Mathematics And Nature

[Hilda Bagnoli]

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Many different homogeneous metrics on Lie groups, which may have markedly different short-distance properties, are shown to exhibit nearly identical distance functions at long distances, suggesting a large universality class of definitions of quantum complexity.

An algorithm is developed to design a shape, a trajectoid, that can trace any given infinite periodic trajectory when rolling down a slope, finding unexpected implications for quantum and classical optics.

A framework through which machine learning can guide mathematicians in discovering new conjectures and theorems is presented and shown to yield mathematical insight on important open problems in different areas of pure mathematics.

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.

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.

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.

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.

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.

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.

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.

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.

Consultants:

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

Have you ever stopped to look around and notice all the amazing shapes and patterns we see in the world around us? Mathematics forms the building blocks of the natural world and can be seen in stunning ways. Here are a few of my favorite examples of math in nature, but there are many other examples as well.

We also see hexagons in the bubbles that make up a raft bubble. Although we usually think of bubbles as round, when many bubbles get pushed together on the surface of water, they take the shape of hexagons.

Another common shape in nature is a set of concentric circles. Concentric means the circles all share the same center, but have different radii. This means the circles are all different sizes, one inside the other.

A common example is in the ripples of a pond when something hits the surface of the water. But we also see concentric circles in the layers of an onion and the rings of trees that form as it grows and ages.

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Mathematics is present throughout nature. Many natural phenomena exhibit geometric shapes, symmetry, Fibonacci spirals, the golden ratio, and fractal patterns. For example, beehives form hexagonal cells, volcanoes form conical shapes, sunflower seeds arrange in Fibonacci spirals, and coastlines display self-similar fractal patterns across scales. Nature demonstrates that mathematics is a language used to describe the physical world.Read less

Dr. Britz works in combinatorics, a field focused on complex counting and puzzle solving. While combinatorics sits within pure mathematics, Dr. Britz has always been drawn to the philosophical questions about mathematics.

Pi is mostly used when dealing with circles, such as calculating the circumference of a circle using only its diameter. The rule is that, for any circle, the distance around the edge is roughly 3.14 times the distance across the center of the circle.

"When you look into other aspects of nature, you will suddenly find Pi everywhere," says Dr. Britz. "Not only is it linked to every circle, but Pi sometimes pops up in formulas that have nothing to do with circles, like in probability and calculus."

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