The Legend of Sessa

The king Check-Rama, marveling at the invention of chess, offered to its inventor who chose his own reward. This asked for a grain of wheat for the first square, two for second, four for third and so on, doubling each time the number of grains of the previous square. What seemed like a modest request was impossible to fulfill, since the total number of grains was 2 raised to 64, or what is the same: 18,446,744,073,709,551,616. An amount far greater than the capacity of all the granaries of the vast Persian Empire. •

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Plato knew that there are only five regular convex polyhedra: The regular tetrahedron composed of four equilateral triangles. The cube or regular hexahedron formed by six squares. The regular octahedron, consisting of eight equilateral triangles. The regular dodecahedron, composed of twelve pentagons. The regular icosahedron consisting of twenty equilateral triangles. All of them can be represented in a plane and are easily constructible in cardboard

The five platonic solids

Homogeneous tilings

There are eleven types of “homogeneous” tessellations (regular + semiregular), ie those that are made exclusively with regular polygons and can be constructed from equilateral triangles, squares, hexagons, octagons and dodecagons. Only one of them is presented in two different forms of reflection (the two that are placed at center), resulting in all these twelve combinations you see in this figure.

Fermat's Last Theorem This is one of the most famous theorems in the history of mathematics. It states that: “no three positive integers x, y, and z can satisfy the equation at right for any integer value of n greater than two”

Euler's Formula

It is considered as one of the most “beautiful” formulas, since it links together some of the most important numbers in mathematics, as we see at left. It also provides a powerful connection between analysis and trigonometry. Just as a small curiosity: you can imagine which is the favourite formula for the main character in “The Housekeeper and the Professor” a beautiful book by the Japanese writer Yoko Ogawa

Cycloid curves

On this model featured in the animation we see a curve being plotted from a spinning wheel over a straight base, without slipping. If the generator point would be located in the edge of the wheel we would get a common cycloid, but in our model the radio can vary, giving place to elongated or shortened cycloids. These kind of curves are very beautiful, and with many applications in engineering and construction.

Galton Box

This is a device developed by Francis Galton, used to demonstrate the Central Limit Theorem. So that upon release of a bunch of pellets by the upper funnel, all of them are finally distributed in a manner which approximates the famous â€œGaussian Bell Curveâ€? at the base.

Anamorphosis

Is a deformation of an image that, when viewed from a certain angle or using some optical device (such as a curved mirror) provides the original image. It has been used often throughout the history of the paint. In fact, one of the postcards that also appears in the animation, "The Ambassadors" also features this trick.

Three spheres II

This is (another) nod to Escher, who also created a small picture with these items. At the same time is a kind of homage to 3D computer graphics, since the sphere is often used as a basic element to represent the color, reflection, refraction and other material properties.

Spirals

Here is a 3D representation of another work by Escher. In the animation we can see how the initial form of torus is transformed into this set of spirals turning in on themselves.

Solitary

This is a very old game (its origins are not known with certainty). One author says it's a game of Roman origin and Ovid described it in detail.

Newton's cradle

This is a device which demonstrates the conservation of momentum and energy. We have seen some of these in many films as a typical toy or gadget for desktops.

Leonardo Bridge

This is an ingenious construction designed by Leonardo da Vinci, in which stability is achieved throughout the structure without using nails, rope or other type of fastener.

Aerial screw by Leonardo

Abacus

Another model based on a famous drawing by Leonardo da Vinci, which has always been regarded as a vision of the helicopter.

It's considered the oldest calculating device, adapted and used by many cultures around the World. Its origin is uncertain although it's usually accepted that could be in China, where they still used frequently, as in Japan

Puzzles of Sam Loyd

Tangram

This is one of those puzzles which can cost considerably more to get than it appears at first sight. But once someone shows you the method for solving (sorting the pieces in a certain way) is very easy.

It is an ancient Chinese game that is to form figures with seven pieces resulting from cutting a square sheet. So usually appears within a box with that form to sell and keep

Pentominoes puzzle

The seven bridges of Kรถnigsberg

Reptiles

Is a simple game created with the 12 possible pentominoes, which when arranged in a certain way they fit perfectly into their box. There are exactly 2339 different ways to combine them.

In the city of Kรถnigsberg (now Kaliningrad, Russia) the Pregel river branched into two channels. It formed an island that was communicated with the banks through seven bridges, as shown in the above model. The tradition said that one of the distractions of its inhabitants was to try to cross the seven bridges without passing more than once for the same. The Swiss mathematician Leonhard Euler, who lived at the court of Russia, showed that it was impossible to get it

This is the main protagonist of the animation. One of the best known works by Escher, where he plays with the two-dimensional complex combination of tiles and their transformation into three-dimensional elements, the crocodiles. Escher also applies here a sense of humor, looking for the paradox, but without attempting to transcend philosophical explanations of any kind.