THE HIDDEN BEAUTY OF MATHEMATICS

An object is symmetrical when there is "harmony in the proportions" of its parts in relation to the whole: height, width and length are balanced. Strictly associated with harmony and beauty, symmetry is also a decisive concept in theories about nature. Ancient Greece was apparently the first place where this idea had room to develop.

Some say that the size and proportion of perfect solids described by Plato are related to each other - the sides of particles of fire, water, and air could be combined together because they are proportional. They were described as having a "golden proportion" among themselves - or a type of symmetry that marks the growth rate in the development of several species. The leaves of a tree, for example, multiply more or less at this speed after they sprout: 2, 3, 5, 8, 13, 21, 34, 55 ... and so on. The last number is always the sum of the two preceding numbers - and when we divide each number by its predecessor, the result will be very close to 1.6180, or what mathematicians like the Italian Leonardo Fibonacci (1175 - 1250) consider as "the golden proportion”. When applied to a succession of proportional squares within a rectangle, this sequence of numbers - always with the golden ratio between them - generates a "golden rectangle”. If we draw a line, formed by quarters of a circle, following the progression of the figures formed in it, we have the "golden spiral", as you can see in the drawing on the side.

We can see several other forms of symmetry in nature. There is a form of bilateral symmetry, like the reflection of an image in a lake that can be divided into two identical parts; and it can also be radial when the image forms around a central point and "radiates" to all sides, such as an open flower or a yellow dandelion. Symmetry also manifests itself in complex forms such as fractals, in which a structure looks similar to the whole on any scale. Also, in the case of sounds and waves of the same frequency, we can say with certainty that sounds and lights are also symmetrical. In the natural world, symmetries are not completely perfect and harbor some visible imperfections.

One of the main symmetries in nature is bilateral. We see how one side of the body of a plant or animal is a very close copy of the other, as if it were a plane, able to split the image into two sides - or two almost perfectly reflected images. Not infrequently, this morphology has a clear function: for example, it would be very difficult for a bird to fly straight if its wings weren't the same size.

A body is radially symmetrical if you can cut it several times and generate equal pieces. Or if it is possible to "rotate" it around a central axis and get a circle effect. The main difference compared to spherical shapes is that in the case of spheres there is no "up" or "down" side, as in a more or less flat plane. In radial forms, these sides exist.

There are shapes or species that combine more than one type of symmetry. Bi-radial species, for example, combine radial and bilateral symmetries. These are not very common in nature, and perhaps one of the best representatives of this type of format are the comb jellies. Resembling jellyfish, these marine animals have symmetrical opposite sides, but each side is different from its adjacent one. What does that mean? If it were a geometric figure, a comb jelly could easily be represented by a rectangle: the top and bottom sides are the same. However, these differ from the right and left sides (which are also the same). If all sides were exactly the same, the figure would no longer be a rectangle, but a square.

“Fractal", a term coined by the French mathematician Benoît Mandelbrot in the mid-1970s, comes from the Latin word "fractus", or "broken”. This explains the logic of a fractal's geometry: it is a structure with a symmetrical scale. Any part of a fractal, no matter how small, has the same shape as the whole figure. A good example is the cube you see, better known as Menger's Sponge. The figure is named in honor of the Austrian mathematician Karl Menger, who in the last century studied the topology of geometric objects.

Not all the symmetries we know happen in the spatial dimension, in the form of geometric figures or in forms found in nature. Symmetries also exist in the natural world in other ways that we can see, hear, and feel. Light and sound, for example, behave as a wave - and we can say that these are symmetrical when their wavelength is regular. Its symmetry does not occur in space the way a geometrical figure visibly does- its pulsation, light and sound are symmetrical in time. Some stars, for example, have regular variations in brightness, or pulsations. RS Puppis, located near the center of our Milky Way, is one of these: its frequency of pulsation is approximately 40 days.