Talk:De Rham curve

Thanks
thankyou for making this page, whoever did. The trouble I am having with it is those unaware of degree level maths facts and jargon have to spend ages looking up other pages. I often have this problem with maths pages of Wikipedia. Can someone please help, because fractals are pretty and I would like to understand them. Edit this page please. —Preceding unsigned comment added by 81.99.106.40 (talk • contribs) 11:24, 8 December 2007‎

Needs minor rewriting
This introductory paragraph:

"Let x be a real number in the interval [0,1], having binary expansion x = Sum from 1 to ∞ of b_k * 2^(-k) Here, each b_k is understood to be an integer, 0 or 1. Consider the map given by [blah blah blah] where o denotes function composition. It can be shown that [blah blah blah]. The collection of points p_x, parameterized by a single real parameter x, is known as the de Rham curve."

Here, first defining the number x rather than first referring to the sequence b_1, b_2,. . . of 0's and 1's is putting the cart before the horse.

Instead, the article should read something like "Given a sequence {b_k}, k = 1,2,3,... of 0's and 1's, ... etc.

Only then should the number x be defined and described as a way of parametrizing the collection of points, which can now be called {p_x}.

And by the way, as is well known, any number whose binary expansion ends in a sequence of all 0's can be re-expressed in a binary expression ending in a sequence of all 1's, and vice versa. SO: Do both sequencs of binary coefficients give the identical p_x ? Whether the answer is Yes or No, this issue needs to be addressed.Daqu (talk) 02:32, 23 August 2008 (UTC)


 * I think you're right about the order, to be rigorous we should:
 * introduce $$d_0$$, $$d_1$$ and their fixed points $$p_0$$, $$p_1$$,
 * introduce the sequence $$b_k$$,
 * introduce the contraction mapping $$c_{(b_k)} = d_{b_1} \circ d_{b_2} \circ \cdots \circ d_{b_k} \circ \cdots$$, maybe saying a word about this construction, because the composition of an infinite sequence of functions is not something natural (and the subject is not even mentioned in the linked page about function composition),
 * show that the contraction mapping $$c_{(b_k)}$$ has an unique fixed point $$p_{(b_k)}$$,
 * define the number $$x_{(b_k)} = \sum_{k=1}^\infty b_k 2^{-k}$$, and tell something about the two possible sequences for dyadic numbers (and link to the appropriate page).
 * Introduce the condition $$d_0(p_1)=d_1(p_0)$$ and use it to show that we can now map $$x_{(b_k)}$$ directly to $$p_{(b_k)}$$, which defines a map from $$[0,1]$$ to the plane,
 * Show that this map is continuous (it can stay in the properties section).
 * But this would be long and technical. A more intuitive solution could be to:
 * introduce $$d_0$$, $$d_1$$ and their fixed points $$p_0$$, $$p_1$$,
 * introduce the condition $$d_0(p_1)=d_1(p_0)$$,
 * define recursively for each dyadic rational ($$q=k/2^n$$, 0≤q≤1), a point $$p_q$$ by: $$p_q=d_0(p_{2q})$$ if q≤1/2 and $$p_q=d_1(p_{2q-1})$$ if q≥1/2, which is well defined because $$p_{1/2}=d_0(p_1)=d_1(p_0)$$.
 * For every number $$n$$, consider the broken line $$L_n$$ formed by joining all the points $$p_{k/2^n}$$, k=0,1,…,2^n, in this order (This would benefit from a nice illustration!).
 * Prove that the family of broken lines $$L_n$$ converges uniformly (because of contraction mappings) toward a continuous curve (because of uniform convergence), that we will define to be the De Rham curve.
 * Show how the construction relates to iterated function systems.
 * Additionally, I think this article lack of historical background.Clément Pillias (talk) 00:35, 11 November 2008 (UTC)
 * Additionally, I think this article lack of historical background.Clément Pillias (talk) 00:35, 11 November 2008 (UTC)

A mistake in Minkowski's question mark function?
I think the definition of the maps that generate the Minkowski's question mark function are wrong.

With $$S(x)=x/(x+1)$$ and $$R(x)=1-x$$, so $$T(x)=(R\circ S\circ R)(x)=1/(2-x)$$, it is true that every rational whose continued fraction is $$[0;a_1,a_2,\cdots ,a_n]$$ is equal to $$(S^{a_1}\circ R\circ S^{a_2}\circ R\circ\cdots S^{a_n})(1)=(S^{a_1}\circ T^{a_2}\circ S^{a_3}\circ \cdots)(1)$$.

So if we choose $$d_0(x)=S(x)$$ and $$d_1(x)=T(x)$$ (and not $$R(x)$$!), it is true that $$c_x(1)=?(x)$$, but this is not what is requested from a De Rham curve: we need to have contraction maps from $$\mathbb{R}^2 \to \mathbb{R}^2$$, when S and T are maps from $$\mathbb{R} \to \mathbb{R}$$. Moreover, $$c_x(1)=?(x)$$ is not the fixed point $$p_x$$.

The relations that we need are $$?(S(x))=?(x)/2$$ and $$?(R(x))=R(?(x))$$ from which we conclude that $$?(T(x))=(1+?(x))/2$$. In the plane, the point $$(x,y)$$ belongs to the curve if $$y=?(x)$$, so the points belonging to the curve verify that (S(x),y/2) and (T(x),(1+y)/2) are also on the curve. Then the following mappings generate the 2-dimensional curve corresponding to the graph of the Minkowski's question mark function:

$$d_0\left( \begin{matrix} x \\ y\end{matrix}\right)=\left( \begin{matrix} x/(x+1) \\ y/2 \end{matrix}\right)$$

and $$d_1\left( \begin{matrix} x \\ y\end{matrix}\right)=\left( \begin{matrix} 1/(2-x) \\ (1+y)/2 \end{matrix}\right)$$

Clément Pillias (talk) 02:35, 11 November 2008 (UTC)


 * Yes, that looks correct. However, in the article, $$z=x+iy$$ so one has to fiddle about to re-obtain the above relationships. Note that ALL of these curves have the period-doubling symmetry, NOT just the question mark. That is, there are appropriate variants of S and T that describe ALL of these curves! 67.198.37.16 (talk) 04:12, 21 September 2020 (UTC)

Inappropriate Title?
In the literature, the De Rham curve expression most often design "curves [that] are obtained from a polygonal arc by passing to the limit in repeatedly cutting off the corners: at each step, the segments of the arc are divided into three pieces in the ratio $$\omega:(1-2\omega):\omega$$, where $$\omega\in(0,1/2)$$ is a given parameter" (see, e.g. http://www.insa-rennes.fr/~jmerrien/articles/drla3.pdf ). So maybe the `De Rham curve´ page should be dedicated to that curve, and the content of the current page should be moved to `De Rham curve (Fractal)´?

Clément Pillias (talk) 12:40, 24 December 2009 (UTC)


 * But this is the same thing. de Rham himself gives both of these constructions in his original paper (the 1957 paper cited in the article). He spends some time computing the length of the cut-polygonal-arc construction, as that is the construction where it is easy to compute the length. 67.198.37.16 (talk) 04:10, 21 September 2020 (UTC)