Tag Archives: Euclid

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.

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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)

Three more mathematical documents in the Neues Museum

Photography by Carlos Dorce

The Berlin Papyrus 6619 s not the only mathematical “paper” of the Neues Museum of Berlin because we can see two more documents on exhibition. The first of them is a Greek papyrus (139 AD) with some geometrical problems and their solutions (first picture).
The second is a table with Greek fractions from the Byzantine epoch (7th century):

Photography by Carlos Dorce

Finally, we find this ceramic piece which is part of a more complete catalogue composed of pieces P 11999, P 12000, P 12002, P 12007, P 12008, P 12609 and P 12611 from the 3rd and 2nd centuries BC, all of them found in Elefantina.

Photography by Carlos Dorce

These pieces contain one of the most difficult problems of the Greek mathematics: the construction of a regular icosahedron. This P12609 was translated and analised by Jürgen Mau and Wolfgang Müller (`Mathematische Ostraka aus der Berliner Sammlung’, Archiv für Papyrusforschung XVII (1962), 1-10.), and we find some words which help us to understand the text. For example, word σφαιρας suggests that we are studying a tridimensional figure, τριγωνων πλευρον refers to equilateral triangles and δεκαγων is used by Euclid in some propositions of the Elements.
We can think about the transmission of the Greek science from Alexandria to other Greek cities because of these pieces were found in Elefantina and not in Alexandria. After Euclid’s Elements, the only reference to a construction of a regular icosahedron is found in the work of Hypsicles (c. 190 BC-c.120 BC) who explains that his father and Basilides of Tyrus discussed in Alexandria about Apollonius’ construction of the regular dodecahedron and icosahedron.
LocationNeues Museum (map)

Pythagoras in Temple Bar Moument

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Temple Bar (map)

Omar Khayyâm’s doodle

Omar Khayyam was born in Nishapur on May 18, 1048. He went to school and two of his best friends there were Nizam al-Mulk and Hassan al-Sabbah. The three boys agreed that the first of them who could get a rich position will help his friends in a future. Al-Mulk became vizier in Alp Arslam’s palace and al-Sabbah and Khayyam also benefited of his nrew position although Khayyam got a job which let him to study Astronomy, Literature and Mathematics. Khayyam’s life wasn’t easy but his astronomical studies and his participation in the reform of the calendar were so decisive that he always had a city or a place where being able to life and work. According to al-Arudî al-Samarcandî, Khayyâm died on December 4, 1131.

Stamp issued by Dubai

Khayyâm studied Euclid’s Elements and Data, Apollonius’ Conics and al-Khwârizmî’s Algebra and wrote his major work on Algebra around 1074 whre was able to solve geometrically the cubic equation. The treatise begins with three introductory lemmas:

  1. To find two segments x and y which a/x = x/y = y/b (Khayyâm finds a point (x,y) which is the crossing point of the parabolas x2 = ay and y2 = bx).
  2. To determine the height of a parallelepiped with known squared base b if we know that its volume must be the same as another parallelepiped with squared base a and height (Khayyâm determines a line k such that a/b =b/m and the searched height K is k/a = h/K).
  3. To determine the side of the base of the second parallelepiped.

Khayyâm solves fourteen canonic cubic equations (he didn’t know the negative numbers!) from these three geometric lemmas. For example:

Lemma 1. From lemma 1, we can find x and y such that 1/x = x/y = y/c and this point (x,y) satisfies x3 = c.

The other thirteen cases are solved crossing parabolas, circles and hiperbolas.

In 1857 E.Fitzgerald discovered Khayyam’s Rubâyyât in the British Museum and trabslated some verses from this “new” manuscript. The translation to English was so popular since 1861 and the Khayyâm’s name was very famous in the literary circles. The Rubâyyât contains over 400 quatrains written in Persian and were translated to English again in 1970s by Robert Graves:

Wake! for Morning in the Bowl of Night

Has flung the Stone that puts the Stars to Flight:

And Lo! the Hunter of the East has caught

The Sultán’s Turret in a Noose of Light

Stamp issued by the Federated States of Micronesia

Stamp issued by Guyana

Dreaming when Dawn’s Left Hand was in the Sky

I heard a Voice within the Tavern cry,

Awake, my Little ones, and fill the Cup

Before Life’s Liquor in its Cup be dry.

The mathematical doodle was published by Google in the Arabic countries two years ago to commemorate Khayyâm’s birthday.

Magritte’s Euclidean Walks (1955)

Euclidean walks (1955). Source. Wikimedia Commons

Euclidean walks (1955). Source. Wikimedia Commons

René Magritte (1898-1967) painted in his Euclidean Walks a surrealist comparison between the actual geometry of the roof of a tower and the projective illusion of a street. Perhaps, this was the reason why he thought of the great Euclid for the title of this charming and surprising picture.

Oh Euclid! If you come back to our World you’ll see what provocations are inspired by your name!

Location: Minneapolis Institue of Arts (map)

Hall of the former Faculty of Sciences in Zaragoza

Photography by Carlos Dorce

Photography by Carlos Dorce

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Photography by Carlos Dorce

Photography by Carlos Dorce

Other important mathematicians are:René Descartes…

Photography by Carlos Dorce

Photography by Carlos Dorce

…Galileo Galilei…

Photography by Carlos Dorce

Photography by Carlos Dorce

…the great Euclid…

Photography by Carlos Dorce

Photography by Carlos Dorce

…Hipparchus of Rhodes…

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Pedro Sánchez Ciruelo (1470-1550)

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Photography by Carlos Dorce

Photography by Carlos Dorce

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

A Surrealist Euclid

Euclid (1945). Painter: Max Ernst (1891 – 1976). The Menil Collection (Houston)

This curious portrait of the great mathematician can be seen in the Menil Collection The first proposition in book I of Euclid’s Elements is about the construction of the equilater triangle and this could be the reason why Euclid’s face is something similar to a tetrahedron. German Max Ernst was very impressed by the surfices which he saw i his visit to the Henri Poincaré Institute of Paris and this mathematical discovery opened his mind to include geometrical things in his works. For now, if you ever visit Houston, don not miss the eccentricity of one of the most important Dadaist and Surrealist Masters.
Location: The Menil Collection in Houston (map)

A Trigonometric lesson

 

Teacher drawing a geometrical figure (1658). Painter: Ferdinand Bol (1616-1680).  Source: Wikimedia Commons

This is one of the mathematical pictures which can be seen in Louvre Museum! Dutch Ferdinand Bol painted a maths teacher explaining a lesson about Trigonometry with the aid of the famous trigonometric circle and their geometrical representation. I think that there will be a good idea travelling to Paris with all my students and explain the sinus, cosinus, tangent,… in the second floor of this wonderful museum. What do you think?

Location: Louvre Museum in Paris (map)

Aljafería Palace (Qasr al-Jaʿfariya)

Aljaferia Palace Photgraphy by Carlos Dorce

Aljaferia Palace
Photography by Carlos Dorce

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.

Source: Wikimedia Commons

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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?

Photography by Carlos Dorce

Photography by Carlos Dorce

Or praying in the octogonal Oratory?

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Aljaferia palace in Zaragoza (map)