...no matter how much we learn, whatever is left, however small it may seem,
is just as infinitely complex, as the whole was to start with.
That, I think, is the secret of the Universe. - Isaac Asimov.


We are biased to look for patterns. All things may be purely accidental, we don't know, but that does not imply that they are disconnected; in connections we find patterns. Consider for example symmetries, geometrical structures, or spirals: these are methods we, as Nature, use to go for higher levels of organization by using simpler structures. Patterns are in the basis of any human activity: agriculture, research, programming, art, philosophy, history, cooking.

We tend to project onto the world around us the models we build in our mind, conferring them existence.
Like that, the patterns we create to understand the universe are not helpful anymore, as they prevent us
from seeing alternative possibilities.

Euclid of Alexandria wrote around 300 BC the 'Elements', one of the most stable and influential works in history, used as a textbook until the late 19th century. In this work, all the principles of what is now called Euclidean geometry were deduced from only five simple axioms: 1) you can connect two points with a straight segment; 2) a segment can be extended in an infinite line; 3) a circle is determined by a center and a radius; 4) all right angles are equal to one another; 5) for every straight line, there is only one other line across any external point that is parallel to it.

We used the geometry derived from these simple axioms to build cathedrals, understand optics, calculate the best way to pack different shapes, and describe the whole universe, among other things. Isn't it remarkable that these principles seem so obvious, despite none of us have ever seen a point? Can you try to define what is an infinite straight line, and tell when was the last time you encountered one?

Only in the 19th century some mathematicians found, by accident, that the denial of the fifth postulate generates many alternative, self-consistent geometries. For example, there is a family of geometries in which there are not parallels: all straight lines extended to the infinity always cross at some point. It is easy to make a triangle that is made of three straight angles. Can you imagine that?

Sure you can. Take the spherical geometry. For a long time we though the Earth was flat because our Euclidean notions were enough to describe local effects, but the shortest distance between Amsterdam and Antananarivo is not a straight line. You have to go along a curve to get from one point to the other.
You may argue that this is because tunnels are expensive, but that does not change the practical conclusion: in fact, we describe our living place, the surface of the Earth, using a spherical geometry better than a flat one.

A more exotic family of geometries is generated if you let any line have infinite parallel lines through an external point: hyperbolic geometries, that we use now for example to understand certain natural patterns.

All these geometries reveal new possibilities of describing the world, and we were blind to them for centuries because we thought one model was enough. The real nature of space is not for mathematical reasoning to settle, but for Physics to measure. (Even in Physics there is not one truth, but at most a few a practical answers to the questions we have right now). Different experiments confirm Einstein's predictions that space is curved, not "flat", so having these alternative geometries to describe it has become very handy.


There are many phenomena around us that are clearly non geometrical. Can we find also patterns in there? Take molds, a type of fungi. Molds look like quite chaotic to us (by chaotic people normally mean "random"), growing without a determined shape. But all living creatures are made of cells that eat, metabolize, reproduce and repair themselves, fight invaders, work in teams to do very complex tasks, and even die at a scheduled moment. Most of these activities are programmed inside the cells themselves, and many cells acting together build up larger structures, like molds or humans.

Although molds grow on dead organic matter everywhere, their presence is only visible when they grow in colonies, in an interconnected network of branched filaments. Also plants are composed of parts that are repeated: think of the branches of a tree or the leaves of a fern. The case of the romanesco is probably the most impressive one. This kind of broccoli is an awesome natural example of a mathematical structure called "fractal": something in which any of the parts looks like the whole.

Fractals were studied systematically only in the 20th century, starting with the works of Mandelbrot.
This mathematician gives name to a set of numbers with remarkable graphical properties. It does not matter how much you zoom in, you always find the same structures with little variations (see http://www.neave.com/fractal/).

Nature is not like that, of course: when you zoom in too much, you hit atoms, and then any pattern ends (where others possibly begin). But inside certain range many natural objects are fractal. Take a picture of some sand dunes without any other reference in it (no plants or animals), and you will find that it is very difficult to estimate their size: they could be kilometers long, or just a few meters, as a small part of a dune looks as a whole dune itself.

Even in chaos we find recursions. 'Chaos', as studied nowadays, is not random but infinitely detailed and self-similar. Fractal patterns arise everywhere (like in some Pollock's paintings, but also in arbitrary dripping), without any particular intention. It is tempting to see a plan behind a pattern, but that could be just our projective mind again, wired through evolution to model our surroundings and give us some certainty in a Universe that just is like it is.

...no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex, as the whole was to start with. That, I think, is the secret of the Universe. - Isaac Asimov.


We are biased to look for patterns. All things may be purely accidental, we don't know, but that does not imply that they are disconnected; in connections we find patterns. Consider for example symmetries, geometrical structures, or spirals: these are methods we, as Nature, use to go for higher levels of organization by using simpler structures. Patterns are in the basis of any human activity: agriculture, research, programming, art, philosophy, history, cooking.

We tend to project onto the world around us the models we build in our mind, conferring them existence. Like that, the patterns we create to understand the universe are not helpful anymore, as they prevent us from seeing alternative possibilities.

Euclid of Alexandria wrote around 300 BC the 'Elements', one of the most stable and influential works in history, used as a textbook until the late 19th century. In this work, all the principles of what is now called Euclidean geometry were deduced from only five simple axioms: 1) you can connect two points with a straight segment; 2) a segment can be extended in an infinite line; 3) a circle is determined by a center and a radius; 4) all right angles are equal to one another; 5) for every straight line, there is only one other line across any external point that is parallel to it.

We used the geometry derived from these simple axioms to build cathedrals, understand optics, calculate the best way to pack different shapes, and describe the whole universe, among other things. Isn't it remarkable that these principles seem so obvious, despite none of us have ever seen a point? Can you try to define what is an infinite straight line, and tell when was the last time you encountered one?

Only in the 19th century some mathematicians found, by accident, that the denial of the fifth postulate generates many alternative, self-consistent geometries. For example, there is a family of geometries in which there are not parallels: all straight lines extended to the infinity always cross at some point. It is easy to make a triangle that is made of three straight angles. Can you imagine that?

Sure you can. Take the spherical geometry. For a long time we though the Earth was flat because our Euclidean notions were enough to describe local effects, but the shortest distance between Amsterdam and Antananarivo is not a straight line. You have to go along a curve to get from one point to the other. You may argue that this is because tunnels are expensive, but that does not change the practical conclusion: in fact, we describe our living place, the surface of the Earth, using a spherical geometry better than a flat one.

A more exotic family of geometries is generated if you let any line have infinite parallel lines through an external point: hyperbolic geometries, that we use now for example to understand certain natural patterns.

All these geometries reveal new possibilities of describing the world, and we were blind to them for centuries because we thought one model was enough. The real nature of space is not for mathematical reasoning to settle, but for Physics to measure. (Even in Physics there is not one truth, but at most a few a practical answers to the questions we have right now). Different experiments confirm Einstein's predictions that space is curved, not "flat", so having these alternative geometries to describe it has become very handy.

There are many phenomena around us that are clearly non geometrical. Can we find also patterns in there? Take molds, a type of fungi. Molds look like quite chaotic to us (by chaotic people normally mean "random"), growing without a determined shape. But all living creatures are made of cells that eat, metabolize, reproduce and repair themselves, fight invaders, work in teams to do very complex tasks, and even die at a scheduled moment. Most of these activities are programmed inside the cells themselves, and many cells acting together build up larger structures, like molds or humans.

Although molds grow on dead organic matter everywhere, their presence is only visible when they grow in colonies, in an interconnected network of branched filaments. Also plants are composed of parts that are repeated: think of the branches of a tree or the leaves of a fern. The case of the romanesco is probably the most impressive one. This kind of broccoli is an awesome natural example of a mathematical structure called "fractal": something in which any of the parts looks like the whole.

Fractals were studied systematically only in the 20th century, starting with the works of Mandelbrot. This mathematician gives name to a set of numbers with remarkable graphical properties. It does not matter how much you zoom in, you always find the same structures with little variations (see http://www.neave.com/fractal/).

Nature is not like that, of course: when you zoom in too much, you hit atoms, and then any pattern ends (where others possibly begin). But inside certain range many natural objects are fractal. Take a picture of some sand dunes without any other reference in it (no plants or animals), and you will find that it is very difficult to estimate their size: they could be kilometers long, or just a few meters, as a small part of a dune looks as a whole dune itself.

Even in chaos we find recursions. 'Chaos', as studied nowadays, is not random but infinitely detailed and self-similar. Fractal patterns arise everywhere (like in some Pollock's paintings, but also in arbitrary dripping), without any particular intention. It is tempting to see a plan behind a pattern, but that could be just our projective mind again, wired through evolution to model our surroundings and give us some certainty in a Universe that just is like it is.