The golden number in Cosmocaixa
Photography by Pol Casellas and Eric Sandín
In the wonderfull wall full of formulas (already mentioned in this blog) that you can see in the Cosmocaixa in Barcelona, there also is the sacred equation which solution is the famous golden ratio:
Photography by Pol Casellas and Eric Sandín
Of course, one of the solutions of x2 = x + 1 is the number x = 1.6180339887498948482… (the other is -0.6180339887498948482…). At first sight it may seem a regular solution for a regular equation, but this number has revealed to the world of mathematics a whole new conception of nature and proportionality and this is the reason why it is interesting to know the history of this number and who dared to study its wonderful properties.
Since the golden ratio is a proportion between two segments, some mathematicians have assigned its origin to the ancient civilizations who created great artworks such as the Egyptian pyramids or Babylonian and Assyrian steles, even though it is thought that the presence of the ratio was not done on purpose. We can go forward on history and find the paintings and sculptures in the Greek Parthenon made by Phidias, whose name was taken by Mark Barr in 1900 in order to assign the ratio the Greek letter phi. So we can associate the first conscious appearance of the golden ratio with the Ancient Greece because of its multiple presence in geometry. Although it is usually thought that Plato worked with some theorems involving the golden ratio as Proclus said in his Commentary on Euclid’s Elements, Euclid was the first known person who studied formally such ratio, defining it as the division of a line into extreme and mean ratio. Euclid’s claim of the ratio is the third definition on his sixth book of Elements, which follows: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser”. He also described that the ratio could not be obtained as the division between two integers, referring to the golden ratio as an irrational number.
Definition VI.3 in Oliver Byrne’s edition of the Elements (1847)
In the 13th century, Leonardo de Pisa (also known as Fibonacci) defined his famous serie in the Liber abaci (1202) although he wasn’t aware that phi is asymptotically obtained by dividing each number in the serie by its antecedent, thus, lots of natural phenomena which follows the Fibonacci sequence in any way, are related to the golden proportion.
Another important work from the 16th century is De Divina Proportione (1509) by Luca Pacioli, where the mathematician and theologian explains why the golden ratio should be considered as “divine”, comparing properties of our number like its unicity, immeasurability, self-similarity and the fact its obtained by three segments of a line, with divine qualities as the unicity and omnipresence of God and the Holy Trinity.
Dodecahedron from De Divina Proportione attributed to Leonardo da Vinci
In the Renaissance, the golden ratio was chosen as the beauty proportion in the human body and all the painters and artists used it for his great masterpieces, like Leonardo da Vinci in his Mona Lisa or his famous Vitruvian Man.
Leonardo’s Vitruvian Man
The golden ratio was known in the world of mathematics as the Euclidean ratio between two lines and it wasn’t until 1597 that Michael Maestlin considered it as a number and approximated the inverse number of phi, describing it as “about 0.6180340”, written in a letter sent to his pupil Johannes Kepler. Kepler, famous by his astronomical theory about planetary orbits, also talked about the golden ratio and claimed that the division of each number in the Fibonacci sequence by its precursor, will result asymptotically the phi number. He called it a “precious jewel” and compared its importance to the Pythagoras theorem.
About one century later, the Swiss naturalist and philosopher Charles Bonnet (1720-1793) found the relation between the Fibonacci sequence and the spiral phyllotaxy of plants andthe German mathematician Martin Ohm (1792-1872) gave the ratio its famous “golden” adjective. If we want to talk about artists who introduced the ratio in their paintings in the modern times, a good example would be Salvador Dalí, whose artwork is plenty of masterpieces structured by the golden ratio.
This is just a brief summary of the history behind the golden ratio, which suffices to show that the interest induced by this number over the minds of the greatest mathematicians hasn’t ceased since the Ancient Greece, and even people non-related with mathematics have used it in their own work, which shows the importance and the multiple presence of mathematics and this special number in places that one could not imagine
This post has been written by Pol Casellas and Eric Sandín in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
Location: Cosmocaixa in Barcelona (map)
The Dalí Theatre-Museum
Source: Wikimedia Commons
The Dalí Theatre-Museum, opened in 1974, is the largest surrealistic object in the World. It was built on the ruins of the ancient theater of Figueres and hosts the most important collection of Dalí’s pictures and sculptures.
Salvador Felipe Jacinto Dalí i Domènech, Marquis of Púbol (11 May 1904 – 23 January 1989) was born in Figueres. Although his principal mean of expression was the painting, he also made inroads in different fields such as cinema, photography, sculpture, fashion, jewellery and theatre, in collaboration with a wide range of artists in different media. His wife and muse, Gala Dalí was one of the essential characters in his biography. His public appearances never failed to impress and his ambiguous relationship with Francisco Franco’s regime made of this multifaceted character an icon of the 20th century and more than an artist. During his life he lived in Madrid, Paris and Catalonia and for this reason he was influenced by other important artists. He died in Barcelona and was buried in his own museum against his desire.
Dalí’s tomb. Source: Wikimedia Commons
Why did I say that he is more than an artist? If you visit the Dalí’s Theatre-Museum in Figueres, you will see his art based on mathematics and physical laws. Dalí’s relationship with science began in his teens when he started reading scientific articles and this passion for science was preserved all his life. In the museum you can find a great reflection of that passion. Furthermore, the painter’s library contains hundreds of books with notes about various scientific topics: physics, quantum mechanics, life’s origin, evolution and mathematics. In addition to that, he was subscribed to several scientific journals to be informed about the new scientific advances.
To show this relation between Mathematics and his masterpieces, I will explain three artworks which are exhibited in the museum from a mathematical point of view. The first one is Leda Atomica (1949). He created it from studying Luca Pacioli’s De Divina Proportione (Milan, 1509) Dalí made different computations for three months with the help of Matila Ghyka (1881-1965). Ghyka wrote some mathematical treatises related with the golden number like Le nombre d’or: Rites et rythmes pythagoriciens dans le development de la civilisation occidentale (1931), The Geometry of Art and Life (1946) or A Practical Handbook of Geometry and Design (1952).
Matila Ghyka
Source: Wikimedia Commons
The painting synthesizes centuries of tradition of Pythagorean symbolic Mathematics. It is a watermark based on the golden ratio, but making the viewer not appreciate it at first glance. In 1947’s sketch, it can be noticed the geometric accuracy of the analysis done by Dalí based on the Pythagorean mystic staff, which is a five-pointed star drawn with five straight strokes:
Source: Wikimedia Commons
You can see that Gala, in the centre of the painting, is enclosed in a regular pentagon and her proportions are according the golden ratio. The picture depicts Leda, the mythological queen of Sparta, with a swan suspended behind her left. There also are a book, a set square, two stepping tools and a floating egg. Dalí himself described the picture in the following way:
Dalí shows us the hierarchized libidinous emotion, suspended and as though hanging in midair, in accordance with the modern ‘nothing touches’ theory of intra-atomic physics. Leda does not touch the swan; Leda does not touch the pedestal; the pedestal does not touch the base; the base does not touch the sea; the sea does not touch the shore…
Leda atomica (1949)( Source: Wikimedia Commons
Another mathematical example is Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors from 1973. This is a stereoscopic work which is an example of the experiments conducted by him during the seventies. Dalí wished to reach the third dimension through stereoscopy and to achieve the effect of depth.
Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors (1973)
Photography by Roger Pijoan Català
The last example is Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln (1975). In this case, Dalí used the double image techinque for creating akind of illusion which is very common in his work.
Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln
Photography by Roger Pijoan Català
So, Dalí was more mathematician than one can imagine.
This post has been written by Sara Puig Cabruja in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).
More information about Dalí’s scientific motivation: Salvador Dalí and Science and Salvador Dalí and Science. Beyond a mere curiosity.
Location: Dalí Theatre and Museum in Figueres (map)
A Platonic Dalí in Washington
Salvador Dalí (1904-1989) is one of the most important surrealistic painters. This eccentric Catalan painter finished in 1955 the Sacrament of the Last Supper after nine months working in it and this painting is mathematically remarkable because of the dodecahedric form over Christ and the twelve apostles. The thirteen sacred figures are having supper when it’s not at night and we can see a beautiful Catalan landscape behind the main scene. If Dalí wanted to capture this Biblical moment, why the dodecahedron was painted over it? The dodecahedron and its twelve sides may be related to the twelve apostles. Furthermore, a philosophic point of view can show to us a platonic idea for this famous painting. Plato (c.428 BC-c.347 BC) wrote in the IVth century BC the Timaeus where he associated the four basic elements to fours regular polyhedra: tetrahedron was the fire, hexahedron was the earth, octahedron was the air and icosahedron was the water. Thus, the universe was in perfect harmony with Mathematics!
Firstly, Plato described two kinds of triangles as the base of all the nature:
In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes ; and every plane rectilinear figure is composed of triangles ; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles ; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies
With these two triangles, Plato built the squares and triangles which had to be the sides of the regular polyhedra:
Now is the time to explain what was before obscurely said : there was an error in imagining that all the four elements might be generated by and into one another ; this, I say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds: three from the one which has the sides unequal ; the fourth alone is framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. […]. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles ; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles ; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more ; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles ; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases.
And then…
There was yet a fifth combination which God used in the delineation of the universe.
Great Plato! Which is this fifth combination? The answer is now so easy: the dodecahedron! The platonic solid which was used by God to draw our universe. It’s easy to understand the reason why this harmonious figure is presiding the scene.
Location: National Gallery of Art in Washington (map)
THE MATHEMATICAL TOURIST 