Tag Archives: Archimedes

Some caricatures of famous mathematicians

Archimedes

Archimedes

In my last post about the Hewelanium Centre of Gdansk, I must show you the caricatures of the famous mathematicians and astronomers which you find on the walls (and you also can buy as a puzzle in the shop of the museum). You have pictures of Archimedes, Pascal, Copernicus:

Copernicus

Copernicus

Halley and Hevelius:

Halley and Hevelius

Halley and Hevelius

Galileo:

Galileo Galilei

Galileo Galilei

Sir Isaac Newton:

Newton

Newton

and Albert Einstein:

Einstein

Einstein

These aren’t good pictures but the posters are in 3D and my camera is not the best camera in the World!

Location: Hewelianum Centre in Gdansk (map)

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Playing with Maths in the Hewelianum Centre

Photography by Carlos Dorce

Photography by Carlos Dorce

The “puzzle” exhibition isn’t the only place in the Hewelanium Centre where you can discover mathematical facts. For example, in the exhibition about the History of the Centre there are cannons in a defensive fortress with which you can learn a lot about parabolic shots…

Photography by Carlos Dorce

Photography by Carlos Dorce

…or how many cannonballs you have in a pyramid… Is Kepler’s theorem right? Do you think about a better way of stacking cannonballs?

There also is space for optical illusions, technology,… and a very modern Archimedes screw:

Photography by Carlos Dorce

Photography by Carlos Dorce

You can also play with the Galilean experiments about movement and see how a piece of wood climbs a path down:

Photography by Carlos Dorce

Photography by Carlos Dorce

In a hidden corner of the museum, a sextant tells you goodbye:

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Hewelianum Centre in Gdansk (map)

Copernicus Science Centre

Photography by Carlos Dorce

Photography by Carlos Dorce

We went to the Centrum Nauki Kopernik in our last day in Warsaw which is a very interesting science museum. The building design was developed by young Polish architects from the firm RAr-2 in Ruda Śląska, who won an architectural competition in December 2005.

There are a lot of different rooms and interactive exhibitions and… there are also a lot of mathematical objects which you can touch and play with them. For example, you can see the Archimedes screw:

Photography by Carlos Dorce

Photography by Carlos Dorce

Water flows forwards and upwards in this simple hand pump, which works just like the rotating blade in an old-fashioned meat mincer. Many places around the world still use such a device to pump water, and it is frequently used to pump sewage in modern sewage systems. It was used for reclaiming land from under sea level in the Netherlands, and it was even used instead of traditional caterpillar tracks on Soviet armoured vehicles! Its key advantage is very simple: it doesn’t contain any complicated mechanisms that may break down.

You can also play with a Möbius band…

Photography by Carlos Dorce

Photography by Carlos Dorce

…or discover the conics rotating a cone full of blue water:

Photography by Carlos Dorce

Photography by Carlos Dorce

Here you have a beautiful parabola:

Photography by Carlos Dorce

Photography by Carlos Dorce

You can also play with the parabola using it as a communication device. Outside the museum there are two parabolas: you talk in one of them and you listen the message in the other:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are models of the Solar system, astronomical and optical experiments… and in the cinematic corner, the cycloid is very important because its property of… play with it! I’ve talked about it before!

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, the museum receives the visitors with this big Foucault pendulum:

Photography by Carlos Dorce

Photography by Carlos Dorce

It was a very nice experience!

Location: Centrum Nauki Kopernik (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!

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)

Pi Day’s doodle

Today is Pi Day!

Google published this doodle four years ago. We can see some formulas in it related with circles, spheres, trigonometry and the Archimedian value of pi.

Immortal Books, essential instruments (I)

Photography by Carlos Dorce

Photography by Carlos Dorce

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).

Photography by Carlos Dorce

Photography by Carlos Dorce

[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:

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Photography by Carlos Dorce

Photography by Carlos Dorce

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

Photography by Carlos Dorce

Photography by Carlos Dorce

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 wonderful vault of a Royal Library

Library of the monastery
Source: Wikimedia Commons

The King Philip II of Spain decided in 1550’s that he wanted to have a great library near his court in Madrid and he chose the new Monastery of San Lorenzo de El Escorial to place it in spite of other bigger villages. He didn’t want that the new library was a regular room inside a monastery so it had to be a very important place. Therefore, the library was placed on the second floor of the monastery just above his royal chambers but never above the basilica. Between 1565 and 1576, the king bought almost 5.000 books and manuscripts and the library became one of the most important libraries in all Europe.

Imaginary portrait of Juan de Herrera (1791) from
the book “Retratos de Españoles ilustres
Source: Wikimedia Commons

The mathematician and architect Juan de Herrera (1530-1597) designed a large room (54 m. long x 9 m. wide x 10 m. high) with big windows in both sides under a great barrel vault. This vault had to be decorated by an important painter and Philip II decided that Peregrino Tibaldi (1527–1596) had to be the right artist to do the work. Philip II was advised by Juan de Herrera and other humanists and he decided that the main subject of the paintings of the vault had to be the Liberal Arts. Furthermore, the seven arts would be together with the Philosophy and the Theology on both ends of the room. The Philosophy represented the compendium of the Human knowledge and she is accompanied by Aristotle, Plato, Seneca and Socrates:

The Philosophy

The Philosophy

The Theology is on the side next to the convent and she represented the Divine knowledge. Therefore the vault represented the way from the Human Philosophy to the Divine knowledge through the seven Liberal Arts: the Arithmetic, the Geometry, the Astronomy (Astrology), the Music, the Rhetoric, the Grammar and the Dialectic. We can see a mathematical detail on the fresco below the Philosophy: it represents the School of Athens and there is a discussion between the Academics leaded by Socrates and the Stoics leaded by Zeno of Elea.

The School of Athens

The School of Athens

The scholars aren’t listening to the speakers because each of them is “playing” with something different. We can see at the lower left corner a man measuring something with a compass and two books, a sphere and an armilar sphere, a dodecahedron and a compass in the middle of the picture:

Detail of the School of Athens

Detail of the School of Athens

Going from the Philosophy to the Theology, we arrive at the Arithmetic after admiring the Grammar, the Rhetoric and the Dialectic. The Arithmetic is a woman turned to a table with simple mathematical operations rounded by muscled young men with tablets with arithmetical operations ans counting with their fingers:

The Arithmetic

The Arithmetic

There is also a representation of the Queen of Saba talking with King Solomon  According to the Book of the Kings (I,10,1), the Queen of Saba went to meet Solomon to ask some enigmas to him so we can see a ruler, a balance and a tablet with some numbers written on it. In the red tablecloth we can read “Everything has number, weight and measure” in Hebrew:

King Salomon and the Queen of Saba talking about numbers

King Salomon and the Queen of Saba talking about numbers

The other panel next to the Arithmetic represents the school of the Gymnosophists who lived near the Nile and thought their philosophical theories from the numerical computations. In the middle of the picture we can see one of the gymnosophist with a compass looking at a triangle with the word “Anima” and the arithmetic progression 1, 2, 3 and 4 and the geometric 1, 3, 9 and 27 written on it. The other gymnosophists are computing with numbers written on the sand:

The Gymnosophists

The Gymnosophists

Finally, at both sides of the Arithmetic on the roof we find four people related with this subject: Archytas of Tarentum (c.428–c.347 BC) and Boethius (c.480-c.525) in one side and the Platonic Xenocrates (c.396/5 – 314/3 BC) and Jordan in the other. They are writing numbers in their tablets.

There is the Music after the Arithmetic and we find the Geometry after it:

The Geometry

The Geometry

She has a compass in one of her hands and the young men around her have different geometrical instruments. The two scenes which are on the corresponding walls next to her are dedicated to some Egyptian monks drawing geometrical figures on the sand…

Egyptian monks measuring the lands

Egyptian monks measuring the lands

and Archimedes’ death:

Archimedes' death

Archimedes’ death

Notice that Archimedes is drawing the demonstration of the Theorem of Pythagoras made by Euclid!

Finally, the four chosen figures are the Astronomer Aristarchus of Samos (IIIrd c. BC) and the Persian astrologer Abd del Aziz also known as Alcabitius (Xth century) in one side and Archimedes (c.287-212 BC) and Regiomontanus (1436-1476)  in the other. Aristarchus is measuring angles and has a dodecahedron at his feet, Alcabitius has a carpenter’s square, Archimedes has a compass and a sphere to measure the Earth and Regiomontanus is pointing at a dodecahedron.

The last Liberal Art is the Astrology. She is backed on a terrestrial globe and her eyes are looking at the sky. She has a compass in one of her hands and the little boys around her have an armilar sphere and some astronomical books:

The Astrology

The Astrology

In one of the two panels on the walls we can see Dionysius the Areopagite observing a solar eclipse the day of Jesuschrist’s death in Athens (Luke, 23,45) We can notice a quadrant and an astrolabe in the hands of the amazed men!

Dionysius the Areopagite observing a solar eclipse

Dionysius the Areopagite observing a solar eclipse

The other fresco represents King Ezekiel resting in bed and looking how time is delayed 15 years by God because of the repentance of his sins:

astronomia2

Ezekiel resting in bed

The four famous men are Euclid, Ptolemy, Alfonso X and Johannes of Sacrobosco. Euclid is represented here meaning the relationship between Astrology and Geometry. He has drawn three geometrical schemes. One is a triangle and a square inscribed in a circle and another square. Another scheme seems to be two overlaid squares partially hidden by Euclid’s name. In the middle of both pictures there is a man measuring the stars. Johannes of Sacrobosco has a quadrant in his right hand.

Euclid and Johannes of Sacrobosco

Euclid and Johannes of Sacrobosco

King Alfonso X of Castile (XIIIth. c) is the author of the Libros del Saber de Astronomía (“Books of the Astronomical knowledge”) and on the tablet which he has in his hands we notice a compass and the Ursa Maior (the compass is anachronistic!). His left hand has an open book with a horoscope

Alfonso X

Alfonso X

So you can see that this wonderful vault is an open mathematical book designed by Tibaldi and Juan de Herrera. I’ve been twice in the library and now I am waiting for the next time that I could enjoy this artistic part of the monastery of San Lorenzo de El Escorial.

Location: San Lorenzo de El Escorial (map)

Da Vinci in Peter and Paul Fortress

Poster about the exhibitionPhotography by Carlos Dorce

Poster about the exhibition
Photography by Carlos Dorce

Last August, I was lucky for visiting an exhibition about Leonardo da Vinci inside the Peter and Paul Fortress. It’s obvious that you don’t have the aim of visiting this kind of exhibitions when you are walking through the streets of Saint Petersburg but… Leonardo needed to be visited! I bought the ticket and my visit was fast because my wife and children decided to wait for me outside. Nevertheless, the exhibition seemed very interesting to me. The “secrets of da Vinci” are more technological than mathematical but I found this delicatessen: the Archimedes screw!

Archimedes screwPhotography by Carlos Dorce

Archimedes screw
Photography by Carlos Dorce

Next to the model, the explication of the device:

The device, capable of lifting water without application of human force was known in ancient times already. For the first time such a device was described by the Greek mathematician Archimedes (287-212 years BC). Leonardo developed several advanced versions of this device. He studied the correlation between the inclination of an axis and the necessary number of spirals. Thanks to improvements by Leonardo it became possible to pump over a larger amount of water with smaller loss.

Furthermore, I met Leonardo starting to paint his famous “Last Supper” with his Vitrubius man behind him and a representation of the sacred scene opposite to him.

Characterization of Leonardo da VinciPhotography by Carlos Dorce

Characterization of Leonardo da Vinci
Photography by Carlos Dorce

I hope that this exhibition wouldn’t be temporary and everybody could enjoy it!

Location: Peter and Paul Fortress (map)

The Polytechnical Museum in Moscow

The Polytechnical Museum in Moscow is not the most popular museum in the Russian capital but it’s an interesting place if you love Science and Technology. I recognize that it can’t be compared with the Kremlin, the Red Square, the Pushkin Museum or other important sites located Moscow but you will have to go there if you have more than a few days to spent. It’s not very big and we find a lot of things related with the History of Technology and its applications in Russia and former U.S.S.R. There is a very complete exhibition about rockets, satellites and Space:

Exhibition about SpacePhotography by Carlos Dorce

Exhibition about Space
Photography by Carlos Dorce

Nevertheless, it’s not a very interesting place if you only look for Mathematical objects: all the Mathematics are hidden behind the technological advances so we must satisfy with a little bit more than a ruler and a compass:

Photography by Carlos Dorce

Photography by Carlos Dorce

There are some posters hung from the roof of the main stairs and in one of them we find a reference to the great Archimedes (c. 287 BC – c. 212 BC) and his famous screw:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, in the exhibition about the Space, we find another poster with the portraits of Bruno, Galilei, Copernicus, Kepler, Newton, Lomonosov and Einstein who are seven of the greatest names in the History of Philosophical Astronomy:

Photography by Carlos Dorce

Photography by Carlos Dorce

I enjoyed the Museum a lot! I hope that you will be able to enjoy it too!

Location: Polytechnical Museum (map)