# Ptolemy and Pythagoras in Ulm

The cathedral is one of the main attractions of Ulm because its tower is the tallest in the World (more than 161 metres!). However, one of the most interesting work of art which can be admired inside the church is the 15th century choir stalls crafted by Jörg Syrlin the Elder. There are 89 seats arranged in two rows with 90 busts of saints, Old Testament figures and classical philosophers and scholars as Ptolemy…

…and Pythagoras:

We must remember that Ptolemy represent the Astronomy in the Liberal Arts and Pythagoras usually represents the Arithmetic although his bust here is related with the Music.

Most of the people who visit the cathedral don’t know that this is one of the most wonderful medieval work of art which can be seen in Germany although this beautiful picture woul remain in their minds in a lot of years.

**Location**: Cathedral of Ulm (map)

# The golden number in Cosmocaixa

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:

Of course, one of the solutions of *x*^{2} = *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.

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.

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.

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

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.

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).

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:

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 thehierarchized 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…

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.

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.

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)

# Hewelianum Centre in Gdansk

I visited the Hewelianum Centre when I was in Gdansk and I discovered a new science museum which must be located in all the tourist guides:

The Hewelianum Centre is an educational and recreational centre for all age groups situated on the grounds of the Fort Góra Gradowa. The view from the top of the hill is the panorama of the historic town and the industrial landscape of the shipyard grounds. A picturesque park and a complex of restored 19th-century military remains hosting interactive exhibitions – this is today’s image of the Fort of Góra Gradowa.

Science popularization is the main objective of the Hewelianum Centre. Interactive and multimedia exhibitions and popular science events disclose the mysteries of physics and astronomy, transfer the visitors to the past, making the historic events better understandable in the present, teach how to be sensitive to the beauty of nature, and strengthen in visitors the belief that we are all responsible for our planet. In Hewelianum Centre you can perceive the world, learn about it, and relax yourself in an interactive, creative, and innovative way!

One of the exhibitions is called “Puzzle” (why not “Maths”?) and it’s a place where people can play with Mathematics:

Break the code and discover a new dimension of mathematics!

The “Puzzle” exhibition is a three-dimensional space: mathematical, interactive, and unconventional. It consists of more than 20 stations for experimenting – where mathematics governs, but in an unprecedented way!

By crossing the mathematical “puzzle” threshold, we enter the world of geometry, symmetry, and numbers. The mathematical setting, however, is only a backdrop for interactive learning and fun. A collection of the exhibition’s main attractions includes the multiplication tower, the Pythagorean theorem in liquid form, and the Möbius strip. Here you can also see what your face would look like if it were composed of two left or two right halves or check whether a meter is the same length for all. Visiting the mathematical “Puzzle” is a perfect idea for a unique scientific experience.

The exhibition is located in the Guardhouse over the Mortar Battery postern

The room is small but all the walls and corners are full of Maths experiments:

For example, there is a Galton box (or Bean machine) where Pascal’s triangle and the Gaussian function can be observed perfectly.

You can also play with the Towers of Hanoi and discover that the minimum number of moves required to solve the puzzle is 2^{n} – 1, where *n* is the number of disks (this problem was first publicized in the West by Édouard Lucas in 1883):

Did you know that it’s possible to construct a byke with squared wheels? Yes, of course. The path for this bike must be formed by contiguous series of inverted catenaries!

And had you ever seen such a wonderful way to demonstrate the Theorem of Pythagoras? Water inside the square constructed on the hypothenusa fills perfectly in the two squares constructed on the other two sides:

Obviously, there are Möbius strips and Klein’s bottles:

And you can play with the light to discover the four conics:

There are poster about a lot of mathematical subjects but tha puzzle that fascinatd so much to my son and daughter was this experiment with volumes. They discovered that the volume of a prism is three times the volume of the corresponding pyramid although they played with the red sand preparing cornflakes for breakfast!

If you visit Gdansk you must go to Hewelianum Centre and really enjoy Maths!

**Location**: Hewelianum Centre in Gdansk (map)

# Pythagoras in Temple Bar Moument

The Temple Bar Memorial (1880) stands in the middle of the road opposite Street’s Law Courts marking the place where Wren’s Temple Bar used to stand as the entrance to London from Westminster.

The monument has two standing statues dedicated to Queen Victoria and the Prince of Wales because both were the last royals to pass through the old gate in 1872.

The reliefs round Queen Victoria contains some allegories which includes the first picture about the Euclidean demonstration of the theorem of Pythagoras. We also find a ruler and a globe with the ecliptic.

**Location**: Temple Bar (map)

# Hall of the former Faculty of Sciences in Zaragoza

One of the most beautiful buildings which can be visited in Zaragoza is the hall of the former Faculty of Sciences. Itwas constructed by Ricardo Magdalena in 1893 and is decorated with 72 statues and roundels designed by Dionisio Lasuén (1850-1916). These allegorical sculptures are dedicated to Medicine and Science and we find some very important mathematicians among all the scientifics represented on them. For example:

We also fins a representation of the Theorem of Pythagoras next to these two great names:

Other important mathematicians are:René Descartes…

…Galileo Galilei…

…the great Euclid…

…Hipparchus of Rhodes…

We also find Spanish scientific and mathematicians as the Andalusi Abû al-Qâsim al-Zahrawî (Al-Zahra, Cordova,936-Cordoba,1013), also known as Abulcasis. He was an important physician, surgeon and doctor who wrote the *Kitab at-Tasrif* (Arabic,**كتاب التصريف لمن عجز عن التأليف**) or *The Method of Medicine *(compiled in 1000 AD) which had an enormous impact in all Medieval Europe and the Islamic World.

Pedro Sanchez Ciruelo (Daroca,1470 – Salamanca, 1550) was an important Spanish mathematician of the 16th century who wrote some mathematical treatiseslike the *Cursus quattuor mathematicarum artium liberalium *(1516) thorugh which Bradwardine’s Arithmetic and Geometric work was taught in Spain.

Jorge Juan (1713-1773) and Antonio Ulloa (1716-1795) were two Spanish scientifics who participated in the measurement of the Terrestrial Meridian organized by the Academy of Sciences of Paris:

Gabriel Ciscar (1759-1829) wrote the* Curso de Estudios Elementales de la Marina,* divided in a volume dedicated to Arithmetics and another dedicated to Geometry.

Finally, José Rodríguez González (1770-1824) and José Chaix (1765-1811) participated in the triangulations of the meridian arc from Dunkerque to Barcelona.Furthermore,Chaix wrote the *Instituciones de Cálculo Diferencial e Integral* and publicó the *Memoria sobre un nuevo método general para transformar en serie las funciones trascendentes* which were so popular in Spain because of the explanations of the differential calculus.

So, the building is so beautiful and you can learn History of Mathematics while walking around it. Do you want anything else?

**Location**: Hallof the Faculty of Science in Zaragoza (map)

# The Museum of Mathematics of Catalonia (MMACA)

Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.

The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…

For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.

Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!

Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.

Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!

Another example: look at these three wooden pieces…

The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:

There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.

And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?

In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:

Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his* Zîj-i Ilkhanî* (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his *De Revolutionibus* (1543). Al-Tûsî’s pair was very famous until the 15th century.

In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:

I didn’t know that it could be described only with a multiplication table! Is its equation *z = xy*? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.

Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:

Furthermore, one of the rooms of the palace is decoratd by a chess lover!

So… you must go to MMACA and enjoy Mathematics in a way ever done!

# The Pythagoras theorem in Berlin

Alexanderplatz is one of the most emblematic squares in Berlin. It’s a very importnat center of communications and transponts of the German capital although it was created for hosting a cattle market until the end of the 19th century. The current square was designed by the former DDR rulers in the 60’s and nowadays it’s always full of people eating, buying, walking or simply having a beer or a coffee.

If you walk to Alexanderstrasse 9 you will see the great mosaic in the Haus des Lehrers (the Teachers’ House).

The rest of the mosaic is wonderful. It’s a pity that it was only a propagandistic set of pictures!

**Location**: map

# A mathematical tomb in Bath Abbey

This musical tomb or burial monument is located in one of the inner walls of Bath Abbey. Unfortunately, there are some needless pictures which cover one part of it but we can notice our mathematical interest in the carved drawing. It consist in an organon with a curious rectangular triangle in its base:

Harmony represented by the theorem of Pythagoras! The triangle is the classical (3,4,5) in its version (60, 80, 100) so it’s easy to identify the longitudes of the rest of the segments. The grave has the name of the composer Henry Harington who also studied medicine and founded the Bath Harmonic Society. We can read:

Memoriae sacrum HENRICI HARINGTON. M:D: verè nobili HARINGTONORUM stirpe de Kelston. In agro Somerset: oriundi: Qui natus Septembris 29 A.D. 1727, obiit Januarii 15. A.D. 1816. Per sexaginta annos suae Bathoniae saluti. Omnibus officiis afsidue studebat. Optimas artes ad municipum suorum. Nelectacionnem et militatem excolens…

One of the books in the right lower corner of the picture is from Euclid:

Bath Abbey is a wonderful construction and it’s possible that almost everybody don’t notice this particular monument. So you have a very good oportunity to admire a mathematical objecti inside a church. If not, you can always follow the angels climbing on its beautiful facade:

**Location**: Bath Abbey in Bath (map)