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The perfection of the snail
Geometry in nature
Taking a walk in the woods or in the countryside, surrounded by trees, mountains, flowers and herbs, you immediately
notice, in addition to the indisputable beauty of the landscape, the curious geometrical property that governs nature.
Nature, in fact, appears to our eyes as different forms that have certain regularities, i.e. symmetries. The sun and the
moon are round; raindrops are spherical and snow is made of hexagonal crystals; trees and mushrooms have rotation
symmetries around a central axis; flowers often have square, pentagonal or hexagonal rotation symmetries; many
animals such as insects and mammals, but also leaves, have bilateral symmetry.
The word symmetry derives from the Greek "syn metria" meaning the same size. Indeed symmetry corresponds to the
concept of proportion, that is symmetrical is synonymous with well-proportioned, balanced and harmonious.
The Little Prince discovers symmetry in nature
In the book "The Little Prince" by Antoine de Saint-Exupery, there is a short dialogue whose subject is precisely
symmetry in nature. Below is the piece in question:
“How beautiful you are!”
“True”, replied the flower softly, “and I was born together with the sun…”
The little prince thought that it was not very modest, but it was very touching! “How can you be so beautiful?”
“Well you see, I am a flower and am a creation of nature, and as such I am perfectly symmetrical…”
“I don't understand” replied the little prince, taken aback by what the flower had said.
“Now I'll explain it to you” said the flower haughtily.
“In nature there are many symmetries”
“And what is their purpose?”
“Well, to make flowers beautiful, there is no doubt. A symmetry of nature is something that the sun has given us and that
no one will ever be able to imitate.
Everything in nature, is born from a symmetry. Many things in nature are symmetric, did you know that?”
“What?”
“For example, starfish, snowflakes, the cells of the beehives and crystals... man!”
“Never been snow nor bees on my planet” The little prince, however, was attracted by what the flower was saying.
“All living things are beautiful and symmetrical from different points of view... I, for example, am coloured and the
symmetries of the colours of my petals make me beautiful”
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The Little Prince and his friend the flower
Perfect and non-perfect symmetries
In reality, in nature symmetries are almost never completely perfect: for example, the leaves of a four-leaf clover are not
exactly equal to each other, symmetry in this case is not precise as in a square. Nevertheless, we say that a starfish has
a pentagonal symmetry even though one of its arms is broken or bent.
A particular example in nature, but with regard to the inorganic world, is fluorite and pyrite crystals. Fluorite has an
octahedral form while pyrite cubic or pentagonal-dodecahedral. It is probable that the very abundance of pyrite crystals in
Sicily suggested the form of the dodecahedron to the ancient Greeks.
Why symmetry in nature?
A system evolves by interacting with the environment and if the external conditions that impose a specific effect on it
possess any symmetry, also the effect can completely or partially retain the symmetry of the cause that generated it. The
Earth is not perfectly round because in rotating it is flattened at the poles, but continues to maintain rotation symmetry
around its axis and reflection symmetry with respect to the equatorial plane.
In the passion flower there is a perfect symmetry.
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So what is the advantage of symmetry? Symmetrical structures and living organisms have several advantages: they are
compact, stable, homogeneous and interchangeable; exactly as in an assembly line, they can be more easily and more
quickly reproduced. For example, if living organisms are spherical in shape, they disperse less heat. In fact, the amount
of heat that an animal disperses through the skin is directly proportional to its surface area. The relationship between the
volume and the surface area of the body is such that the bigger the animal, the lower is the surface area per unit of
volume and, hence, the slower is its heat loss.
Evolution and symmetry
In general, simpler living organisms tend to be more symmetrical than those more evolved. In fact, many single cell
organisms, such as the Volvox green alga, have the most symmetrical shape that exists, i.e. the sphere. More complex
marine organisms, however, such as jellyfish, starfish and vegetables fixed to the bottom where they grow, are
influenced by the direction of the force of gravity: their top part is different from the lower part, but they retain rotational
symmetry around an axis. Even more complex organisms that move autonomously, such as fish or mammals, possess a
front part where the eyes are located, different from the rear, while retaining bilateral symmetry, so as not to move like a
screw but in a straight line.
The human body also has bilateral symmetry; in fact, the embryo initially develops with a repetitive translatory structure,
i.e. that moves all the points of a fixed distance in the same direction. As the foetus grows, however, bilateral symmetry
is lost when the organs with a single copy, such as the liver or the heart are formed, arranged laterally, in fact, to the
central axis in order to occupy as little space as possible, or when the intestine, in order to extend, begins to fold on itself.
Some male crustaceans, on the other hand, break bilateral symmetry with higher growth of one of the two claws. If they
lose the smaller claw, over time this grows again without any problems; but if they lose the bigger claw, the smaller
grows rapidly until it reaches the maximum size and then the mutation takes place and thus also the smaller claw will
reappear. Therefore the two possible forms are mutually symmetric.
The asymmetric claws of the fiddler crab.
A structure that grows in the same direction in modules that are always the same, like a worm or a blade of wheat, is
potentially symmetrical by translation: segments always the same over time are continually born.
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The architect of nature: the golden ratio
Numbers are not always arid and cold, sometimes they describe the beauties of nature; this is the case of the golden
number, associated with the golden ratio, also called, not surprisingly, the Divine Proportion. The golden number is
indicated with Φ (phi) because its exact value is difficult to write being, in fact, an irrational number, a little greater than 1
but composed of an infinite number of digits:
Φ = 1.6180339887498…
In order to learn more about the golden number, it can be approximated geometrically through the golden ratio, which
can be visualised and therefore better understood.
The golden ratio of any segment can be found by identifying a point inside it such that the greater part is the proportional
mean between the entire segment and the smaller part.
AB : AC = AC : CB
The segment AC is called the golden part of the segment AB.
With a few mathematical steps, it can be demonstrated that from the proportion the ratio between a segment and its
golden part is obtained:
And also
This ratio is precisely the number Φ
When two segments are in a golden ratio, they create a sense of harmony and balance and appear incredibly appealing
to the eye. It almost seems that the human brain is particularly susceptible to this ratio.
The golden ratio is found everywhere in nature, in the spiral shape of a snail, in the position of the seeds in the sunflower
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head, in the bark of the pineapple or the pine cone and even in the logarithmic spiral of the swoop of a hunting falcon or
in the natural expansion of billions of galaxies.
The perfect spiral of a snail
Kepler was quoted as saying: “Geometry has two great treasures: one is Pythagoras' theorem; the other is the golden
ratio of a segment. The first we can compare to a gold object; the second we can define as a precious jewel.
Golden ration around us
In reality, the golden ratio can also be found in everyday life. Without going too far, suffice it to cut through an apple to
discover that the pericarp, which contains the seeds, is shaped like a five-pointed star or pentagram. And in this
geometric figure we happen to come across the golden ratio, the side and the base of each of its five triangles in fact
constituting a golden ratio. In fact, the Pentagram, or five-pointed star, stems from the construction of a regular pentagon
with its five diagonals which in turn intersect forming another regular pentagon. An intersection point of the diagonals
divides a diagonal into two segments such that the ratio of the entire diagonal to the larger segment is equal to the ratio
of this segment to the smaller segment.
In order to learn more about the golden number, it can be approximated geometrically through the golden ratio, which
can be visualised and therefore better understood.
The golden ratio of any segment can be found by identifying a point inside it such that the greater part is the proportional
mean between the entire segment and the smaller part.
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The pericarp of an apple is in the form of a five-pointed star
The “divine” rectangle
There is a special rectangle whose proportions correspond to the golden ratio. It is in fact called the golden rectangle. To
construct it, you must draw a square with side a and we will call its vertices, starting from the upper left and going
clockwise, AEFD. Then you must divide the segment AE in two and find the midpoint A'. With a compass pointing on A’,
you must draw an arc that from F intersects the extension of the segment AE in B. Then draw a segment BC
perpendicular to AB. The resulting rectangle ABCD is precisely a golden rectangle in which AB is divided in the point E
exactly in the golden ratio:
AE:AB=EB:AE
If you subtract from this rectangle a square whose side is equal to the smaller side of the rectangle, you obtain a small
rectangle that is again golden. Proceeding in the same manner, increasingly smaller rectangles are formed. Repeat at
least 5 times. Pointing the tip of the compass on the vertex of the square lying on the long side of the rectangle, you can
draw the arc which joins the ends of the two sides forming the selected angle. Repeat the operation for each square
drawn so as to create a continuous line.
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The golden ratio is directly related to the Fibonacci number, but that's another story…
Soon we will also explain this ratio in a simple way: stay tuned to eniscuola and soon you will also learn more about the
Fibonacci number.
By Tiziana Bosco
Bibliography:
Valeria Cannata, Università degli studi di Palermo, Analisi del cartone animato Paperino nel mondo della matemagica
Valentina Argnani, Liceo classico Evangelista Torricelli, Le trasformazioni tra natura e arte: rosoni, fregi e frattali
IIS Ettore Majorana di Avezzano, La sezione aurea, la serie di Fibonacci e la natura
Gabriele Andreoni, Uca un granchio dalle mille sorprese
matematita.it