Monthly Archives: May, 2014

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)

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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)

An astrolabe in Berlin

Photography by Carlos Dorce

The first object which you can see in the second floor of the Pergamon Museum in Berlin is this Iraqi astrolabe made by the astronomer and poet Hibât Allâh al-Bagdâdî and designed by the great Abû Jacfâr al-Khâzin (c.900-c.970). This bronze piece is unique!

Al-Khâzin was a very important Persian mathematician and astronomer who worked in Ray and wrote a Commentary on the Almagest. He also wrote one of the most important work on the construction of astrolabes which was very appreciated by his colleagues.

Location: Pergamon Museum (map)

Chen Jingrun’s Birthday

Chen Jingrun (22 May 1933 – 19 March 1996) made very important contributions in Goldbach’s conjecture like the demonstration of Chen’s theorem (1966): every suffiently large even number can be written as the sum of a prime and a product of two different primes.

In 1999 China issued the stamp “The Best result of Goldbach conjecture”:

The doodle was published five years ago.

The Palmerston Number Sculpture

The Palmerston Number Sculpture. Photography by Loxias (TravelPod)

This is a new mathematical sculpture found in the net and dedicated to numbers. It was designed by Anton Parsons and represents a great monument to our loved Mathematics.

Location: The Palmerston Number Sculpture (map)

Pythagoras in Temple Bar Moument

Photography by Carlos Dorce

Photography by Carlos Dorce

The Temple Bar Memorial (1880) stands in the middle of the road opposite Street’s Law Courts marking the place where Wren’s Temple Bar used to stand as the entrance to London from Westminster.

The monument has two standing statues dedicated to Queen Victoria and the Prince of Wales because both were the last royals to pass through the old gate in 1872.

Photography by Carlos Dorce

Photography by Carlos Dorce

The reliefs round Queen Victoria contains some allegories which includes the first picture about the Euclidean demonstration of the theorem of Pythagoras. We also find a ruler and a globe with the ecliptic.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Temple Bar (map)

Forty years of Rubik’s Cube

Today Google celebrates puzzle’s 40th anniversary with an interactive Doodle. The game was invented by Hungarian professor of architecture Ernö Rubik in 1974 and won the German Game of the Year six years later. All the people born in the 70s had at least one of these cubes at home and tried to solve this almost impossible game.

To solve it, you must know taht there are 8!=40.320 ways to arrange the eight corner cubes and each of the seven first cubes can be orientated in 3 different ways (the eighth cube depends on the other seven!). So, you can arrange the eight corner cubes in 37 · 8! ways. For arranging the edges there are 12!/2 ways and there also are 211  ways to orientate these 12 edges.

So if you multiply the four quantities you will see that there are approximately 43.252.003.274.489.856.000 ways to solve Rubik’s cube.