# The geometric pattern of beauty

The basic approach of Ars Qubica is to **transmit the presence of geometry and mathematics in art**. The figure of a cube is used as a conductive thread, a geometric figure that, when cut by a plane, can give rise to a square, an equilateral triangle, a non-regular pentagon or a hexagon. We will also see how these sections are present in different artistic and ornamental works.

In this blog post you can learn more about how this project originated and you can also visit the official website. Here we are going to focus on what is “behind” it, from a theoretical-technical point of view.

But before, I don’t want to forget to mention the group of people who made it possible and with whom it has been a pleasure to collaborate:

Script

**Fernando Corbalán and Luis Rández**

Pictures

**Cristóbal Vila**

Music

**“Atmostra III – Mars” Cedric Baravaglio**

Project Management

**Pedro Miana**

Communication and Distribution

**Beatriz Rubio**

*And of course, the sponsors and friends who supported us with crowdfunding (see credits)*

For the analysis of the elements that appear in Ars Qubica we will not follow the order of the animation, but **we will group them by styles or typologies**.

In Ars Qubica we find several works belonging to two avant-garde movements that appeared in Russia at the beginning of the 20th century: Constructivism and Suprematism. Let’s take a brief look at these artistic manifestations.

# Constructivism and Suprematism

Constructivism is a philosophical, artistic and architectural movement that emerged in Russia around 1919. It defends that the work is related to the environment and uses profusely elements with geometric, linear and flat forms. He conceives art not as something isolated, but at the service of society and seeks the abstract and functional, related to industry and technique.

Suprematism is another Russian artistic movement, founded by Malevich, which was in force between 1913 and 1923. It was nourished by the most basic and fundamental geometric forms, especially the square and the circle, and searched for the “supremacy of nothingness” representing a universe without objects.

And which are the differences between the two movements? The truth is that they have many points in common and most artists moved between both ones. But one could sum it up by saying that for Constructivism art has a social purpose and the artist must subordinate his freedom to achieve the common good. Suprematism also admits the ethical purpose but does not renounce the creative freedom of the artist. It defends that art is a more personal experience than a public one, no matter how universal its application may be.

And which are the works of these two trends that appear in Ars Qubica? To begin with, we have the first of them all, which opens the animation and introduces us to our “protagonist”, the **cube**, an element that will reappear throughout the animation to be cut and give rise to different sections:

#### “Proun”

*Collage with ink and watercolour, El Lissitzky, 1925*

El Lissitzky is a pseudonym of Lazar Markovich Lissitzky (1890-1941), a Russian artist, designer, photographer, teacher, typographer and architect. One of the most important figures of Russian abstract and avant-garde art, pioneer of constructivism and suprematism.

The “proun” (or rather “prounen”, in plural) are a series of geometric and abstract paintings made by Lissitzky. He himself defined this concept as “an intermediate state between painting and architecture”.

With the prounen he introduced three-dimensional illusions through the use of forms with a certain architectural effect. Proun was essentially Lissitzky’s exploration of the visual language of suprematism with spatial elements, using changing axes and multiple perspectives; both were unusual ideas in suprematism.

*(text extracted from Wikipedia. Here you have the complete article about El Lissitzky)*

From a technical point of view, this project did not pose any problems. Just mention that after creating the 3D model of the elements that appear in the work had to “put” literally the camera into the picture, pointing directly to the black face of the cube, and then make it leave the work. This can be seen in the following image:

Our cube is cut by a plane and we obtain the first geometric figure: the square. And with it we make appear the following work, also belonging to the Russian Vanguards:

#### “Black square”

*Oil on canvas, Malevich, 1913*

Kazimir Severinovich Malevich (1878-1935) was a Russian artist and theoretician (born in today’s Ukraine, of Polish parents), pioneer of the Russian avant-garde, forerunner of abstract art and maximum representative of suprematism. And this work is the great symbol of suprematism.

Malevich said: “By suprematism I mean the supremacy of pure sensibility in the figurative arts” and also: “The authentic and stable value of a work of art, what distinguishes a masterpiece from a mediocre one, is the sensibility that is expressed in it .

The square is the best method to capture pure art, it is considered the most elementary and fundamental artistic element, and therefore, “supreme”. The square represented the non-objective sensibility, the perception of the non-objectivity and the white background would be the Nothing.

As with the previous work, from a technical point of view, this project posed no special challenges to be digitally recreated. Just take care of the detail of the surface finish of the paint (already aged) to give it that “old” patina.

At a certain time our cube is cut again by a plane and this occasion we end up generating the figure of an **equilateral triangle**. And with it we take the opportunity to show another work of these same avant-garde:

#### “Red triangles in Round”

*Ink and watercolour, Popova, 1923*

Lyubov Sergeyevna Popova (1889-1924) was a Russian painter and designer, framed in the avant-garde of the revolutionary era: futurism, constructivism and suprematism.

In 1921 she renounced to easel painting and promulgated the need for artists to devote themselves to creating utilitarian art. Consequently, from 1922 she devoted herself to graphic and textile design, as well as theatrical scenography.

For this work it was decided to vectorize the whole. The first of the following images is a scanner of the original work and the last one shows the result after recreating all the elements using bézier layouts:

This vector information was then passed to a 3D application to animate its components and give life to the growth of all structures, while we animate the movement of the camera:

So far the three works belonging to the Russian avant-garde of Constructivism and Suprematism. Let’s now see another set of works, which occupies most of our animation, and that we could fit into the world of crafts or decorative arts:

# Tesselations

Tessellations (together with calligraphy) are part of the “arabesques”, a characteristic element of Islamic art. These repeated geometric forms often have hidden meanings. For example, a simple square: its four equal sides symbolise the equal importance of the elements of nature: earth, air, fire and water. However, circular shapes illustrate the infinite uniqueness of the Creator, “Allah”.

Geometric composition was used in Islamic Art to avoid any human representation of divinity. Geometric intertwining was the way in which the “idea of divine unity proclaimed by Mohammed and underlying the infinite variety of the world” was shaped according to Islam. The harmony of the world is expressed for Islamic Art in the complexity of geometric interlacing, because “in unity multiplicity is shown and multiplicity is found in unity”.

#### Nasrid Nail

The Nasrid dynasty, descendant of Yusuf ben Nazar, reigned in Granada from the 13th to the 15th century. The Alhambra in particular, and the whole city in general, lived a period of splendour that has been reflected, among others, in its mosaics. The one known as “Clavo Nazarí” (Spanish for “Nasrid Nail”) is one of the many tessellations developed at that time.

Starting also from a **square** (that we remember, came from the section of a cube) we can obtain this base module known as “Nasrid Nail”.

At right we can see an animated gif with the first steps to obtain it, starting with some axes and diagonals and continuing with simple turns.

And then we have the result after setting it all up in two dimensions, using vector paths:

Once we have everything planned in 2D we can export it to our 3D tool to give it volume and make the pieces evolve in an organic way, at the same time that we control the animation of the camera:

#### Nasrid Paper Bird

This is another tessellation that can also be found profusely in the Nasrid Palaces of the Alhambra. In this case our cut in section of the cube had given rise to an **equilateral triangle** (already used in Popova’s work) and that also serves us as a starting point for this tessellation:

In the animated gif we start from an equilateral triangle, we divide in two each one of its sides, generating another triangle “mirror”, having line segment bisectors of one and another until we find the centres to trace circular arcs that allow us to generate a base wave (I believe that it is understood much better in the images… ;-)

The same steps must be repeated for all sides, until we reach our “pajarita” base. *NOTE: in Spanish “pajarita” (female little bird) is a common kind of paper figure, similar to the popular Japanese origami cranes.*

All these forms were made with precision in a 2D drawing program, using bezier layouts, and then exported to our 3D tool.

Once inside our 3D application, we generate the surfaces in a modular way, define rotation centers, motion attenuators and finally animate all the elements.

#### Cairo Tesselation

For this case, our starting point is a **non-regular pentagon**, also product of the cut produced by a plane in a cube. It’s not possible to tessellate the plane using regular pentagons, but it is possible to get it using non-regular ones. In fact 15 types of convex pentagonal tessellations are known today and it’s unknown if there are more or not (the most recent has been discovered in 2015).

The tessellation of Cairo (so called because it is found in abundance in the pavement of that city) starts from non-regular pentagons with four equal sides, and from those we can find infinity of variations, with different angles. In our case we have used two angles of 112º, two others of 90º and one of 136º.

We could also say that this tessellation would fall within the category of “hexagonal”, since with four tiles we obtain a greater module that is a non-regular hexagon, as we can see in the animated gif on the left.

Once we have that major module, formed by four non-regular pentagons and in the form of hexagon “flattened” we can tessellate the plane. And we also prepare everything to create a square module, as we can see in the following graph. All this, as always, working in 2D and using bezier paths.

And as in the previous cases, once we have everything planned in 2D, we can export it to 3D, define hierarchies, rotation angles, automate processes… to grow the whole set.

#### Gaudí Tile

These hexagonal tiles with vegetal motifs were originally designed by Antoni Gaudí for the paving of the Batlló house, although he finally placed them in La Pedrera. Later, this design was recovered to decorate the sidewalks of the Paseo de Gracia in Barcelona.

In this case, therefore, we start from a regular hexagon, which has also been obtained through the section of a cube. In the animated gif on the right you can see how the ornamental structures of its interior have been created, always using bezier layouts.

In the following animated gif we have a simplified schematic representation of the interior structure of the tile, to better understand how tessellation occurs. The symmetry is that of group **p3** of the 17 crystallographic groups of the plane.

And as in the previous cases, once we have everything planned in 2D, we can export it to 3D, define hierarchies, rotation angles, automating processes, etc. until the whole set grows.

#### Mudéjar Façade of La Seo

Mudéjar art is an artistic style developed in the Christian kingdoms of the Iberian Peninsula, but which incorporates influences, elements or materials of the Hispanic-Muslim style. It is the consequence of the coexistence of cultures existing in medieval Spain (between the 12th and 16th centuries), as a mixture of the Christian and Muslim artistic trends of that times.

The exterior wall of “La Parroquieta” in La Seo (Cathedral of the Savior of Zaragoza) is one of the most striking examples of this style. It was built by the master builder Miguel del Cellero, and Mudéjar artists from Aragón and Seville worked on it. His work with glazed ceramics and brick in geometric reticules produced one of the masterpieces of Aragonese Mudéjar.

In our animation, to recreate this stage, everything emerges from the figure of a white square, a 45º rotated copy is produced to form the Mudéjar star, and from there the rest of the elements are generated. As always, everything was first planned with 2D vectorial layouts starting from a couple of superimposed photographs that served as a guide, as can be seen in the following animated gif:

Once the base repetition module for the bottom and top cloth was obtained, all that was left was to produce copies of the same that would fit perfectly, intertwining with each other:

And once everything is planned in 2D, we can move on to 3D, extruding to form volumes and animating the growth of each of the pieces, as well as the camera:

After all this long series of pieces nailed inside the group “Tesselations”, we pass to a unique and very special work:

# Iconography and Symbolism

#### “Melancholy I”

*Engraving, Albrecht Dürer, 1514*

Considered Dürer’s most mysterious work, it is characterized by its complex iconography and symbolism, which have given rise to various interpretations.

It also has many details related to geometry, arithmetic and the measurement of time: on the wall there is a turned wooden sphere, a truncated polyhedron formed by **irregular pentagons** (the figure we use to enter the work) and triangles, a hourglass, a scale… and the most interesting piece: a “perfect” magic square of 4×4.

In the following graphic we can see all the possible combinations of this famous magic square: it incorporates the first numbers, from 1 to 16, and the sum of each of its rows, columns, diagonals, subquadrants, etc, always gives us the same amount: 34. It also incorporates the date of completion of the work (below, the two central squares).

# Epilogue

#### “Magic Square Zaragoza 2015”

*Luis Rández, 2015*

Created by Luis Rández, professor of mathematics at the University of Zaragoza, it proposes a way of saying goodbye to animation by winking at the most characteristic element of the engraving “Melancholy I” by Dürer, his magic square.

We incorporate the date of the short film and suggest the location in an “indirect” way (morphing from numbers to letters).

In Luis’ own words:

“It is not really a magic square 4×4 in the strict sense, since the numbers are not 1, 2, 3…16, as in Dürer’s square. It also has ligatures for ZA-RA-GO-ZA (first block 2×2, two first rows and columns) and for the year, 20-15. Therefore the best way to create it is to raise the equations sum rows = sum columns = sum diagonal = sum counterdiagonal (…) all this = to the first block 2×2 which is 192”

*Cristóbal Vila, September 2015, Zaragoza, Spain*