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!

# A masonic geometric symbol in Barcelona

There are a lot of hidden enigmas and misteries in the modern buildings of Barcelona and some of these details are unknown by almost everyone. For example, on the facade at 11 Portaferrissa Street in the Old Town (Barri Gòtic), we can see the sculpture of two little boys above the main door on the top of a pile of bricks between them and the boy on the right holds some kind of square, whereas the one on the left holds a compass and a paddle. At first sight, it may seem another figure of the architectural style of the 19th Century. Nevertheless, it is a masonic sign indeed. The square and the compass were some of the most remarkable symbols of Freemasonry since both appear in the Masonry emblem.

In ancient times, the compass symbolized the Heavens inasmuch as it was used to study the starry Heavens, while the square represented the Earth because it was used to measure it. Nowadays both might have some philosophical and ethic connotations such as boundaries, so as to keep the equilibrium, and morality respectively. So, according to the Barcelona Historical Archive, the construction was made in the late 19th Century since there is a file with a works license of 1867, projected by Domingo Sitjas. It is interesting to notice that, in the original plans of the project, there is no sign of this sculpture. Due to Masonry persecution, everything related to it had to be kept secret. So, logically, the sculpture doesn’t appear in the plans!

Freemasonry describes itself as a beautiful system of morality, veiled in allegory and illustrated by symbols. Traditionally, masons are fond of architecture and are dedicated to the seven Liberal Arts: Grammar, Rhetoric, Arithmetic, Logic, Music, Geometry and Astronomy. Therefore, since Arithmetic, Geometry and Logic belong to mathematics, there is a link between Masonry and mathematics somehow. For instance, number 1 is represented by a point, which has no dimensions and turns to be the generator of any imaginable figure. According to the Masons, one is the arithmetic symbol of the Unity as well as the point is the geometric image of the Being. On top of that, if there are two points, they can be joined with a compass considering the straight segment connecting them like the one-dimensional projection of the geometrical link. It is important to note that the symbol of the Unit is the generator of duality, ternary, etc. using the compass. In addition to that, Geometry is the basis on which the Masonic superstructure stands and is considered by them as the mother of Science. However, Arithmetic has an important role in Masonry philosophybecause under Freemasons’ point of view, each of the four fundamental operations corresponds to a present value in their lives. For example, the sum is related to adding knowledge to our cognizance. In regard to such importance of the Unity as the genesis of the rest, it should be noted that the quaternary arises just like the ternary does. The last one would be represented in the form of a square. Mathematically, this generating method from the Unit would be considered as *N* = 1 + *n* (0<*n<*9 integer number), where *N* is associated with geometric figures representing values or facts of their daily life. It must be kept in mind that this is a cycle whose culmination is the following expression: 9+1=10=1+0=1. Let’s look at some examples of the corresponding symbolism to certain geometrical figures identified, as we have seen, with numbers. Remember that the “construction” of these is given by using a compass and a square like the ones in the image:

- The triangle, which represents number 3, is the geometric figure par excellence. The rest of figures can be represented as a set of triangles. It is known among Masons as Radiant Delta and each of its vertices represents space, time and energy. The union of these vertices turns out to be the force that gives structure to the universe, which is the Great Architect of the Universe (G), i.e., God.
- The square, which represents number 4, symbolizes at the same time two squares or the union of two triangles, that is, harmony and balance. If these two squares are within a circle, then they represent the harmony between the Earth and the Sacred World.
- The 5-pointed star (or pentagrammon) represents number 5 and symbolizes the man and life, what is masculine and what is feminine, and the union represents androgyny.

It’s imperative to comment that, although there may be no direct link between Pythagoras and Freemasonry, the teachings of Pythagoras have greatly influenced its structure and its teachings: numbers (especially numbers from 1 to 10) are symbols representing the philosophical universal organization and the only way to reach God. Furthermore, every degree of the Masonic initiation corresponds to one number.

Finally, it is necessary to remark that this is just a selection of the whole mathematics they studied, since the 47th Euclid problem and the golden proportion should be also mentioned.

In conclusion, the sculpture emblems a fraternal community full of mysteries which has always been passionate about mathematics. One last mathematical fact: if you come in the building, you will see 7 steps followed by 7 steps and 14 steps more!

*This post has been written by Paula Arrebola and Abel Hernandez in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).*

**Location** Casa Domingo Sitjas (map)

# A plaque in the Cloth Hall in Krakow

In the ground floor of the Cloth Hall of Krakow there is this modern plaque (2007) designed by Czeslaw Dźwigaj and the selection of the text was made by Jerzy Wyrozumski and Alexander Kravchuk:

1257 Krakow city rights advocates by the German tradition and the situation of the market and the houses and affected courts

It was Bolesław V, “the Chaste One”, (1226–1279), Duke of Sandomierz in Lesser Poland from 1232 and High Duke of Poland from 1243 until his death, who introduced in 1257 the city rights modelled on the Magdeburg law allowing for tax benefits and new trade privileges for the citizens and this plaque was designed to conmemmorate the 750th anniversary of this moment.

The Cloth Hall is one of the most emblematic symbols of Krakow because is probably the oldest shopping mall in the World:

It’s in the middle of the main square of Krakow and everybody visit it and buy something in the shops of the ground floor.

So next time that you visit Krakow look at the floor to this conmemorative plaque:

**Location**: The Cloth Hall in Krakow (map)

# Museum of Nicolaus Copernicus, Frombork

In a previous post I began to talk about this museum located inside Frombork castle. You can learn almost everything about him, his life and his works on medicine, economies and, of course, astronomy, including the replicas of his instruments (we saw them also in Warsaw). For example, it’s possible to see some facsmile editions of his works and also a recreation of his desk:

Among the references about his publication of his works, we can find this engraving showing Copernicus in a lecture for the Cracovian scientists in 1509:

Or this other wonderful one (1873) with Copernicus in he middle of the picture talking about his heliocentric system:

How proud he is of his heliocentric theory!

And who are his guests? First of all, Hipparcus (with the armillar spher) and Ptolemy (with his geocentric system) are listening the theory which will finish theirs. Ptolemy looks askance at Tycho Brahe meanwhile Newton is looking at Laplace:

Galileo Galilei is behind Copernicus looking at him with great reverence:

And Hevelius, the other great Polish astronomer, agrees Copernicus’ theories although he never had the telescope to check them.

Finally, Johannes Kepler seems to be bored of listening this obvious theory although his ellipses will be the curves which will change the astronomy.

A beautiful picture for a beautiful museum. Next step: the cathedral!

**Location**: Frombork castle (map)

# Copernicus and Hevelius in the Museum of Technology

My trip to Poland and Praghe finished yesterday and I remember that in my first post about the Museum of Technology of Warsaw I didn’t talk about the astronomical room in the second floor of the museum. It has some telescopes, reproductions of satellites and a lot of information about the space and we can also find the corner dedicated to Copernicus and his *De Revolutionibus Oribium Coelestium* and Hevelius.

There are three reproductions of Copernicus’s astronomical instruments which we can imagine in the hands of this Polish astronomer. First of all, the armillary sphere…

…the paralactic triangle (triquetum) for measuring the angular heigh of the Moon…

… and the solar quadrant used by Copernicus in 1510-1520 in order to watch the Sun:

Johannes Hevelius’ instruments are represented by some old images…

…and there is a representation of his observatory in Gdansk:

**Location**: Museum of Techonolgy in Warsaw (map)

# Bernard Bolzano’s birthhouse

The Municipal Library of Prague (Marianské Square No.101) is in the same place where was Bernard Bolzano’s birthhouse. The house was detroyed but we have a photography of the main door from 1906:

The grey building is not beautiful although is full of allegories and symbols and among them we can see a ruler and a compass:

**Location**: Bolzano’s birthhouse in Prague (map)

# Copernicus’ House in Torun

Copernicus’ house was built in the 14th/15th century in the Gothic style of the Hanseatic towns. It has a high entrance hall with an open kitchen on the ground floor and the merchant’s office. The rooms are upstairs.

The building was the property of Copernicus’ family in the years 1463-1480 and it may well be here that he was born in 1473. In an old picture it’s possible to see Napoleon Bonaparte walking in front of the present Copernicus Street nr 40. Till the end of the 19th century there was the wrong opinion that Copernicus had been born there although it seems to be the present numbers 15 and 17 the right houses. In the end of the 15th century, Copernicus’ father owned these three houses and another one in 36 and 37 Old Market Square.

Nowadays, there is a restaurant in 40 Copernicus Street…

…and a new building is a department store (“Dom Towarowy”):

The interior of the house (17 Copernicus Street) is reconstructed and the imagination is the only way to see Copernicus familiy in it.

The museum is full of portraits (I’m going to write another post about them!), the replicas of the instruments, scultures about the heliocentric system… and information about Copernicus’ life and works. For example, here you have a very modern design for Copernicus’ office:

And the room dedicated to his *De Revolutionibus*:

Finally, hthe replicas of Copernicus’ instruments:

A few days ago I went to Frombork where Copernicus died and today I’ve been in the house where he was born. I’ve been in Olsztyn too so… what about going to Krakow where he studied? Thus… next step: Krakow!

**Location**: Copernicus’ house in Torun (map)

# The Long Market in Gdansk

The Long Market (*Długi Targ*) is one of the most important touristic attractions of Gdansk. It was a merchant road in the 13th century. After the massacre of Gdansk citizens on 13 November 1308 by Teutonic Knights, the place became the main street of the city and is name “Longa Platea” was first written in 1331. Nowadays it’s a very beautiful long square full of typical shops and restaurants which are the soul of this cosmopolutan city. One of its most representative houses is the town hall from the 16th century and Neptune’s Fountain, the main symbol of the city, is also there. This fountain was constructed in 1617 from Abraham van den Blocke’s designs.

Thus, if you visit Gdansk, you must have time to take a beer or a coffee in one of the cafes or have a typican Polish dinner in one of the restaurants which fill all the beautiful houses which can be admire in the square.

Among all these houses we also find a lot of mathematical symbols which allow me to talk of them in this new post. For example, Radisson Blue hotel is located in number 19 and the allegorical paintings of the facade are a joy for the mathematical freak:

On both sides we have some of the most important men in the history of astronomy like Hipparcus of Rhodas,

Ptolemy:

Copernicus:

Galileo Galilei:

and Hevelius:

Approaching the town hall, there is another red house which is full of artists ans it’s coronated by a replica of Aristotle and Plato from Raffaello’s “School of Athens”:

In another house there also are the allegorical Astronomia rounded by Cellarius’ heliocentric systems:

And finally we find other allegories like the Architecture, the Geometry or the Geography in the opposite side of the square:

For example…

…or:

As you can see, this is an excuse to admire the beautiful facades of the houses in this square which I never tire of walking through it.

By the way, there is a beautiful sundial in the town hall:

**Location**: Długi Targ in Gdansk (map)

# Pythagoras in Temple Bar Moument

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.

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.

**Location**: Temple Bar (map)