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What surrounds us (including us!) is governed by precise and fascinating mathematical rules. Mathematics lives in the abstract world of ideas, but in this room you will be able to visualize it and put it into practice concretely.
The famous theorem applies not only to squares, but also to hexagons and all other figures: let’s discover the principle together with a simple mechanism.
Through the “choices” that the marbles make along the way, you can understand how, sometimes, math can help you predict the future.
“Could a butterfly flutter in Brazil cause a tornado in Texas?” In a chaotic system, very small initial differences are enough to obtain completely different evolutions: experiment with the pendulum of this station.
Divisible only by 1 and by themselves, with prime numbers we can build all integers (and even program computers). They were well known since ancient times: more than two millennia ago the Greek mathematician Eratosthenes developed a simple and powerful method to find them that you can experience live with this location.
The water level shows different curves according to the inclination of the cone: circumference, ellipse, parabola and hyperbola are a family of sister curves with the same origin. This is why they are called conics.
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Water falling from a shape to another shows you directly that some figures have the same area, as the well-known theorem states: “In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs.”
Although its celebrity, fewer people know that the Pythagorean theorem does not hold just for squares, but also for hexagons and, as a general rule, for any figure; squares however are mostly used because with squares you also get an algebraic equation: c2=a2+b2.
When the fall is over we will find that the majority of marbles are in the central bins and just a few of them are in the lateral ones. The curve that the marbles form, after the fall, is something similar to what mathematicians call Gaussian Curve or Normal Distribution.
The reason is linked to the probability theory. When a marble falls and hits a pin, it has a 50% chance to fall to the right and 50% to the left. Marbles in the left bins are there because during the fall they mostly turned on the left. Same thing happens to the marbles arrived in the right bins. However, while the marbles are falling it is much more likely that they sometimes turn on the right and sometimes on the left, so that at the end they are found in the central bins.
Although the initial conditions of the pendulums are quite the same, and they initially move the same way, shortly after their motion is no longer alike. This happens because the system is chaotic and the time evolution strongly depends on initial conditions.
In a non-chaotic system, two bodies which start from slightly different initial conditions will also have a slightly different time evolution. On the contrary, in a chaotic system small differences in the initial conditions become huge differences in the evolution of the system. It is never possible to obtain the very same initial conditions with two objects in a dynamic system.
Chaos theory has its relevance in meteorology: climate is indeed a chaotic system; that's why weather forecasts are hard to make and are reliable only for a short time.
The uncovered numbers are prime numbers. They are natural number greater than 1 that have no positive divisors other than 1 and themselves. The method you’ve just used is called the sieve of Eratosthenes, in honor of its inventor, the Greek mathematician Eratosthenes of Cyrene (III-II century b.C.). It’s an ancient algorithm, yet so efficient and simple that is still used today by a lot of computer softwares. These numbers have an important role in Mathematics, since all natural numbers can be obtained from them simply by multiplication. They are also used in modern cryptography.
Two prime numbers can sometimes be separated only by an even number: in that case they are called “Twins”. Apart from the pair formed by 2 and 3, all the primes are never consecutive, since they must necessarily be odd not to be divisible by 2.
Your reflection will multiply several times.
Inside the kaleidoscope your image will be reflected by the mirrors that build it in order to create a hexagonal tessellation plane. This happens because the kaleidoscope is internally made up of three mirrors attached together with 60° angles and each one reflects the image produced by the others. Plane tessellation is a series of geometric figures repeated endlessly without overlapping, which cover all space and leave no empty spaces.
