One of the main attractions in Warsaw is the royal castle. It was the royal residence of the Polish kings sincs the 16th century and the place where the first Parliament of Poland was located. It was destroyed during the Swedish wars in the middle of the 17th century but one hundred years later it regained its magnificence.
The castle was bombed by the Germans in 1939 and blown up by the German army five years later. So there wasn’t any castle in Warsaw after the Second World War until th Communist authorities decidied to rebuild it in 1971. It was reopened in 1984.
So we have visited this emblematic building of the city (before enjoying a very good ice cream!) and the mathematical tourist has found the Knights’ Hall (1786) which should be explaied in all the touristic guides.
It was the most important ante room leading to the Throne Room intended to perform the functions of a National Pantheon. During the royal audiences, all the senators and diplomats accredited to the Court gathered here.
The array of paintings and sculptures renowned Polish men and historic events and the statue of Chronos-Saturn symbolizes the lasting memory of great statesmen. And now… if you look at the painting on the great World… you can see…
Copernicus! But he is not the only important astronmer in the room. There also is a bronze bust of Johannes Hevelius (1611-1687), sculpted by Giacomo Monaldi. Hevelius was the founder of lunar topography and after Copernicus, he is the second most important Polish astronomer!
We also find the painting “The Establishment of Krakow Academy, 1400” by Marcella Bacciarellego (1783-1786) which is one of the set of paintings dedicated to the events of Polish History:
Finally, we can enjoy some very beautiful mosaics represented on the floors like this one:
I would have never said that this castle hid these mathematica joys. Enjoy them!
Location: Castle of Warsaw (map)
This little “Giralda” built in the Catalan town called L’Arboç was projected by Joan Roquer i Marí after a trip in Andalusia. He loved Andalusian architecture and decided to copy this Spanish style in his home. The Giralda was built between 1877 and 1889 designed by Roquer although measures half the height of the real Giralda of Seville. If you visit it you will also find a replica of the Court of the Lions of the Alhambra of Granada.
The decoration of the building was designed from several photographies taken by Roquer in Seville and therefore we have a lot of mosaics which can be admire in this mathematical post:
Location: La Giralda in L’Arboç (map)
Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.
The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…
For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.
Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!
Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.
Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!
Another example: look at these three wooden pieces…
The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:
There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.
And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?
In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:
Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his Zîj-i Ilkhanî (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his De Revolutionibus (1543). Al-Tûsî’s pair was very famous until the 15th century.
In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:
I didn’t know that it could be described only with a multiplication table! Is its equation z = xy? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.
Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:
Furthermore, one of the rooms of the palace is decoratd by a chess lover!
So… you must go to MMACA and enjoy Mathematics in a way ever done!
My last post about my visit to Birmingham is dedicated to St. Martin’s Church. The building is one of the most known churches of the city because of ii’s next to Bullring. Inside the church there is one of the sides of the nave which is full of flowered mosaics. Thus, the mathematical tourist must leave the Bullring for a moment and visit it for a few minutes:
Location: St. Martin’s church in Birmingham (map)
Bullring is a very big shopping center in Birmingham downtown which is impossible to not visiting if you are in this great English city. In one f the corners of the mall we can see this curved structure which shows a metallic mosaic full of circles. There are a lot of restaurants and shops below it so here you have a very good place to start your day in the shopping center.
Location: Bullring Shopping Center (map)
Birmingham Central Library is one wonderful example of a mathematical building. Look at their beautiful three mosaics (the white one, the blac one and both together) which we can admire in Centenary Square!
So don’t forget to visit it if you are going to Birmingham!
Location: Birmingham Central Library (map)
Last August I could visit Madrid alone and I walked unhurried through the old town. After Plaza Mayor (Main Square) I took a coffee in a bar near Tirso de Molina Square from which I saw this mathematical building. Can it be real? It’s an Escher tessellation!
The facade seems quite new and it’s based on Escher’s Metamorphosis:
It was a very good surprise!
Location: Conde de Romanones 14 (map)
Segovia is one of the most beautiful Spanish cities. The staple of the city is the Aqueduct (Ist century AD) located in the Plaza del Azoguejo but all the old city is a monument which must be visited if you come to Spain. Another emblematic building of the city is the Alcazar (XIInd century) which was the Royal palace of the Kings of Castile in Medieval times:
Segovia is also a very mathemtical place. There is not any particular museum or palace but a walk around the old city displays the reason why I am writing this post: it is full of mosaics on the facades of the buildings:
There are a lot of buildings with this kind of decoration and we can find some examples of the 17 symmetry groups in which we can clasify all the mosaics represented on the plane. For example:
If you want to take pictures of mosaics you must go to Segovia and enjoy its mathematical facades. I think that I must come back to Segovia to o a mathematical study about all these wonderful mosaics.
Location: Segovia (map)