Tag Archives: Cycloid

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)

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The Museum of Mathematics of Catalonia (MMACA)

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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”.

Mercader Palace Photography by Carlos Dorce

Mercader Palace
Photography by Carlos Dorce

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,…

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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!

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Photography by Carlos Dorce

Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.

Photography by Carlos Dorce

Photography by Carlos Dorce

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!

Photography by Carlos Dorce

Photography by Carlos Dorce

Another example: look at these three wooden pieces…

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The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:

Photography by Carlos Dorce

Photography by Carlos Dorce

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?

Photography by Carlos Dorce

Photography by Carlos Dorce

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”:

Source: Wikimedia Commons

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

Furthermore, one of the rooms of the palace is decoratd by a chess lover!

Photography by Carlos Dorce

Photography by Carlos Dorce

So… you must go to MMACA and enjoy Mathematics in a way ever done!

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Location: MMACA in Cornellà de Llobregat (map)

Christiaan Huygen’s birthday

Today is  Christiaan Huygens’s birthday and this is the doodle dedicated to Huygen’s birthday in April 16, 2009.

Huygens (1629-1695) as one of the first mathematicians to study Probability. He published his De ratiociniis in ludo aleae in 1656 in which he established the foundations of the Calculus of Probabilities after Pascal and Fermat’s letters. Furthermore, he worked on the cycloid, the rectification of curves, the pendulum,…

He also made important contributions to Phisics and Mechanics.

Source: Wikimedia Commos