Albrecht Dürer, born in Nuremberg on May 21, 1471, is universally considered one of the great pioneering painters of the Renaissance —such as Giotto, Piero de la Francesca or Alberdi—, who developed a new way of painting based on perspective, or “legitimate construction”, as they called it. To understand why this new way of representing reality worked so well, it was necessary to create a new geometry.
The son of a prominent goldsmith, Dürer showed, already as a child, an amazing disposition for drawing, which allowed him to enter as an apprentice in a prestigious engraving workshop in Nuremberg. Being very young he traveled to Bologna where he learned, possibly from the mathematician Luca Pacioli and from the painter and architect Bramante, the principles of linear perspective.
These were times of great discoveries, and Italian artists had turned their attention to mathematics and anatomy to represent space and the forms of the human body. Dürer held that geometry and precise measurements are the key to understanding classical art. His publications include The four books of measure and The four books on human proportionswhich contain various constructions of regular polygons and polyhedrons, as well as, as stated therein, the four different types of female and male figures, whose body dimensions are expressed as fractions of the total height.
Dürer held that geometry and precise measurements are the key to understanding classical art; they were times of great discoveries, and Italian artists had turned their attention to mathematics and anatomy
Mathematics are very present in his work. your engraving Melancholy I contains a 4×4 magic square in which all the numbers between 1 and 16 are arranged so that the sums of their rows, columns, and diagonals is always 34.
Although Dürer did not make any original geometric discoveries, he was among the first to describe perspective in terms of Euclid’s Elements: the painted image is a projection of reality onto the canvas, the center of which is situated in the painter’s eye. All the depth lines are found in that point of view and the painting is the result of the intersection of the plane of the painting with the visual cone, formed by the lines that join the point of view with the represented figures.
In this approach, the parallels, regardless of their orientation, join at their vanishing point located on the horizon, that is, on the horizontal line that passes through the point of intersection with the square of the perpendicular drawn from the point of view.
The method of Dürer and other geniuses of the Renaissance was later enriched by adding perspective in the vertical, or aerial direction.
In this pictorial process, both the lengths and the angles are distorted, however, the original scene is always very recognizable. To explain it, great mathematicians such as Girard Desargues and Blaise Pascal, first, and Charles Brianchon and Jean-Victor Poncelet, later, created a new geometry —projective geometry—, characterized precisely because, in it, the parallel lines have a point in common : the point of infinity.
Desargues considered a very simple mathematical object, such as a triangle, and wondered what was enough for two triangles to be in perspective. In answer he gave a nice theorem: “The projection of a triangle of vertices ABC from a point of view O is the triangle A´B´C´ if, and only if, the lines containing the corresponding sides intersect in aligned points ”.
!['The hare' (1502), is the symbol of Dürer, one of the most famous of his watercolors is perfect as a natural portrait.](https://imagenes.elpais.com/resizer/l7j5X7KaCjUVf5y7ICZjFq-_W20=/414x0/cloudfront-eu-central-1.images.arcpublishing.com/prisa/IO5VDTIOITZU4QVJFICQIVNVRU.jpg)
That by prolonging the corresponding sides AB and A’B’ intersect at a point P, AC and A’C’ in Q, and BC and B’C’ in R, it is easy to see, but what is surprising is that the three points P, Q and R lie on the same straight line in space and that, furthermore, this is a sufficient reason for the two triangles to be in perspective.
In the following years, the method was developed, obtaining very beautiful and interesting theorems that have given rise to projective geometry. In projectivities, lines are transformed into lines, but, in general, distances and angles are not preserved. It happens, however, that the so-called double ratio of four aligned points is maintained: (A, B, C, D) = (CA/CB)/(DA/DB). So if we have another quadruple of points located on the same straight line A’, B’, C’, D’, and we want to know if the latter can be the image of the former by a projectivity, it is enough to verify that two numbers coincide , namely, that they have the same double ratio.
Mathematicians tried to find other projective invariants, such as the double ratio, which led, centuries later, to the abstract formulation of geometry by Felix Klein’s Erlangen program: space is now any set on which a group of transformations, the search for invariants being the main task of geometers. That effort, on the shoulders of giants like Carl Friedrich Gauss, Bernhard Riemann and many others, would lead us to general relativity and modern cosmological theories.
The method of Dürer and the other geniuses of the Renaissance was later enriched by adding perspective in the vertical direction, or aerial perspective. Then the cubists deconstructed it by introducing different points of view in different parts of the same painting, which would lead us to Picasso and the modern concept of topological variety.
Anthony Cordoba He is a professor emeritus at the Autonomous University of Madrid, a member of the Institute of Mathematical Sciences and an honorary academic at the Academy of Sciences of the Region of Murcia.
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Edition and coordination: Ágata A. Timón G Longoria (ICMAT).
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