THE HIDDEN BEAUTY OF MATHEMATICS

Museu do Amanhã

Mathematics is everywhere. It is in the objects we create, in the works of art we admire. Although we may not notice it, mathematics is also present in the nature that surrounds us, in its landscapes and species of plants and animals, including the human species. Our attraction to other humans and even our mobility depend on it. But how does this happen?

From the structure of buildings to the discovery of new planets, from trade to fashion and new technologies, mathematics has always served as an important tool in the advancement of science and technology, in fields as diverse as Engineering, Biology, Philosophy and Arts. And it is also present in nature, concealing- and revealing- its charms in various forms, intriguing researchers and inspiring poets. One of the ideas that best embodies mathematics in all its elegance is the concept of symmetry.

The roof of the Lotfollah mosque in Isfahan (Iran) is a great example of symmetry endowed with beauty in architecture.

Inside the mosque there are several rooms with diverse symmetrical motifs.

Harmony and beauty
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.
The Stanford Dictionary of Philosophy reminds us that in Timaeus, the work of Greek philosopher Plato (429-347 BCE or Before the Common Era), for example, regular geometric forms take center stage in the doctrine of natural elements because of the proportions they contain and the beauty of their forms. The four elements - Fire, Water, Earth and Air - could be represented by regular geometric shapes (with polyhedrons of four, twenty, six and eight equal sides, respectively). The Universe could also be represented by a 12-sided polyhedron - or a symmetrical dodecahedron. When particles with these different forms are combined, they give rise to all the natural elements we know. Although the word or concept of "symmetry" did not exist in Greek vocabulary in the days of Plato, the concept was already developing. The Greek noun "summetria", which literally means "of the same size", was already being used to refer to "proportion”.
The golden ratio
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.
In nature, this type of symmetry marks the growth rhythm in the development of several species - and is also perceptible to the naked eye, fitting into the rules that determine the conception of "beauty" in art. The greatest example of the materialization of the golden spiral in nature is perhaps the nautilus, a prehistoric mollusk that still has living 'relatives' in the Pacific Ocean.

The nautilus is a surviving species of the archaic subclass of nautoloids which appeared at the beginning of the Paleozoic - long before the dinosaurs and even before the appearance of the first terrestrial animals. The subclass of ammonoids included the extinct species of ammonites - still much appreciated by fossil aficionados - that also displayed the golden proportion in their shells.

Same, but different?
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.

According to IME-USP professor Eduardo Colli,: "our eye looks for symmetries, even if these are not perfect in nature. In fact, the greatest beauty in the symmetries of nature lies in these little imperfections.”

Bilateral beauty
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.
Spheres, spheres
An object is spherically symmetrical if it can be cut into two equal halves - regardless of the direction of the cut, as long as it passes through its center. Fruits like oranges and some lemons have a shape that is very close to being spherical.

For Plato, the sphere was the most symmetrical and homogeneous form that existed. And therefore the most beautiful and perfect form of all. He said that the Cosmos had a spherical shape - as well as the celestial bodies, like the planet Jupiter we see in the image.

According to Graham Smith, a mathematician from IM-UFRJ, "today's physicists believe that the so-called 'cosmological constant' is positive, which means that at the scale of the universe, the cosmos really could be a sphere. It would be a fourth-dimensional space-time sphere - but even so it seems that Plato was not that wrong after all!”

Radial shapes
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.

Take this conifer pine, for example: viewed from above, it has radial symmetry; when seen from the side, it has a more or less spherical symmetry.

A mixture of symmetries
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.

The morphology of comb jellies is not easily explained, but they are beautiful to observe. Their translucent glow changes all the time. The images in the video were filmed at the Shedd Aquarium in Chicago, USA.

Breakable Shapes
“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.

You can create a Menger Sponge by removing the central part of a cube and repeating the process a few times on an increasingly smaller scale.

Probably the best representation of fractal forms in nature is the Roman cauliflower.

Symmetries in another dimension
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.

In this video of the American (NASA) and European (ESA) Space Agency, we see a kind of time-lapse of the pulsation of RS Puppis.

Symmetries are everywhere all the time. Just look around to see that they surround us. In addition to endowing our daily life with more grace and beauty, they also have many functions of which we are unaware. Nature hides numbers, equations, and proportions that can be unraveled by anyone who is curious enough. As the celebrated physicist Richard Feynman once said, "knowledge of science only enriches the excitement, mystery, and admiration" for nature. It does not take away its beauty.

Credits: Story

Museum of Tomorrow, 2018

Chairman of the IDG Board of Directors: Fred Arruda
CEO: Ricardo Piquet
General Curator: Luiz Alberto Oliveira
Executive Director: Henrique Oliveira
Director of Programming: Adriana Karla Rodrigues
Director of Fundraising: Renata Salles
Director of Scientific Development: Alfredo Tolmasquim
Manager of Content: Leonardo Menezes
Research and Writing: Meghie Rodrigues
Editing: Emanuel Alencar

Consultants:
Eduardo Colli (Department of Applied Mathematics, Institute of Mathematics and Statistics - University of São Paulo) and Graham Andrew Craig Smith (Institute of Mathematics, Federal University of Rio de Janeiro)

Photos and videos: Phillip Maiwald, Marie-Lan Nguyen, H. Zell, Niabot (Wikimedia Commons), NASA/SwRI/MSSS/Kevin M. Gill, NOAA Photo Gallery, NASA, ESA, G. Bacon (STScI), the Hubble Heritage Team (STScI/AURA)-Hubble/Europe Collaboration, and H. Bond (STScI and Pennsylvania State University)

Credits: All media
The story featured may in some cases have been created by an independent third party and may not always represent the views of the institutions (listed below) who have supplied the content.
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