MATHS IN NATURE

Mathematics gives up hope that every problem has a solution.

MATHS IN NATURE

"MIE" MATHS IS EVERYWHERE

GOLDEN RATIO IN NATURE

THE TEAM WORK OF GROUP 1

Where can we find the "Golden Ratio"?

The "Golden Ratio" is a unique mathematical relationship. Two numbers are in the golden ratio if the ratio of the sum of the numbers (a + b) divided by the larger number (a) is equal to the ratio of the larger number divided by the smaller number (a/b). The golden ratio is about 1.618, and represented by the Greek letter phi.

FRACTAL GEOMETRY

As illustrated in the Nautilus shell below, the distance from Point 1 to Point 2 divided by the distance from Point 2 to Point 3 is quite close to a golden ratio for the complete rotation of the Nautilus spiral shown below. This is indicated by the golden ratio ruler below (in blue/white), which has a golden ratio point at the division between the blue and white sections. When the blue section has a length of 1, the white section has a length of 1.618, for a total length of 2.618.

Nautilus shell

The golden ratio seems to represent the standard of reference for perfection, grace and harmony, both in architecture, sculpture and painting, and in nature itself.

Face of the Parthenon

The photo below seems to show a Golden Rectangle with a Golden Spiral overlay to the entire face of the Parthenon.

Using this approach, the actual spiral expansion rates for the above Nautilus shell, taken every 30 degrees of rotation were: 1.572, 1.589, 1.607, 1.621, 1.627, 1.622, 1.616, 1.573, 1.551, 1.545, 1.550 and 1.573. This averages to 1.587, a 1.9% variance from 1.618. This is not exactly a golden ratio, but then it’s not hard to see why it would appear to be one.

This illustrates that the height and width of the Parthenon conform closely to Golden Ratio proportions. This construction requires a assumption though, that the bottom of the golden rectangle should align with the bottom of the second step into the structure and that the top should align with a peak of the roof that is projected by the remaining sections.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.A galaxy is a gravitationally bound system of stars, stellar remnants,interstellar gas, dust, and dark matter.Galaxies are categorized according to their visual morphology as elliptical,spiral, or irregular.

Galaxies

SOURCES

THE MENTORS OF GROUP-1

• Melissa Esposito

Iti Medi Italy

• Cojocari Ludmila

Gaudeamus High School Republic of Moldova

• Bénédicte Leduc

Cité Scolaire Brocéliande France

• ELVAN İNAN

BANDIRMA BİLİM VE SANAT MERKEZ Turkey

GEOMETRY IN NATURE

THE TEAM WORK OF GROUP 2

OUR Video

FRACTAL GEOMETRY

Fractals are complex and dazzling shapes that are formed by many similar geometric shapes from large to small, going towards infinity. The word fractal is derived from the Latin word "fractus", meaning broken and fragmented. Fractal is a system of geometry; When fractals are examined closely, it is seen that the small shapes that make up the big shape and become proportionally smaller resemble the big shape, and this repeating phenomenon extends to infinity.

The fern is one of the examples of fractals in nature. British mathematician Michael Barnsley gave an example of fern in his book "Fractals Everywhere". Hence the fern fractal is called the Barnsley fern fractal.

Snowflakes are an other example that can also be given.

A piece of broccoli is very similar to the whole broccoli.

EXAMPLES OF FRACTALS

EINSTEIN

YOU WILL FIGURE OUT BETTER EVERYTHING WHEN YOU LOOK CLOSER TO NATURE

A sequence of numbers formed as a result of the addition of each number with the previous one is called the Fibonacci sequence. In this series, which continues in this way, the golden ratio emerges when the numbers are proportioned to each other. In other words, when a number is divided by the number before it, a series that approaches the golden ratio is obtained.

FIBONACCI SEQUENCE

EXAMPLE OF FIBONACCI SEQUENCE

EXAMPLES OF FIBONACCI SEQUENCE

Geometry in Nature - The succession of Fibonacci in Nature.

TURING PATTERN

The fact that the tiger has a pattern of evenly spaced, vertical, parallel lines and spaces in places is based on Alan Turing's 60-year equation, which is explained by the Turing reaction-distribution model.

In this photo, we can see a perfect circle reflection in the lake. This phenomenon occurs when a bridge is found just above water, and a reflection take place.This is a geometry that is often found in nature.

EXAMPLES OF GEOMETRY IN NATURE

EXAMPLES OF GEOMETRY IN NATURE

The honeycomb is a mass of hexagonal prismatic wax cells built by honey bees to contain their larvae and stores of honey and pollen.Beekeepers may remove the entire honeycomb to harvest honey.

In this picture we can see straight lines. These lines form some triangles, rectangles and squares to separe the differents greens (light and dark). The lines are created by the tractors. To add something, we can see a very straight road on the middle to separe all the fields.Thus, in this image, geometry is realy present in nature.

In the photo we see several examples that geometry surrounds us everywhere.

In this photo we can see a solar farm in the south of France. The farm is made up of 112 780 solar modules. The solar modules are symmetric and long. It allow of made a lot of energy. It is extended. It spread out over more than 500 acres.

Kepler said that snowflakes commonly adopt a geometric shape that is a hexagon, although depending on the humidity and temperature conditions it can also be different geometric figures as it appears in the imagen suchs as triangles, circunference, decagons etc ..

OUR POSTERS

OUR POSTERS

OUR POSTERS

On this poster, we can see how lenticular clouds are formed and their beauty in the sky.

Kamilla Stefańska

Marina Nikolić

Astghik Karapetyan

Esra Atalay

The mentor teachers are:

These slides have been created by Group-2. Thanks everyone who contributed.

FRACTALS IN NATURE

THE TEAM WORK OF GROUP -3

A fractal is a geometric object in which the same pattern is repeated at different scales and with different orientation. The expression fractal comes from the Latin fractus, which means fractured, broken, irregular. A fractal is something that is relates to a mathematical model that describes and studies objects and frequent phenomena un nature that cannot be explained by classical theories and that are obtained through simulations of the process that creates them.

FRACTAL GEOMETRY

Where can we find the fractals in nature?

Trees are perfect examples of fractals in nature. You will find fractals at every level of the forest ecosystem from seeds and pinecones, to branches and leaves, and to the self-similar replication of trees, ferns, and plants throughout the ecosystem.

TREES

This block of plexiglass was exposed to a strong current of electricity that burned a fractal branching pattern within. This can be best thought of as bottled-lightning.

ELECTRICITY

When ice forms on panes of glass, it is sometimes refered to as window frost, fern frost, or ice flowers.There are many types of snow and ice, all of which have certain characteristics common during the formation process.Snow crystals take the shape of a 6-sided dimension, similar to lava that also solidifies into columnar basalt matrix with the same number of sides. Yet, ice crystals take shape into a more dendritic fractalform.Dendritic simply means a multi-branching tree like form, while fractal refers to a mathematical pattern that is common to much of the structure throughout nature.From bird feathers, geologic landscape formations, to ice crystals, nature has an orderly signature.

Ice crystals

The profile of a river is another example of a natural fractal.

RIVERS

Lightning bolts

Another naturally occurring fractal pattern is a lightning bolt. As Benoit Mandelbrot noted in the opening quotation, lightning does not travel in straight lines. Rather, it follows a chaotic, jagged path, formed as the huge charge separation built up in the sky suddenly breaks down. The majority of a lightning bolt is generally hidden in a cloud, much as an iceberg hides beneath the ocean. Lightning can be very large, spanning several kilometers, but it is formed in microseconds. Thunder is a fractal sound. It is caused by the superheating of air. Because the pathway of the lightning bolt is a jagged fractal in 3D space, the time it takes to reach your ear varies, and the thunder rumbles in a corresponding fractal pattern.

The heart is filled with fractal networks: the coronary arteries and veins, the fibers binding the valves to the heart wall, the cardiac muscles themselves.Fractal branching makes available much more surface area for absorption and transfer in bronchial tubes, capallaries, intestinal lining, and bile ducts.The fractal structure of the human circulatory system damps out the hammer blows that our heart generates.Fractals may save our lives every minute.

The human body is full of fractal networks

OUR POSTER

Luminita Moise

Vesna Kushevska

Inmaculada Illán Gómez

Özlem Kahraman

GROUP-3

Maths on Planet Earth

THE TEAM WORK OF GROUP -4

The earth do always the same trajectory, a perfect round trajector. Futhermore the earth turn on it-self. It’s because in sapce there is nothing (apart from asteroids) for stop or slow down the celestials bodys, so they always do the same things.

tHE trajectory of planet eartH

Gravity is the force that attracts a body to the centre of the earth, or any other physical body having mass. Mass is how we measure the amount of matter in something. How much weigth ​something is, the more of a gravitational pull it exerts. As we walk on the surface of the Earth, it pulls on us. But since the Earth is so much more massive than we are, the pull from us is not strong enough to move the Earth, while the pull from the Earth can make us fall. The formula of the gravity is: F=G×m¹+m²/r2 The explanation of the formula is what I am going to put next: Pull :Two objects with masses m1 and m2, with a distance r between their centers, attract each other with a force F equal to: F = Gm1m2 / r2 where G is the gravitational constant.

Gravity in Earth

Biomimetics – a science created not long ago - allows, thanks to XXI century technology, a new relationship between man and nature. As the role of mathematics is to provide tools for modeling some natural phenomena, Mathematics may comprise in a few formulas several geometrical shapes, dynamic structures and even collective intelligence. Intelligent behaviors seen in colonies of ants or termites, swarms of bees, flocks of birds, herds of animals, schools of fish etc. are called Swarm Intelligence, name taken by artificial intelligence as well to describe a similar collective behavior of self-organized distributed systems.

Biomimetics and collective intelligence

Albedo is a kind of ability to reflect in the space the light sent by the sun. But it is not actually the same everywhere on Earth, indeed the light colours grounds like north and south pole have a strong albedo and then reflect a lot of light in space. On the contrary, dark grounds like desert has a low albedo and then absorb a lot of light then warm up the temperature of the Earth.

Albedo on planet Earth

It is the Eratosthenes experiment, which more than 2,200 years ago measured the radius of the Earth through the shadow of a stick at a certain time of day. At the time of day in which there was no shadow in Siena when the Sun shone from a zenith (solar noon, which is not the same as noon), he who lived in Alexandria took a stick and placed it upright on the ground and measured its shadow. To calculate this experiment we make a rule of 3: The degrees (360) times the km (whatever I calculate) between the angle (which I also calculate) that way I will get the longitude of the earth once I have the longitude I calculate the radius (it is 2ΠRT.) And I would already have it .

Eratosthenes' experiment

Any object or person is somewhere on Earth. The geographical coordinate system is used to find out its position. In addition, it is also used to determine the distance between two points. The units of this system are: degree, minute, second. In order to specify an absolute point on or above the Earth, the elevation must also be specified. This defines the vertical position of the point relative to the planet's surface. It can be expressed as the vertical distance from the Earth below, but due to the ambiguity of the terms "surface" and "vertical", it is preferred to express it relative to a more precise data set, such as sea level. The distance from the center of the Earth is also a practical coordinate for both deep and space positions. The usual coordinates of elevation / height from the surface or other data are altitude, height, and depth.

Geographic coordinate system

If a planet does not have an atmosphere, it is enough to look at the planet Mars to understand what happens to the planet. In the atmosphere, the air near the Earth's surface consists of 77% nitrogen (N) and 21% oxygen (O). .there is also a low amount of carbon dioxide (CO2) and other gases (helium, hydrogen, etc.). With the increase in the oxygen ratio, the way for the formation of gigantic animals, thousands of times larger than single-celled organisms, has been paved. .Oxygen provided ready-made energy for these gigantic animals to use.The increase in the Sun's brightness, the gradual changes in the Earth's orbit, or, rarely, the colossal, as are the effects from within the Earth, such as continental movements, heat flow, gas and vapor exchange, or chemicals released by living thingsThere are also extraterrestrial effects such as meteor strikes.

The terrestrial axis, wich has an extension of 12,713 kilometers, present an inclination of 23 ° 5 with respect to the perpendicularity to the plane of the ecliptic.

atmosphere OF THE EARTH

Axis of the Earth

These slides have been created by Group-4. Thanks everyone who contributed.

THE MENTOR TEACHERS:

Ursu Elena

Ana Francisco

Nada Sokolović

GROUP-4

our project vıdeo

"MIE"Maths Is Everywhere Team

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