# The Padovan Sequence in Menorca

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 *x*^{3} = *x* + 1 which is:

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

The other two complex solutions are:

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 B ara 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

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.

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!).

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.

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 “Taula” in Torretrencada

The name of the Talaiotic culture comes from conic towers built with stones, probably used as a dwelling, watch towers and defense towers. These tables (“Taules” in Catalan) consist in a vertical rectangular stone and another one placed horizontally on its top, so the name of the table is given by the form of “T”. But… why these old monuments are mathematic? The front of most of them is oriented to the south! This orientation is related to the possible use as calendar in this former culture. The construction of the first monuments in Balearic islands dates from the end of the 2nd millenium BC to the beginnings of the 1st millenium BC. At this moment, these monuments began to proliferate on Mallorca and Menorca (there are 31 only in this small Mediterranean island!) appearing in isolated fashion as a territorial boundary stone.

The tables served as sanctuaries next to other monuments and all of them were built in almost the same latitude (and longitude?). For example, Sa naveta des Turons (latitude = 39.99º and longitude = 3.93º), Torretrencada (latitude = 40.003º and longitude = 3.89º) and Torre d’en Gaumès (latitude = 39.93º and longitude = 4.12º) seems to be aligned!

In 1996, Vicente Ibáñez Orts published his hypothesis on the Table explaining that their design was very well computed and not the result of chance. Regarding Torretrencada, it seems that the monument was built from some mathematical computation indicating that Talaiotic men had a system of writing numbers and a deep knowledge of arithmetic and geometry)

*This post has been written by Laura Barber and Anabel Luís in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).*

**Location**: Torretrencada (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 first stellation of the octahedron

This first stellation of the octahedron can be seen in Plaza Europa in Zaragoza.

As you can see, the obelisc is rounded by these polyhedrical lights and also bysome little stellations more.

So here you have a very good mathematical complement if you visit Aljaferia Palace in Zaragoza because this rounded square is next to it!

**Location**: Europa Square in Zaragoza (map)

# Aljafería Palace (Qasr al-Jaʿfariya)

Aljaferia Palace is one of the most beautiful Islamic palaces which can be visited in Spain. It was built in the second half of the 11th century in the Moorish taifa os Saraqusta (present day Zaragoza) by the King al-Muqtâdir Bânû Hûd.

I’m sure that you are wondering why I am talking about this building now. The building is wonderful but this is not the reason. Do you know who King al-Mu’tamân is? No? King al-Mu’tamân (1081-1085) grew in this palace and was educated under teachers and philosphers. Before 1081, he began to write an encyclopaedic work about Mathematics (*Kitâb al-Istikmâl* or Book of the Perfection) with his collaborators’ contributions. Al-Mu’tamân wanted to write the most important mathematical treatise until that time. Only four hundred propositions about Classic Geometry have survived: some results from Euclid’s *Elements* and *Data*, Apollonius’ *Conics*, Archimedes’ *On the sphere and the cylinder*, Theodosius’ *Spherics*, Menalaus’ *Spherics* and Ptolemy’s *Almagest*. There also are Arabic contributions as Thâbit b. Qurra’s treatise on amicable numbers, some of the Bânû Mûsâ’s works, Ibrâhim b. Sinân’s *The Quadrature of the Parabola* and Ibn al-Haytham’s *Optics*,* On the Analysis and the Synthesis* and* On the given things*. One of the most interesting results is the demonstrarion of Ceva’s Theorem (attributed to the Italian mathematician Giovanni Ceva (d. 1734) ). Unfortunately, al-Mu’tamân became King of Saraqusta in 1081 and the* Book of Perfection* was never finished so the sections about Astronomy and Optics weren’t writen. The *Book of Perfection* was commented by Maimonides (1135-1204) some years later.

In 1118 King Alfonso I of Aragon conquered Zaragoza and after a lot of years, the palace became the royal residence. Nowadays, we can visit most of its rooms included Catholic Monarchs‘s throne room. Can you imagine young al-Mu’tamân playing with his friends in this idilic place?

Or praying in the octogonal Oratory?

Visiting the Palace, we can see a very good quotation about the importance of the Geometry in the Islamic art:

The preference of the Islamic culture for abstract art developed a type of decoration based on geometric order, its main argument being repeated themes and the objective of suggesting infinity. Of great importance in this concept was the development of mathematics in the Muslim civilization, which were then skillfull applied to construction and decoration. Starting off with a few examples of symmetry, Hispano-Muslim and then Mudejar art was capable of developing complex decorative themes that were always based on repetition.

**Location**: Aljaferia palace in Zaragoza (map)

# Immortal Books, essential instruments (and III)

Descartes (1596-1650) and his *Discours de la méthode* (1637) is exhibited here although there is no word about his *Géométrie*. I could cry for it!

D’Alembert and Diderot are also here but the important D’Alembert’s mathematical works are not mentioned either. They compiled the first *Encyclopèdie ou Dictionnaire raisonné des sciences, des arts et des métiers *(1751-1772) and we have to settle this:

Finally, a little mention to Albert Einstein (1879-1955) and his *Über die spezielle und die allgemeine Relativitätstheorie: Gemeinverständlich* (1917):

As you can see it’s a very good opportuniry to learn some things about all these great scientists and their works! The other scientist are Lavoisier, Lyell, Darwin, Bernard, Maxwell, Ramón y Cajal, Curie, Dirac and Morgan.

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

# Immortal Books, essential instruments (II)

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 workDe Revolutionibus Orbium Coelestiuminthe 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):

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 RudolphinaeandAstronomia Nova(1609). InAstronomia Novahe explained two of the three fundamental laws describing the movement of the planets; the third one was explained inHarmonices 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!

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:

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*):

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)

# Immortal Books, essential instruments (I)

This post is about a very interesting exhibition about 26 selected scientific books which I visited in Madrid in August and it can be visited now in A Coruña (from the 17th October). There are explanation of the 26 books and their authors and I am going to talk about the mathematical ones (of course!). Furthermore, there are Eulogia Merle‘s drawings of every scientist exhibited here so this is another interesting attraction to visit it.

The first great mathematician is Euclid (c.295 BC).

[In Spanish:] Es difícil precisar datos de la biografía del más destacado matemático de la antigüedad grecolatina, considerado el Padre de la Geometría. Solo se conocen con certeza dos hechos indiscutibles: vivió en una época intermedia entre los discípulos de Platón y los de Arquímedes, y formó una gran escuela de matemáticas en Alejandría. Según el filósofo bizantino Proclo, Euclides enseñó en esta ciudad del delta del Nilo durante el mandato de Ptolomeo I Sóter, es decir, entre los años 323 y 285 a.C. Murió en torno al año 270 a.C. Su fama radica en ser el autor de los

Elementos, un tratado de geometría que ha servido de libro de tecto en la materia hasta comienzo del siglo XX. Está compuesto por trece libros que tratatn de geometría en dos y tres dimensiones, proporciones y teoría de números. Presenta toda la geometría basándose en teoremas que pueden derivarse a partir de cinco axiomas o postulados muy simples que se aceptan como verdaderos.

There are two different digital editions of the *Elements* and a compass from the 16th or 17th century with all this information:

The next Greek mathematician is Archimedes (287-212 BC) although his book here is *On the floating bodies* which is less mathematical than phisician.

Ptolemy (2nd century) is the next and his *Almagest* was the most important astronomical book since the 16th century.

There is also an interesting wooden astrolabe from 1630 (“Claudii Ricchardi”):

Arsitotle, Hippocrates and Pliny the Younger are the other three Greek scientists represented in the exhibition.

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

# The frontispiece of Museo del Prado

The forntispiece of Museo del Prado of Madrid is full of allegorical figures of the muses and the arts. If we watch it carefully, we’ll notice Urania with a compass and a globe in her hands counting on a parchment:

The building was designed by the architect Juan de Villanueva (1739-1811) and it had to host the Royal Observatory, a Science Room, the Botanic Gardens, schools, laboratories,… The Spanish king yhought that it could be a very good example of the new illustrated Spain. However, it never was used in this way:

Nowadays thousands of tourists visit the pictures in the Museo del Prado and only a few ones visit outside the building. Among all the statues which decorate this neoclassical structure there are the Architecture…

…and the Symmetry:

There are also some medallions with busts of famous Spanish scientist and writers on each of these statues. Of course, Juan de Herrera is also here:

**Location**: Museo del Prado (map)