Enigma Machine in the Museum of Technology of Warsaw

Photography by Carlos Dorce

Photography by Carlos Dorce

The Museum of Technology of Warsaw is located in the Palace of Culture and Science of Warsaw. This building is the tallest one in Poland (231 m.) and one of the most polemical symbols in this country. The construction started in 1952 (to 1955) and it was a present from Joseph Stalin and the U.S.S.R. authorities to the Polish people so it’s easy to imagine why there are a lot of people who don’t like it.

The Palace is decorated with a lot of statues of different allegories and Polish women and men and among them…

Photography by Carlos Dorce

Photography by Carlos Dorce

I’ve been searching for the net but I’ve not been able to find the names and representations of all these sculptures but this image is very familiar for us, isn’t it? It’s Copernicus! And which book is he reading? Look at the orbits!

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum is in one of its corners and it is not so big. There are a lot of old cars, bikes, motorbikes and some old objects as telephones, radios,… and, of course, a calculator:

Photography by Carlos Dorce

Photography by Carlos Dorce

The Enigma machine is in the first floor next to the telephones:

Photography by Carlos Dorce

Photography by Carlos Dorce

Machine cypher was introduced in Germany at the end of the 1920s, and in 1933-34 the cipher communication system was adopted by the Third Reich.

In 1929 the Cipher Department of the Polish Army’s Headquarters organized in Poznan a training for cryptologists, and some time later set up its agency there. Soon the agency’s most clever mathematicians were transferred to Warsaw. Among them were: Marian Rejewski (right), Jerzy Rozycki (left), and Henryk Zygalski (middle).

Photography by Carlos Dorce

Photography by Carlos Dorce

This team set to work on the Enigma’s cipher with the application of mathematical methods, permutations in particular, and their previous experience of working on German ciphers. Cooperation with French radio intelligence was helpful, too. The Enigma’s cipher was ultimately broken at the end of 1933 and the beginning of 1934. In the process of deciphering scientists employed machines of their own design (similar to Enigma) produced by the AVA Fadiotechnical Factory. They also built a cycle counter and a cryptological bomb which were used in solving alterating cipher keys.

Photography by Carlos Dorce

The Enigma Machine. Photography by Carlos Dorce

 In the face of the incoming war the method of breaking ciphers together with the machine and the whole documentation was in July 1939 passed on France and Britain, where Enigma’s cipher was still not deciphered.

In September 1939 secrets of the Cipher Department were scrupulous obliterated, and the whole staff was evacuated to France. Rozycki was killed there and after the war Zygalski remained in England and Rejewski returned to Poland.

The riddle of the Enigma remained unrevealed for a long time. The first publication concerning this subject appeared in 1967 in Poland (W.Kozaczuk’s “Battle for Mysteries”). However, it was a book by G.Bertrand, the former head of French radio intelligence, that won a widespread fame after its publication in France in 1973 and triggered many other publications all over the world. While Bertrand (advocate of the idea of the Polish-French cooperation in the 1930s and 1940s) pointed Polish scientists as the authors of the success in breaking the Enigma cipher, some time later the English credired themselves with this achievement.

So… WAS ALAN TURING THE ENIGMA BREAKER?

The Enigma machine on display dated back to the end of the 1930s and was used in the Wehrmacht.

Location: Museum of Techonolgy in Warsaw (map)

Copernicus and Hevelius in the Knights’ Hall

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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.

Photography by Carlos Dorce

Photography by Carlos Dorce

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…

Photography by Carlos Dorce

Photography by Carlos Dorce

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!

Photography by Carlos Dorce

Photography by Carlos Dorce

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:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, we can enjoy some very beautiful mosaics represented on the floors like this one:

Photography by Carlos Dorce

Photography by Carlos Dorce

I would have never said that this castle hid these mathematica joys. Enjoy them!

Location: Castle of Warsaw (map)

Copernicus in Warsaw

Photography by Carlos Dorce

Photography by Carlos Dorce

Today is our first day in Warsaw! My wife, my son, my daughter and me are going to visit Poland for twelve days and I am sure that the mathematical tourist is going to see a lot of things!

This morning we have been in the city centre and we have walked through Krakowskie Przedmiescie and Nicolas Copernicus was there! His natural position is the middle of Copernicus square which is called with his name and he is rounded by his heliocentric system:

Photography by Carlos Dorce

We can read in wikipedia:

Nicolaus Copernicus (Polish: Mikołaj Kopernik ; German: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance mathematician and astronomer who formulated a heliocentric model of the universe which placed the Sun, rather than the Earth, at the center.

Portrait, 1580, Toruń Old Town City Hall Source: Wikimedia Commons

The publication of Copernicus’ book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), just before his death in 1543, is considered a major event in the history of science. It began the Copernican Revolution and contributed importantly to the scientific revolution.

Copernicus was born and died in Royal Prussia, a region of the Kingdom of Poland since 1466. Copernicus had a doctorate in canon law and, though without degrees, was a physician, polyglot, classics scholar, translator, governor, diplomat, and economist who in 1517 set down a quantity theory of money, a principal concept in economics to the present day, and formulated a version of Gresham’s law in 1519, before Gresham.

The bronze statue was designed by the Danish sculptor Bertel Thorvaldsen (c.1770-1844) in 1822 and was completed eight years later. The phiolosopher Stanislaw Staszic (1755-1826) made important donations for erecting it in Torun because he knew that Napoleon was very surprised for not finding a monument dedicated to the Polish astronomer in his birthplace.

The inscription says “To Nicolas Copernicus [from a] grateful nation” in Latin and Polish although during the German occupation of Poland in 1944, the inscription was translated to German saying “To Nicolaus Copernicus [from] the German Nation”.

copernic02

Photography by Carlos Dorce

In 2007, Copernicus’ solar system was represented rounding him representing an image from his revolutionary De Revolutionibus Orbium Coelestium (1543).

Photography by Carlos Dorce

Photography by Carlos Dorce

The statue is in front of Staszic Palace (1620), the seat of the Polish Academy of Sciences. In 1967, a bronze replica was erected in Chaboillez Square in Montreal during the World’s Fair:

Source: Wikimedia Commons

In 1973, on the 500th anniversary of Copernicus’ birth, another bronze replica was installed in front of Adler Planetarium in Chicago:

Source: Wikimedia Commons

Location in Warsaw: map

Location in Montreal: map

Location in Chicago: map

Ten Barcelonas!

The new advertising campaign in the city of Barcelona is so numerical. The city is divided in ten districts and the first ten numbers are the stars of the new posters which are in all the streets. Enjoy them!

1. The former Barcino which is the focus of the city where the Roman ruins are together with the Mediaeval Gothic architecture, was the origin of the first district: Ciutat Vella (Old Town). Their streets and squares are an open book to the history of Barcelona.

Districte 2 L’Eixample

2. Eixample is the Modernist district! At the end of the 19th century, Barcelona was expanded in this rectangular net with their new wide streets which nowadays are the focus of the urban movement of the city.

Districte 3 Sants-Montjuïc

3. Sants-Montjuïc offers a long walk from the mountain to the sea pointing our attention in the history of the comercial Barcelona.

Districte 4 Les Corts

4. Between Diagonal, full of shops and malls, and the noise of the Camp Nou, Les Corts is a Barcelona full of gardens where you can be lost in this residential town.

Districte 5 Sarrià-Sant Gervasi

5. Old Sarrià-Sant Gervasi is the most residential district of Barcelona. It’s like a quiet town rounded by gardens and museums which ends in Tibidabo mountain.

Districte 6 Gracia

6. Gracia is the opened town to all the cultures, the urban artists, the music, the theatre ans the cinema although the people whi lives here are proud of it and its past.

Districte 7 Horta-Guinardó

7. Horta-Guinardó is the great balcony in the city. It’s full of water and the old houses and “masias” and the gardens are the treasure and the image of this richness.

Districte 8 Nou Barris

8. This different and wide district called Nou Barris (Nine Towns) is the focus of a lot of gardens and green zones to walk and enjoy its cultural life which is independent form the rest of the city.

Districte 9 Sant Andreu

9. sant Andreu is like its neighbours: characters, fight and a deep respect for its traditions. From the Tres Tombs to the Esclat and other parties, tradition is the most important thing in this old town.

Districte 10 Sant Martí

10. Beside the sea, ubiquitous chimneys welcome the Sant Martí district recalling its industrial past. An industry that has led to innovation and new technology in the district and has become the engine of the new Barcelona.

Escher’s doodle

Maurits Cornelis Escher (Leeuwarden, 17 June 1898 – Laren, 27 March 1972) was a Dutch painter whose works can be considered as Mathematical Art.

His drawing hands were the main theme of the doodle published in June 16, 2003:

Source: Wikimedia Commons

A Catalan Giralda

Photography by Carlos Dorce

Photography by Carlos Dorce

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:

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: La Giralda in L’Arboç (map)

Another Sundial… now in Cornellà

Photography by Carlos Dorce

Photography by Carlos Dorce

On May 2, 2014, I told you that I vidited the MMACA with some of my students and we also noticed this sundial walking from the underground station (Gavarra) to Mercader Palace. We were in a hurry so we could’n stop to analise the shadow of the gnomon but this picture must be the first step for coming back in the not too distant future.

Location: map

A sundial next to the motorway

Photography by Carlos Dorce

Photography by Carlos Dorce

Some weeks ago I had to stop to put gas in my car and Destiny led me to a petrol station next to Sidamon (a small village near Lleida). I took the opportunity to have a drink in the bar and… what was that? There is a big sundial in the roundabaut next to the petrol station!

All the people who lives in Sidamon (less than 700 people!) see this sundial all the days of their lifes.  Why don’t they paint it? So it will bright in the middle of this big plain!

One thing more… the coffee in the bar wasn’t nice.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Sidamon (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)

A piece of the Berlin Papyrus

Photography by Carlos Dorce

The Berlin Papyrus 6619 (1800 BC) is one of the only surviving witness which demonstrates that the Egyptian escribes knew how to solve certain quadratic equations.

The first problem in the papyrus says: You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares? That is:

x2 + y2 = 100
4x – 3y = 0

There also is a second similar problem equivalent to the quadratic system:

x2 + y2 = 400
4x – 3y = 0

The solving method is the rule of false position. The escribe assumed that x = 0,75 and y = 1 so x2 + y2 = 1,5625. But the result should be 100 = 64 · 1,5625! Therefore, our two squares must be 64 times bigger and their sides must be 8 times bigger. So the result is x = 0,75 · 8 = 6 units and y = 1 · 8 = 8 units, and  x2 + y2 = 100.

This papyrus becames unnotices in the Neues Museum of Berlin due to its close position to the famous bust of Nefertiti:

Source: Wikimedia Commons

But dont’t leave the museum without giving attention to this important mathematical document.

Location: Neues Museum (map)

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