Filling the curves (or how to make a one-time coloring with pencil)

My daughter does not like when the colored pencils are cut too thin . Yeah what, after it takes longer to color ! I'm explaining that thanks to Filling curves , one can always just coloring even with an infinitely thin pencil, I feel that the argument does not pass. And yet, we'll see in this post that we can actually find curves that completely fill a surface through all points . And even if it goes against intuition !


Why this seems impossible courbe ?
Alors square and let's go , let's take up the challenge : finding a curve that completely crayon square. It sounds difficult , because everyone knows that in mathematics , the lines are not thick. Under these conditions , we have a mind to think that the surface covered by a curve is always zero . It is not ready to come to cover an entire square. Moreover, a curve is a dimension object 1, while a square is size 2. We see that it can not work ! Another way to tell is to count the number of points on a segment and a square . While the segment , there is an infinity . But in the square , we feel that there is an infinity of times more points in a segment ! And yet…

The Peano fractal curves 
 The construction exhibited by Cantor definitely shows that a square is not "bigger" than a segment. Yet as such, this does not constitute proof of the fact that you can color a square with a curve. What he lacks is continuity! Indeed the correspondence established by Cantor is not continuous as would a true curve. With the application of Cantor, two points in very close segment will be sent on very different square areas. Cantor's correspondence can not therefore draw without lifting your hand!
 However, since Cantor brought down the main psychological obstacle to the discovery of a curve filling the square, the solution will come a few years later in the writings of the Italian Peano. The latter indeed proposes to construct a curve which fills a square, and in stages. The drawing below shows the first three steps of the process

By continuing the process "to infinity", a curve that goes through all the square of dots is obtained! This is called a filling curve. Some of you may be here will recognize the principle of fractals: one starts with a pattern, then repeat on the lower level, then again and so on. (For those that this way of constructing curves bothers, I will return to my "To go further ...") Following the discovery of the Peano curve, many other mathematicians propose Filling curves based on the principle of fractals. Hilbert curve is one of the best known and the Lebesgue curve shown above-cons, whose principle is to construct a fractal from a pattern Z. Ah small detail for those who like math, these curves are of a species entirely made exotic; they are continuous everywhere but differentiable nowhere! Basically any point of the curve is an amazing way ... right? In short, if you want to color a square with an infinitely thin pencil possible: choose the path of one of
these curves Filling. Once again you find in mathematics, it does not always have to rely on intuition, especially when it comes to infinity! However Filling the curves are not just entertainment for amateur mathematicians of paradoxes, it is also a useful tool in areas such as data reduction. With the trajectory of a filling curve, you can actually browse effectively a multidimensional space.