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A simple mathematical curiosity?

In our daily routine we have many simple problems that we solve unconsciously by using infinite "formulas". Take as example a storage room: experience tells us that the best way to store objects is putting them in boxes- that way we can sort the contents, pile them up so we can make the most of the space left. The secret resides in the box, with its geometric shape, which lets to create an apparent order within the chaos that we are going to hide inside it.

The obsession for using geometric shapes in order to organise everything comes from a very long time ago:


Cavemen discovering regular shapesOur ancestors lived in natural surroundings where regular forms were rare.

They had their senses adapted to survive in a chaotic world full of dangers, and it is very probable that they would be astonished when finding objects with those features.

Regular shapes were so rare in nature that they became something magical.

Cavemen creating objects with regular shapesSoon they found out the utility of those pure shapes and they wanted to make the most of them.

They created several tools adapted to their needs, i.e. the wheel, one of the most memorable.

Nowdays human being perfecting the use of regular shapesThat way we started to transform our surroundings and we kept doing so for thousands of years till nowadays, ending with the invention of formulas that let us manipulate the regular forms in a more efficient way now. This is what we call classic geometry or Euclidean.

The transformation was so big that everything that couldn't be explained with the Euclidean geometry was considered chaotic and lacked of sense.

However, another kind of geometry exists, one that was always with us, but which never was understood nor acknowledged how to manipulate till a very short time ago, when a new branch of maths came up. It is the geometry of irregular objects and was called Fractal geometry. This new idea has been exploded in many ways within the science where has been used to bring light upon complex problems not really understood till now. In order to understand this new concept we have to go back to nature...

Natural elements are modelled by a series of chaotic phenomenons, that build irregular structures as mountains, clouds, ... Lets focus on the sea and imagine its horizon. If we were asked to draw the landscape we would most likely use a straight line in order to represent the line where water and sky meet. As we can see, we would be using an element form the Euclidean geometry, which would be hiding a chaotic element that we are representing, as well as in the storage room.

Horizon line

If we use a telescope in order to observe the horizon we would see that the line drawn is not that straight as it seems, but it is delimited by several waves.

waves in the horizon

If we fix our vision in each wave we would see each one is created by infinite smaller waves, and so on. This phenomena is called selfsimilarity at different scales and it reflects the capacity of an object to be similar to a portion of itself.

Little waves in the horizon

Each sublevel of similarity is called iteration (or repetition) and as result of this structure the horizon line is longer than the straight line. The more iterations taken into account for the measurement, the more its length tends to infinite within the topologic space it occupies.

COmparison of horizons

Another feature of this type of structures is that they don't have a whole topologic dimension*, but a fractal dimension (or fractionate).

*Let's remember the concept of Euclids' topological dimension, after which the point has zero dimensions, the line has one (length), the surface has two (length and width) and the volume has three (length, width and height).

Now, ignoring its width, imagine an A-4 paper is a two-dimensional object that represents the surface. If we crumple little by little it starts to show up a relieve that comes through the third dimension and makes the object to have volume. If we continue with the process the paper would turn into a ball, with is similar to a 3D object, but its surface is not smooth and the wrinkles don't let the object to complete the third dimension. In this case we would have an object between the 2D and the 3D.

This s why the fractal dimension is the topological dimension plus a dimensional factor that indicates the capacity of occupy more dimensions than the indicated from its own topological dimension.

Coming back to the topic at hand, the horizon is represented with a line, and the Euclidean lines have a dimension equal to one, but the reality is that the horizon line tends to fill up a very small surface. Getting a clear vision of this may be difficult, so let's see an example taken to the extreme:

Hilbert curve

This is the Hilbert curve, a simple line that follows a pattern which is capable to fill up a plane. As we can see, the more iterations in the same space, the more filled up is that same space. The line fills up the whole two dimensions turning into a surface.

With this simple analysis we have verified that the horizon line from the sea is a complex structure based on simple repetitive patterns, in this case it is the wave. AS we see, what seems chaotic hides a specific order, and with this the fractal geometry has become a good approximation to the real structure of many shapes in nature. This branch of maths can be applied to several objects and behaviours as the ramifications of a lightning, the movement of an avalanche though the rocks, the distribution of the stars and galaxies in the Universe, the fluctuations of the stock market, a populations' growth dynamic,...

As conclusions we will highlight the main features of a fractal structure:

  • Its length tends to infinite within the topological space it occupies.
  • It's generated through simple repetitive patterns.
  • It is selfsimilar at different scales.
  • It has a fractal dimension or fractional.

Caveman filed

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What to see now?
If you found interesting this article you may fancy reading the introduction to chaos, or you can see more fractal natural examples in the gallery fractals in nature.
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« La ribera izquierda del río Vístula, medida con precisión creciente, daría resultados diez, cien, y hasta mil veces mayores que la longitud que aparece en los mapas »
Matemático y divulgador Hugo Steinhaus
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