Johannes Kepler Monument in Weil der Stadt

kepler0

Photography by Carlos Dorce

Weil der Stadt is located near Stuttgart. Johannes Kepler was born in this very veautiful town on December 27, 1571 and his memory is still there: this big statue is in the middle of the Market Square…

Photography by Carlos Dorce

Photography by Carlos Dorce

… and Copernicus, Mästlin, Tycho Brahe and Jobst Bürgi are with him in this monumental sculpture.

Photography by Carlos Dorce

Photography by Carlos Dorce

The four scientists are in the corners of the base of the statue and the words “Astronomia”, “Optica”, “Mathematica” and “Physica” are graved on each of the four sides.

Photography by Carlos Dorce

Photography by Carlos Dorce

I must say that here we have the first (imaginary) bust of Bürgi that I know. Bürgi was one of the originators of the logarithms because Kepler said that he had seen Bürgi using logarithms in astronomical calculus (Rudolphine Tables (1627)) before their “official” first occurrence in Napier’s  Mirifici Logarithmorum Canonis Descriptio (1614). Furthermore, Bürgi published his logarithms in his Aritmetische und Geometrische Progreß tabulen (1620) but his “red numbers” and “black numbers” couldn’t never win the “logarithms” which were the first calculator in all history.

Photography by Carlos Dorce

Photography by Carlos Dorce

Notice that this statue is not very similar to this other portrait from 1620:

Joost_buergi

J.Bürgi at the age of 67. Drawing by Aegidius Sadeler (1619)

The base of the statue also have four graved images representing moments in Kepler’s life like thispicture with Kepler in the middle explaining the Copernican system…

Photography by Carlos Dorce

Photography by Carlos Dorce

… with Hipparchus and Ptolemy watching how a central Sun brights in the middle of the universe.

kepler7

Can you imagine Kepler investigating about his elliptical orbits?

Photography by Carlos Dorce

Photography by Carlos Dorce

Next to Market Square there is his bithplace which hosts… no, no, no! Tomowwor will be another day!

Location: Weil der Stadt (map)

Abû al-Wafâ’ al-Buzjanî’s doodle

1075 aniversari del naixement d'Abu al-Wafa al-Buzjani

This beautiful doodle was published by google in the Persian and Arabic countries last 10th June because in 10th June 940 the great Abû al-Wafâ’ al-Buzjanî was born in Persia. Since 959, he worked in the Caliph’s court in Baghdad among other distinguished mathematicians and scientists who remained in the city after Sharâf al-Dawlâh became the new caliph in 983. He continued to support mathematics and astronomy and built a new observatory in the gardens of his palace in Baghdad (June 988) which included a quadrant over 6 metres long and a sextant of 18 metres.

Abû al-Wafâ’ wrote commentaries on works of Euclid, Ptolemy, Diophantus and al-Khwârizmî, and his works were very important in the developement of Trigonometry and Astronomy.

The conoids in Sagrada Família Schools

Photography by Àlvar Pineda

Photography by Àlvar Pineda

It is nothing new that Antoni Gaudi’s constructions are related to mathematics. In this post, we will focus on the conoid, a ruled surface that appears in Sagrada Família Schools (Escoles Provisionals de la Sagrada Família, in Catalan), near the Basílica i Temple Expiatori de la Sagrada Família in Barcelona.

First of all, we have to explain the concept of ruled surfaces: a surface S is ruled if through every point of S there is a straight line (called ruling) that lies on S. This implies that a ruled surface has a parametric definition of the form S(t,u) = P(t) + u Q(t).

Source: Wikimedia Commons

Source: Wikimedia Commons

As you can see, the roof of the schools is one ofthese surfaces which we call conoid: we can generate it by displacing a straight line above another straight line (the axis) and above a curve (often a sinusoid). Consequently, for every point on the conoid there is a straight line that passes trough that point and intersects de axis. If all of those straight lines are perpendicular to the axis, then the conoid is called right conoid. The conoid of this post is not in Sagrada Família but on the roof and the façade of the Escoles provisionals de la Sagrada Família. Antoni Gaudí designed that building on the commission of the entity that sponsored the project of the Sagrada Família, the Associació de Devots de Sant Josep (presided over by Josep Maria Bocabella (1815-1892)), and the school was for the children of the parish and also the children of the building workers of the temple. The building was divided in three classrooms, a hall and a chapel, and was constructed with brick. Its principal promoter was Gil Parés i Vilasau (1880-1936), the first parish priest of the Sagrada Família. He was also the school’s principal until 1930 and he used the Montessori method from 1915.

The building, inaugurated on November 15, 1909, has an amazing story of destructions and reconstructions. In fact, this peculiar school was intended to be demolished because Gaudí located it occupying land reserved for the construction of the Sagrada Família’s Passion façade. However, it was dismantled and rebuilt earlier than expected as a result of the several damages during the Spanish Civil War (1936-1939). Domènec Sugrañes i Gras (1878-1938) designed the restoration that finished in 1940, but the project had few funds and for this reason, in 1943 Fransesc Quintana (1892-1966) directed another refurbishment. Many years later, in 2002, the Passion façade was going to be built, so the building of the Escoles provisionals de la Sagrada Família was dismantled again and reconstructed in the corner between Sardenya and Mallorca streets, where the picture has been taken. In this regard, we can add that the building has become a small museum. It is important to note that the fact of being surfaces generated by straight lines makes the construction of the roof and the façade easier. Besides that, the profile of the roof is highly effective to drain off waterin a rainy day.

The contrast between the simplicity of the building (it was a very cheap and quickly erected structure) and its importance in twentieth century architecture is really remarkable.

This post has been written by Àlvar Pineda in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Sagrada Família Schools (map)

A Fractal garden in Barcelona

Photography by Àdel Alsati

Photography by Àdel Alsati

The Botanical garden of Barcelona, located in Montjuïc, has an extension of 14 hectares. It is specialized in the mediterranean climate and contains a wide range of plants from all over the world. Moreover, it is divided into the five main regions of the planet with this kind of weather, such as Chile, California, South-Africa, Australia and Southern Europe.

It was designed by the architects Carles Ferrater and Josep Lluís Canosa working in an interdisciplinary team whose two main priorities were, firstly, to distribute the plants so that they are placed together with the other ones of the same geographical region, and, in addition to that, that within every region, plants are disposed following their ecological affinities representing the different landscapes existing in those zones. Secondly, they didn’t want to do it making large earthworks.

Photography by Àdel Alsati

Photography by Àdel Alsati

They achieved the design of the park in a mathematical way since they designed the park following fractal structures: they split the land into triangles, so that every triangle contained the plants of a particular landscape, while each of the five regions was represented by a set of this triangles.

Map of the Botanical Garden

Map of the Botanical Garden

If we look at the zigzag shape of the path, and then at the trapezoidal pieces which constitute it, we can found a very good example of fractal geometry.

Photography by Àdel Alsati

Photography by Àdel Alsati

And… if we look more carefully, we’ll find it everywhere around us!

Photography by Àdel Alsati

Photography by Àdel Alsati

This post has been written by Àdel Alsati in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Botanical Garden in Barcelona (map)

The Fibonacci Sequence in Barcelona

Photography by Marc Adillon

Photography by Núria Casals

The picture shows the Fibonacci’s sequence monument in Port Vell in Barcelona. We can see the numbers which are part of the sequence built on the pavement and separated by a proportional distance related on the ratio between them. The Fibonacci sequence was born in a mathematical problem related to the rabbit reproduction proposed by Leonardo of Pisa in his very famous Liber abaci (1202).

Leonardo Pisano, also known as Fibonacci, was born around 1170 in the city-state of Pisa. Leonardo’s father, Guglielmo Bonacci, was a Pisan engaged in business in northern Africa, in Bugia (now Béjaïa, Algeria). This Algerian city was a very important intellectual focus during the eleventh and twelfth centuries. Therefore, it is not surprising that a businessman like Mr. Bonacci realize the possibilities that had the contact of the two cultures.

Guglielmo Bonacci put his son in a calculus positional Hindu course as well, Leonardo started to be interested about mathematics. He took profit of the frequent business trips of his father to know mathematicians of the countries they visited —Egypt, Syria, Provence, Sicily, Greece— and to make a deep study about Euclid’s Elements, that always had as a logical model of rigor and style. Therefore, it was natural that Fibonacci should have been steeped in Arabic algebraic methods, including, fortunately the Hindu-Arabic numerals and, unfortunately, the rhetorical form of expression. Fibonacci decided to write about all the knowledge that he had been collecting and that impressed him so much. He wrote it in a serie of books, the first of which was Liber Abaci (1202, reprinted 1228), Practica geometricae (1223), Liber quadratorum (1225), Flos (1225) and Epistola ad Magistrum Theodorum (1225). In Liber abaci Fibonacci explains the Arabic positional numbering system and how to read numbers, add them, multiply them… and to solve all the kind of problems which could need any trader in the Medieval Ages. The famous Fibonacci sequence is one of the problems of the chapter twelve:

How many pairs of Rabbits Are Created by One Pair in One Year

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months

Fibonacci solved the problem with the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and 377 and here one of the most famous mathematical set of numbers was introduced for the first time.

This post has been written by Marc Adillon and Núria Casals in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Pla de Miquel Taradell in Barcelona (map)

The Catalan Cane

This column measures 1 cane

This column measures 1 cane

In the corner of Santa Llúcia and Bisbe Streets, we find a corious thing that does not call the atention of any of the tourists who walk around the cathedral of Barcelona. On the romanic chapel of Santa Llúcia, erected three decades before the construction of the Cathedral, we find this semicircular column sculpted in stone on the wall which measures exactly one “destre cane”.

Photography by Robert Salla

Photography by Robert Salla

The word “cane” is engraved on the wall next to the column so everybody in the Medieval barcelona could check that this was the standard measure of longitude in the market.

A cane (from the latin qana) was an ancient unit of mesurement used on the Crown of Aragon, part of France and the north of Italy. Before the Internacional Sistem of Units it was a way to have a fixed reference of lenght. This unit was used for building specific sticks of wood that were used on the market tents to have a reference when they were selling. In Barcelona, it was equivalent to eight palms, six feet or two steps, that is about 1.55 meters, although it wasn’t exactly the same measure everywhere. For example, in Tortosa it was equivalent to 1.59 meters but the reference to the whole Catalan countries was the same as in Montpellier, equivalent to 1.99 meters. Furthermore there were the square cane which was used to measure surfaces: in Barcelona it was equivalent to 2.44 square meters, 2.42m. in Girona , 2.43m in Tarragona and in 2,45m in Mallorca. Surprisingly we see that there is less diference between the square canes than in the lenght measures. As we’ve said, it was the unit of longitude used in the markets next to the cathedral in the Middle Ages althought the Catalan “destre cane” was also used. It was equivalent to twelve palms and this is exactly the height of the column that we find next to the cathedral.

This post has been written by Ander Castillo and Robert Salla in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Carrer de la Pietat 2 (map)

The Padovan Sequence in Menorca

Photography by Sara Bagur

Photography by Sara Bagur

This sculpture was designed by the students and teachers (Lluís Cervera Pons and Gabriel Seguí de Vidal) of IES Pascual Calbó i Caldes to promote Maths in the population of Menorca. It represents a spiral of equilateral triangles with side lengths which follow the Padovan sequence and the “plastic number”

The plastic number is the unique real solution of the equation x3 = x + 1 which is:

plastic01

This number is approximately equal to ψ=1,324717957244746025960908854…

The other two complex solutions are:

plastic2

ψ is also the limit of the sequence ol ratios P(n+1)/P(n) where P(n) is the Padovan sequence (which is named after the architect Richard Padovan (born in 1935)). The Padovan sequence is the sequence of integers P(n) defined by the initial values P(1)=P(2)=P(3)=1 and the recurrence relation P(n)=P(n-2)+P(n-3). According to Padovan, the plastic number was discovered at the same time by Hans van der Laan.

In the Van der Laan Foundation webpage we can read:

The plastic number

[…] Through experiments with pebbles and then with building materials, he discovered a ratio he called the plastic number. The basis for the plastic number is the relationship between measures belonging to a group of measures. They increase or decrease according to the ratio four to three. The parallel in music is the ratio that relates whole and half notes to each other within an octave. The analogy between the plastic number and music goes even further: in music we can play chords, combinations of tones, with the plastic number we can compose walls and rooms and spaces that are in harmony with each other because they relate to each other as objectively as the tones within a musical key. The plastic number is not a particular measure: it disciplines the relationship between the measures we choose.

Van der Laan expressed his thoughts on the relationship between architecture and nature in this way:

If the body function of the House consists of establishing harmony between the body and its natural environment, the expression of that function will be based on the harmony between the wall that separates and the separate space. It becomes thus registration extension appreciable by the senses, both space-enclosing the separator element. That harmony will depend on the mutual proportions, who speak to the intelligence through the target language of the plastic number and that are established by the rules of the corresponding architectural ordering.

This post has been written by Bara Bagur in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: The Padovan Sequence sculpture (map)

A solar “eight” in the Cathedral of Palma

Source: Wikimedia Commons

Source: Wikimedia Commons

Palma’s cathedral, also known as La Seu, like almost every building of this kind hide curiosities and mathematical facts that surprise due to the construction’s age. We can find trigonometry relations in a common cathedral or church, some of them even with a religious meaning behind. But in La Seu we can find two particular incidents that catch our eye. The first one happens every 2nd of February (2/2) and 11th of November (11/11), when the sunbeams go throw the east-faced rose window (known as Oculus Maior) and they are projected under the west-faced rose window, tangentially, in such a way that their centers lie on a line that is perpendicular to the ground.

Sorice: XEIX webpage

Source: XEIX webpage

It is not a coincidence the days in which the event takes place, as they are both in a similar position regarding the winter solstice. Thanks to this light  effect and the data currently available on the internet, we can easily calculate La Seu’s orientation. When this phenomenon takes place, the direction of the sunbeams coincide with the nave’s orientation. Then, we only have to set the time and figure out the exact azimuth considering this time and geographic situation. With some of the available programs, we can find out that it has a value of 122,4º and the angle of solar elevation is 10,2º (both calculus with an error smaller than 0,5º!).

Source: XEIX webpage

Source: XEIX webpage

The other incident takes place during the days near to the winter solstice. We can stare at the sunrise going through both rose windows causing a rather impressive light effect. Then again, we can find the value of the azimuth which results in 120,3º. Curiously, the bell tower, which has a square base, doesn’t have an axis of symmetry parallel to the central nave, they are out of place about 10º from each other.

Source: XEIX webpage

Source: XEIX webpage

Like many others cathedrals and churches, this one is built on an ancient mosque. In this culture, it was very important to have the ”qibla” oriented towards Mecca, specifically, towards the Ka’ba, following the precepts of the Koran. The muslim domination in Mallorca happened between 903 and 1231. In this period of time, the solution of the “qibla problem” was already known, solved by al-Khwârizmî in the 9th century. In effect, if one traces the line segment bisector of the east-faced bell tower and lengths it over the terrestrial sphere it matches the Ka’ba with an astonishing precision.

You can read more information about these two facts in this interesting article.

This post has been written by Aina Ferra in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Cathedral of Palma (map)

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)

A very famous magic square in Barcelona

Source: Wikimedia Commons

Source: Wikimedia Commons

If you ever visit one of the biggest Gaudí’s (1852-1926) architectural achievement in the beautiful city of Barcelona, the world-wide known and still under construction Sagrada Família, and you are passionate about maths, you might want to take a closer look at the sculpture of Judas’ betrayal; right by its side you can find embedded on the Sagrada Família’s Facade of Passion a 4×4 matrix, known as the magic square.

Magic squares are square matrices with feature integer numbers, which add up to the same amount in columns, rows and diagonals. That amount is known as the magic constant and the one concealed in Sagrada Família is the number 33. Check it out:

Source: Wikimedia Commons

Source: Wikimedia Commons

Notice that there are more combinations which add up to 33. For example, sum the red numbers and the green ones in each square:

SF3

This magic square is also included as a decoration in one of the main doors of the Passion Façade. Can you find it?

And what does it stand for? While some people argue it might have something to do with the highest degree in the Masonic lodges – and consequently relates the architect to Freemasonry – the truth is that its author is Josep Maria Subirachs (1927-2014) (Catalan sculptor famous for this design) who chose the number 33 since it’s the age at which Jesus died on the Cross. Whether or not there might be other curious legends surrounding it, it’s remarkable how maths has found room in such a masterpiece. For those who never got along with numbers, here they have a whole new and much more artistic rather than scientific perspective that might light up their face whenever they come across a magic square. Thus everyone can fully enjoy the world of maths!

This post has been written by Carles Raich in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Sagrada Família in Barcelona (map)

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