A short movie inspired on Escher’s works

And a free vision on how it could be his workplace

When this animation project started to take their first steps I intended to bring life to a large and extensive still life, flying over it in a manner similar to that fantastic intro created for the opening credits of a French film called Delicatessen.

But then I still had not found the motif, the main characters of the action. So I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular. This, though, from a completely imaginary, free and invented point of view.

And here is the result of that process, accompanied by the beautiful song “Lost Song” composed by Ólafur Arnalds, an Icelandic musician.

Inspirations on Maths

Throughout the course of this animation we see many objects. I imagined that these things could be his travel souvenirs, gifts from friends, sources of inspiration… Some are tridimensional representations of works by Escher and others might be just his tools as artist and engraver. Here you will find some brief explanatory notes about those elements which have a highly mathematical nature, including the works of that great Dutch artist that appears along the film.

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The Legend of Sessa

The king Check-Rama, marveling at the invention of chess, offered to its inventor, the Brahman Sessa, 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 [ +info ]

2012_inspirations_maths_01The five platonic solids

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 [ +info ]

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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 [ +info ]

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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”. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica, where he claimed he had a proof that was too large to fit in the margin. But the first successful proof did not appear until the end of the 20th century [ +info ]

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 [ +info ]

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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 [ +info ]

2012_inspirations_maths_05Galton 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 “Gaussian Bell Curve” at the base [ +info ]

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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. Oh, and do not miss the work of István Orosz, which has some beautiful works using these techniques [ +info ]

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Three spheres II

This is (another) nod to Escher, who also created a small picturewith 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.

2012_inspirations_maths_08Spirals

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.

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Solitary

This is a very old game (its origins are not known with certainty, so that I could find). One author says it’s a game of Roman origin and Ovid described it in detail. In my case I have always called “Berber Solitary” by the simple fact that many years ago I bought one of these in the Atlas Mountains in Morocco, carved in wood in the bud. And I included it in an old work with more than 10 years, but now I’ve gone back to modeling for the occasion. 🙂

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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 [ +info ]

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Planetary

This is another model that has been recycling. Its origin (from where I got the idea, I mean) is in the main room of Fallingwater short movie. I modeled it in order to appear in one of the final shots on that animation. However I have returned to build it for this new project almost from scratch, using Subdivision Surfaces and adding more detail in the gears and chain drives.

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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. I got the idea of the model from an interesting exhibition organized by the Museo de Matemàtiques de Catalunya (MMACA) in my town [ +info about inventions of Leonardo ]

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Aerial screw by Leonardo

Another model based on a famous drawing by Leonardo da Vinci, which has always been regarded as a vision of the helicopter. There are many images on the internet with wooden models based on that drawing, I have relied on them.

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Abacus

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 [ +info ]

2012_inspirations_maths_15Hourglass

This is a simple instrument for measuring a certain lapse of time. His origin is unclear, although it is undoubtedly his almost-hypnotic ability, something which allows it to continue being appreciated today as a decorative object [ +info ]

2012_inspirations_maths_16Three Spheres I

Based on another famous and simple Escher print. Here we find a kind of “multi-level game”, since what appears to be a sphere is actually a flat circle with a print of a reticulated pattern that simulates the volume of the sphere. To make clear the “game” Escher represents the same sphere (a flat disk, in fact) placed in various forms: vertically, stretched and folded in half. The ironic thing is that when we see the original Escher drawing everything is another double game, as the perspective is not real in any of the disks (everything is simulated, since it remains a DRAWING). And as a final curiosity, I had to build a 3D model for animation, a “real CG” tridimensional object, this time, though of course, seeing him on our screens we see something 2D, again.

2012_inspirations_maths_17Spherical kaleidoscope

This element is based on another model I found in that exhibition by the Museum of Mathematics in Catalonia. I think this was one of the objects that struck me from that event. It was big, you could stick your head inside and by looking around you perceived an enormous sphere, formed from reflections of the small central module.

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Reuleaux Triangle

This has been a device that I’ve always found very interesting since I discovered it many years ago thanks to the wonderful Enciclopedia Salvat del Estudiante, an old encyclopedia that my mother started buying in fascicles when I was ten. I was very surprised to find out that it was possible to build rollers with a quasi-triangular section in a way that a platform could roll over them, like with cylinders, with no oscillations. And we could use it as special drill to get square holes (although obviously it should not rotate exactly around his center, but would have to make a kind of movement like the one you see in the animation) [ +info ]

2012_inspirations_maths_19Cube with double dovetail joints

Another little curiosity drawn from the aforementioned Enciclopedia Salvat del Estudiante (like many other ideas on this animation): How is it possible to explain the manufacturing process of a cube of wood like this, with these two dovetails joints flowing through it that way? The solution is simple, as seen in the animation 🙂

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Puzzles of Sam Loyd

Another idea taken from the aforementioned exhibition organized by the MMACA. 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. Sam Loyd was an American chess player, chess composer, puzzle author, and recreational mathematician [ +info ]

2012_inspirations_maths_21Tangram

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 [ +info ]

2012_inspirations_maths_22Pentominoes puzzle

Is another 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. Another one of those wooden games have always liked and I have in a shelf of my living room [ +info ]

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The seven bridges of Königsberg

In the city of Königsberg (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 [ +info ]

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Scissors unleashed

Another detail taken from Enciclopedia Salvat del Estudiante, barely seen in the animation, really: a seamstress (well, in our case, Escher himself, let’s imagine) was in the habit of tying his scissors as shown in the figure, by a string subject to a nail under the table, so that their children could not take and lose it. One day, however, her children got to unleash the scissors without cutting the string or remove the nail. How did they do? [ click here to see the solution ]

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Reptiles

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. Although many people always want to see deep esoteric meanings, even there where there are not at all, as you can read in the Wikipedia article [ +info ]

Inspirations on Arts

Mixed with “cabinet of maths curiosities” we find a bunch of printed sheets and postcards with great masterpieces of painting and drawing. Not sure if Escher would like all of these, but I love them.

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Pieter Claesz, “Still-Life with Oysters”, c. 1633

 

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Gauss, Newton and Euler

 

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Da Vinci, Dürer and Velázquez

 

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Leonardo da Vinci, “VitruvianMan”, c. 1487

 

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Albrecht Dürer, “Young Hare”, 1502

 

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Paolo Ucello, “Persp. Study of a Chalice”, c. 1450

 

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Tughra of Suleiman the Magnificent, 1520

 

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Katsushika Hokusai, “The Great Wave”, c. 1829-32

 

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Goya, “The Sleep of Reason Produces Monsters”, c. 1797

 

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Click to view larger version & read below each postcard for more info.


Cristóbal Vila, February 2012, Zaragoza, Spain