves are by construction self-similar, since
for and
for
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.
The of the curve, i.e. the set of points , can be obtained by an Iterated function system using the set of contraction mappings . But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.
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.
Cesàro curves (or Cesàro–Faber curves) are De Rham curves generated by affine transformations conserving orientation, with fixed points and .
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