Tag Archives: Maestlin

The golden number in Cosmocaixa

Photography by Pol Casellas and Eric Sandín

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

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)

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

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

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)

Advertisements

Immortal Books, essential instruments (II)

Photography by Carlos Dorce

Photography by Carlos Dorce

The Astronomical Revolution is visited after the Greek books and Copernicus (1473-1543) and his De Revolutionibus orbium coelestium are the next couple to study:

He was born in Poland in a very rich family. His parents died and his uncle (bishop of Warmia) took care of him. He went to the University of Krakow and he studied Canonic Law in Bologna some years later. He was under the Italian Humanism there and he began to have interest for Astronomy. He completed his studies and also Mechanics in Padova and read his doctoral dissertation in Canonic Law in the University of Ferrara. After this, he came back to his country and entered the Bishop’s court. In 1513 he wrote the Commentariolus – manuscript which circulated anonymously- where astronomers could read his new astronomical system. He was invited to reform the Julian calendar. He wrote his great work De Revolutionibus Orbium Coelestium inthe last days of his life and he defended the heliocentrical hypothesis in it. His disciple Rheticus brought a copy of the manuscript to the printing in 1542 and it was published in 1543. Copernicus died in Frombork and his theory was condemned by the Church in 1616 and was in the List of Prohibited Books until 1748.

I think that I’m going to go to Poland next holidays!

One of the most important followers of the heliocentrism was Johannes Kepler (1571-1630):

Photography by Carlos Dorce

Photography by Carlos Dorce

The scientist who opened the way to the modern astronomy was born in Weil der Stadt, Germany. He suffered from myopia and double vision caused from smallpox and this wasn’t a problem for him to discover the laws which explain the movements of the planets around the Sun. He studied Theology in the University of Tubingen under his teacher Michael Mastlin and he soon noticed his unusual skills reading Ciopernicus’ heliocentrism. He mainly lived in Graz, Prague and Linz. He met Tycho Brahe in Prague and some years later he became Imperial Mathematician under Rudolph II’s protection. It wa sin this period when he developed his great works: Tabulae Rudolphinae and Astronomia Nova (1609). In Astronomia Nova he explained two of the three fundamental laws describing the movement of the planets; the third one was explained in Harmonices Mundi Libri V (1619). Kepler was the first scientific in needing phisician demonstrations to the celestial phenomena.

Who is the next? Galileo (1564-1642), of course!

Photography by Carlos Dorce

Photography by Carlos Dorce

His book is the Dialogo sopra i due massimi sistemi del mondo Tolemaico, e Copernicano (1632). In this book he defended the Copernicanism against the Ptolemaic system although the book was prohibited by the Inquisition and he was condemned to house arrest.

Galileo died in 1642 and Newton (1642-1727) was born some months after his death. His Philosophiae Naturalis Principia Mathematica was one of the most important scientific books of all the History of Science. I am not going to talk about Newton and his book after my visit to Englang last holidays but here you have his portrait:

Photography by Carlos Dorce

Photography by Carlos Dorce

The other scientists of this epoch are Vesalius (De humani corporis fabrica), Harvey (Exercitatio anatomica de motu cordis et sanguini), Linneo (Systema naturae) and Hooke (Macrographia):

Photography by Carlos Dorce

Photography by Carlos Dorce

There is another important mathematician from the 17th century but… it will be presented tomorrow!

Location: MUNCYT in Madrid (map) and MUNCYT in A Coruña (map)