Spirals :)

(via proofmathisbeautiful)

Programming fractals ! I’m currently using JAVA and it’s canvas, so the results are not optimal. It’s a very funny exercise figuring out how to program them. Now I’ll try to do more complex figures.

On the pictures: Sierpinski Triangle, Cantor Set and Sierpinski carpet.

dailymajordeegan:

Hundreds of spinning blades reveal the invisible patterns of the wind in American artist Charles Sowers’ kinetic installation on the facade of the Randall Museum in San Francisco.

via dezeen

vector fields in real life :DD

(via oplik)

Let’s play a game, a game called Topology. In this game, we manipulate some objects using certain rules:

- You can shrink and stretch objects.

- You can’t paste different parts of the objects.

- You can’t cut different parts of the objects.

So, using these rules we will take an object and convert it in other (this is why topology is sometimes called ‘glue geometry’). Our first example, shown in picture 1, is the Donut-Mug convertion. None part was cutted/pasted, just stretching and shrinking. We’ll say then that donut and mug are topologically equivalent (or using technical language, homeomorphic). 

Now, your goal is determine if the elastic-man shown in picture 2 can separate his hands using the topology rules, and if that is possible, describe how. In other words, we need to determine if the elastic-man of the picture 2 is topologically equivalent to the free elastic-man of picture 3. Think about it, the solution is below.

Finally, we’ll use topology for a mathematical problem: what’s the difference between a torus (a donut) and a sphere? (We are thinking these objects as surfaces, with an empty inside). The answer is obvious: one has a hole, the other doesn’t. But that’s not a very mathematical answer. Let’s see how topology helps here:

The sphere has a property that torus not: every closed loop on it can be shrunk into a point, using the rules described above*. An example is shown in picture 4. Meanwhile, on the torus (picture 5), the loops a and b can’t be shrunk into a point without cutting the torus. Just loops as c can do it. Objects with this property are called simply-connected. 

The Poincaré’s conjecture, recently solved by Perelman (picture 6), asserts that the only simply-connected objects are sphere-shaped. For his work,  he won a million dollars, prize he rejected.

Now, the answer to the glue-man problem: Answer here.

———

*The formal rules are slightly different.

Everybody has ever seen those artistic images of fractals, full of colors and beauty shapes (pic 1), but, what really is a fractal? What is the difference between a fractal, and normal figures, like triangles, squares and circles? One of the first fundamental concepts is self-similarity. Fractals are characterized for being self-similar figures, this is, some parts of the figure are repeted through the whole figure in different scales. One of the best examples of this, is the Koch Snowflake (pic 2 & 3).

Usually, fractals are constructed as the limit of certain iterative process. Koch Snowflake is defined as the limit of the process described in picture 2, so the fractal is not any step shown in the figure, it is the limit of these steps. The result is seen approximately in picture 3.

Some fractals have some other interesting properties. The Cantor Set, described as the limit of the process shown in picture 4 (in each step, divide in three parts each black bar and remove the central part), has multiple properties, setting it as one of the bests examples in topology and measure theory. One can easily show that this set has null lenght, and it has an un-countable number of points

Finally, one of the most important characteristic of fractals, is the concept of Dimension. Mathematicians, trying to distinguish fractals and other geometric objects, construct some invariants which characterizes them. The usual dimension notion, is a number which distinguishes linear, planar and spatial usual objects, associating to each linear object the number 1, to each planar object number 2, etc. In order to classify fractals, mathematicians extended the dimension to fractals, creating the Hausdorff Fractal Dimension. So, using this, one can classify now objects like fractals. 

Associating a dimension to a fractal is not an easy task. In picture 5 we can see (in spanish) the formal definition of the Hausdorff Fractal Dimention. Yes, it is hard. A hard calculation yields that Cantor Set has a Fractal dimension Log(2)/Log(3) = 0.631… < 1, so, this object is not a line nor a point!

Soon: Fractas in Nature!