Monthly Archives: April, 2014

Hamilton’s quaternions in Broombridge

Hamilton plaque at Broombridge. Source: ingeniousireland.ie

Sir William Rowan Hamilton was walking through this brisge in Dublin on the 16th October, 1843, when he has one of his wonderful ideas: his quaternions. Nowadays, this very important moment in the History of Mathematics is recorded on a plaque located in the North-West corner of the bottom of the bridge. We can read:

Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
i² = j² = k² = ijk = −1
& cut it on a stone of this bridge.

William Rowan Hamilton (midnight, 3-4 August 1805 – 2 September 1865)

I’ve never been in Dublin yet but I am sure that I am going to visit it very soon… very soon.

Location: Broombridge in Dublin (map)

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National Library of Belarus

This emblematic and polyhedrical building was designed by Viktor Kramarenko and Mikhail Vinogradov and was finished in 2006. This rumbicuboctahedron is 72 m heigh so we are probably talking about one of the tallest polyhedra ever constructed!

Location: National Library in Minsk (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)

Mental Arithmetic in Russia

Mental Arithmetic. S.A.Ratxinski’s Public school (1895)

Nikolai P. Bogdanov-Belsky (1868 – 1945) painted this classroom with eleven students resolving the problem written on the blackboard: (102 + 112 + 122 + 132 + 142)/365. Professor  Sergei Ratxinski is in a poor public school because of kids’ clothes and he’s listening… what? What is this little blond boy saying? He has probably discovered that 102 + 112 + 122 = 132 + 142.  In this case, the solution is very simply!

Location: Tretiakov Gallery in Moscow (map)

Archimedes in Dresden

Archimedes thinking (1620). Painter:  Domenico Fetti (1588 – 1623)

This picture is in the Gemäldegalerie Alte Meister of Dresden. Archimedes (c. 287 aC – c. 212 aC) is in the middle of a brainstorm reading one of his papers! A compass and a ruler remember us the Geometry and the mirror could be an allegorical detail of his inventions which defended Siracussa from the Roman invaders. There also are a sand clock (the Sand Reckoner?) and a terrestrial globe. As you can see, Rafaello Sanzio’s angels aren’t the only brightning star of the Museum!

The Pythagoras theorem in Berlin

Photography by Carlos Dorce

Photography by Carlos Dorce

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).

Photography by Carlos Dorce

There are some daily pictures in this mosaic in which we can noice a teacher rounded by his four pupils. He marks a point in Africa in a globe meanwhile two students look through a microscope and a telescope, another observes the liquid in a test tube and there is another listening teacher’s lesson. In the background, we can see a Pythagoras theorem in the blackboard.

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

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: map

A cuneiform tablet in Pergamon Museum

Photography by Carlos Dorce

Photography by Carlos Dorce

This map is dated inthe 21st century BC and can be seen in the Pergamon Museum of Berlin. As you can see, there is a plane of some lands and the cuneiform figures tell us the dimensions of it. Of course, this is a small piece of clay rouded by the great Mesopotamian treasures that you can admire in this wonderful museum but… can we forget focussing our attention in this cuneiform tablet? I don’t think so!

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Pergamon Museum in Berlin (map)

Zu Chongzhi’s birthday

Zu Chongzhi (429-501) is one of the greatest ancient Chinese mathematicians. He was taught Mathematics from Liu Hui’s commentary on the Nine Chapters on the Mathematical Art. He was appointed as an officer in the city of Yang-chou and during this time he worked in Arithmetic and Geometry. He gave the rational 355/113 as an approximation of pi in his Zhui shu (“Method of interpolation”) and proved that:

3,1415926 < pi < 3,1415927

Zu also proposed a new calendar (the “Calendar of Great Brightness”) in 462 based on a cycle of 391 years with 144 extra months inserted (=4.836 months).

In the Wikimedia Commons there is this photo of a statue dedicated to him in Tinglin Park in Kunshan.

Source: Wikimedia Commons

Google dedicated this doodle five years ago to celebrate his birthday.

Location: Tinglin Park in Kunshan (map)

The first stellation of the octahedron

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Source: Wikimedia Commons

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

Photography by Carlos Dorce

Photography by Carlos Dorce

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 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)