The Cubic Houses (or Kubuswonig) are one of the symbols of Rotterdam. Their architect was Piet Blom (1934-1999) whose first drafts originate from 1973-1974. Three testhouses were built in Helmond in 1975 and another eighteen were built around a cultural center there in 1977:
According to the official web page, the first drafts of the Rotterdam version were presented in 1978. The construction of the houses started in March 1982 due to financial problems and they were completed in 1984. The first presentation showed 74 cubic houses and a cultural centre next to them but the definitive version accommodated only 38 houses but also a school, shopping centres and a tower of appartaments. All the houses were sold before they were finished and are inhabited since 1984. Blom tried to create a forest by each cube representing an abstract tree. Each cube stands along its diagonal and it’s split into three levels: the lower one contains the living area, the middle level contains the sleeping area and a bathroom and the top level is used as either a living space or a bedroom.
In the British Museum we can find this diptych containing an account written in two columns along the grain. The text is virtually complete and lists cash sums usually for specified goods. The ruling through of some entries implies that they had been paid whilst those not deleted were outstanding debts (we can see that there are some crossed sentences out). The account may have been that of a civilian trader doing business with soldiers.
…, bugler, for the price of… modii 15, denarii 12, asses 1 and 3/4; likewise, for sundries, denarii 2, asses 2; Ircucisso, as part of the price of bacon, denarii 13 and 1/2; Felicio the centurion, bacon, 45 pounds, likewise, bacon lard, 15 and 1/2 pounds, total 60 and 1/2 pounds, denarii 8, asses 2; likewise, he (?) has received for sundries denarii 6, asses 2 and 3/4; Vattus… Victor… For the price of a horse… Exomnius the centurion, denarii… Atrectus the brewer, as part of the price of iron, denarii… for the price of pork-fat, denarii 11, asses2; Andecarus, denarii… Sanctus, denarii…
It’s also interesting to compare the Roman numbers with the translation to see how the Roman figures were written.
This wonderful picture shows Pythagoras (left), Euclid (middle) and Ptolemy (right) sitting in front of the Arithmetic, the Geometry and the Astronomy respectively. We can see the Arithmetic holding a tablet, the Geometry holding a compass and the Astronomy holding an armillar sphere and the three men are holding their books. Of course, Euclid has his Elements and Ptolemy may be writing the Almagest. This section is part of the painting which we find in the Spanish Chapel at Santa Maria Novella. According to Wikipedia, the Spanish Chapel is the former chapter house of the monastery. It is situated at the north side of the Chiostro Verde and it was commissioned by Buonamico (Mico) Guidalotti as his funerary chapel. Construction started c. 1343 and was finished in 1355. The Guidalotti chapel was later called “Spanish Chapel”, because Cosimo I assigned it to Eleonora of Toledo and her Spanish retinue. The Spanish Chapel was decorated from 1365 to 1367 by Andrea di Bonaiuto and the large fresco on the right wall depicts the Allegory of the Active and Triumphant Church and of the Dominican order. It is especially interesting for us the fresco called The Triumph of Saint Thomas Aquinas:
We can see St. Thomas Aquinas holding the Book of Wisdom with the words:
And so I prayed, and understanding was given me; I entreated, and the spirit of Wisdom came to me. I esteemed her more than scepters and thrones; compared with her, I held riches as nothing.
Book of Wisdom 7:7, 8
There are seven figures over him which are the Seven Virtues: the three figures on the top from left to right are the Faith (holding a cross), the Charity (with her arms open and the Hope (holding an olive branch); the four figures on the bottom from left to right are: the Temperance (holding a upright branch of peace), the Prudence (holding a book to educate people in the correct way), the Justice (holding a scepter and the crown of the power) and the Fortitude (wearing an armor and holding a sword and a tower).
Next to St. Thomas sitting in the same row as him there are ten Biblical figures (from left to right): Job, David, Saint Paul, Matthew, John, Luke, Moses (holding the two sheets of the Law), Isaiah and Solomon. Under St. Thomas there are three heretic figures: Nestor, Arius and Ibn Rushd (Averroes).
Finally, the bottom row is full of allegorical figures. On the image’s left (from left to right): the Civil Law(and the Emperor Justinian whose code was the law of the Roman Empire, sitting at her feet), the Canonical Law (and Pope Clement V), the Philosophy (and Aristotle), the Holy Scripture (and Jerome), the Theology (and John of Damascus), the Contemplation (and Dionysius the Areopagite) and the Preaching (and St. Agustine). On the right, the Arithmetic (and Pythagoras), the Geometry (and Euclid), the Astronomy (and Ptolemy) and the Music (and Tubal Cain) represent the Quadrivium. Finally, the Dialectics (and Pietro Ispana), the Rhetoric (and Cicero) and Grammar (and Priscian) represent the Trivium.
We find the Victoria and Albert Museum opposite the Science Museum of London. The V&A is a fantastic exhibition of paintings and sculptures just in the heart of the English capital. Sometimes, people don’t look for a particular thing but this particular thing finds these people. This is the exact situation of this Math’s Lesson. It seems that the boy doesn’t know how to sum the numbers, doesn’t he?
We can observe that the sum of the units is 1+2+9+4=16. Of course, the boy has written a 6 and he must remember 1. Tens digits total 7+9+6+1+1=24 and the boy should write 4 and conserve 2. Is it the correct sum?
I went to a very interesting exhibition in Cosmocaixa (Barcelona Science Museum) last Saturday:
The aim of this exhibition is to show that the contributions to which science and technology have contributed to the progress and social development are endless. In the last decade, these developments have increased exponentially and this is mainly due to the technological convergence. “TecnoRevolució” is an interactive exhibition that seeks to converging technologies: nanotechnology, biotechnology, ICT, and cognitive science. Interconnections between them change the world around us and cause a revolution in different fields as construction, transportation, agriculture, medicine, education or art. The exhibition is very interesting but I’m going to focus my attention in the section dedicated to old calculators.
I am sure that the nanotechnology and the bio-medics are more important than more than 30 old calculators but I am a little “freaky” and walking around that part of the exhibition was very funny for me. For example, can you say that this calculator Ifach (1943) is not interesting? The Ifach was invented by the Spanish pharmacist Genaro Calatayud. The machine doesn’t have any gear. For adding two numbers, the user had to move one of the two concentric circles with a broach to see the total in the right of the central circle.
Another interesting machine is the Stima (1930) which was a portable machine for adding and subtracting. It was made by the Swiss Albert Steinmann based on a Contostyle with an automatic mechanism to see the result in a window at the bottom of the machine. The subtraction was made by the complement number system.
Of course, there also was the adding machine called Contostyle (1847) which is the oldest calculator of the exhibition. Its reduced dimensions (17 cm. x 9 cm. x 1 cm.) were part of its success because of people could take them in their pockets. Furthermore, the Constostyles could add two numbers faster than other calculators because it wasn’t necessary for the user to move up the broach from the machine.
To be continued…
The exposition can be visited until May 6, 2013.
The Two-gatherers is a painting attributed to Marinus van Reymerswaele which we can see in the National Gallery of London and also in the Louvre Museum of Paris (the French copy was executed before the English one) although we have simplified versions in Belgium and Poland. A very good article written by Paul Ackroyd, Rachel Billinge, Lorne Campbell and Jo Kirby gives the details of the painting:
Both the London and the Paris pictures show, behind the two men, a wooden cupboard on top of which are piled documents, an oval deed-box, a ledger, a turned wooden sand-box and a brass candlestick; across its base lies a pair of snuffers. The folded document above the head of the man on our right is a deed issued, according to the inscription, in 1515 by two aldermen of Reymerswale. The name of the first is concealed by the folding of the document; the second, Cornelis Danielsz, was indeed an alderman in 1514-15. The man on the left is writing in his ledger an account of the income of a town over a period of 7 months -from the excise duties on wine and beer, the ‘first-bridge’, the weight-house the ‘hall’, the ferries, fees for deeds, charges raised for specific expenses, loans and the civic mills.
I still continue counting money!
I didn’t know that Marynus van Reymerswaele had painted a similar situation to the last post: a banker or a moneylender counting money. I’m surprised! If I had to say something about this painting I’d notice that here there is the son of the banker and his wife behind her. And… have you seen that the banker is weighting the coins?
I am sure that this painting is not going to be the last financial situation represented by Van Reymerswaele!
Location: Musée des Beaux-Arts in Valenciennes (map)
This is not the first time in which I talk about Justus van Gent’s paintings. Indeed, the first post of this blog was dedicated to the portrait of Euclid of Alexandria made by this Dutch painter. This portrait is in Louvre Museum and it’s part of the serie of portraits of some important people made by Van Gent.
Claudius Ptolemy lived and worked in the IInd century AD in Alexandria. We know little about him and we can place his life from his own astronomical observations recorded in his great work entitled Mathematical Collection. His first observation was an eclipse of the Moon made in Alexandria in the 5 April of 125 AD and the last one was the observation of the maximum elongation of Mercury made in the 2 February of 141 AD. This recorded data mean that Ptolemy worked in his astronomical book in the period 125-141 AD. Furthermore, we know that in the year 147/148 AD he erected a stele in the town of Canopus about 25 Km East of Alexandria. we can also observe that his name Claudius Ptolemy is a good definition of his life: Claudios is a Greek name whereas Ptolemaios could indicate that he came from one of the various Egyptian towns named after the Ptolemaic kings.
Ptolemy’s Mathematical Collection was the most important Greek astronomical work. Later the Arabs called it with the superlative Al-Majistî (Almagest) and with this name the Latin Europe adopted as the referencial astronomical handbook. In the IVth century, Pappus of Alexandria (c.290-c.350) made a Commentary on it and part of the commentary on Book V (the Almagest is divided in 13 Books) as well as his commentary on Book VI are actually extant in the original. Theon of Alexandria (c.335–c. 405) wrote another commentary on the Mathematical Collection in 11 books incorporating as much as was available of Pappus’ work. Theon was assisted by his daughter Hypatia of Alexandria (c.360–March 415) and the whole text was published at Basel by Joachim Camerarius (April 12, 1500 – April 17, 1574) in 1538.
The Mathematical Collection arrived at the Muslim World and it was translated into Arabic, first by translators unnamed at the instance of Yahyâ b. Khâlid b. Barmak, then by al-Hajjâj, the translator of Euclid (c.786-835), and again by Ishaq b. Hunain (d.910) whose translation was improved by Thâbit b. Qurra (c.826–February 18, 901). The first edition to be published (Venice, 1515) was the Latin translation made by Gherard of Cremona from the Arabic, which was finished in 1175. Although there was a previous Latin translation from the Greek, the first Latin translation from the Greek to be published was that made by Georgius of Trebizond in 1451 and the editio princeps of the Greek text was brought out by Grynaeus at Basel in 1538.
According to Sir Thomas Heath in his A History of Greek Mathematics (II, 275), the Almagest is most valuable for the reason that it contains very full particulars of observations and investigations by Hipparchus, as well as of the earlier observations recorded by him. The indispensable preliminaries to the study of the Ptolemaic system, general explanations of the different motions of the heavenly bodies in relation to the Earth as centre, propositions required for the preparation of Tables of Chords, the Table itself, some propositions in spherical trigonometry,… are in Books I and II; Book II deals with the length of the year and the motion of the Sun on the eccentric and epicycle hypotheses; Book IV is about the length of the months and the theory of the Moon; in Book V we find the construction of an astrolabe and the theory of the Moon continued, the diameters of the Sun, the Moon and the Earth’s shadow, the distances between them and their dimensions; the conjunctions and oppositions of Sun and Moon, the solar and lunar eclipses and their periods are studied in Book VI; Books VII and VIII are about fixed stars and the precession of the equinoxes and Books IX-XIII are devoted to the movements of the planets.
We continue counting money! Another moneylender is the guest star in a Flemish painting. Here, the wifer of the moneychanger looks up from the book to see how her husband counts the silver and gold coins. Is he counting how many coins are needed to pay the Greek, Portuguese and Spanish bill?
Nowadays it’s not a secret that economical situation in almost all the countries is not the best of the recent history. This situation is more prominent in Europe than in America because the countries of the Northern and Central Europe are concerned about the public finances of countries such as Greece, Portugal, Spain and Italy. Furthermore, in these countries the problem is the same inside and we are living a rise of independence movements. For example, in Spain there are going to be elections on the 25th November to choose the Catalan Parliament and it seems that the parties which want a referendum for the independence are going to win quite most. In this context, it is easy to take a look to the XVIth century and the scene painted by the Flemish Matsys. He represented the confrontation between a greed of a moneylender who is counting his gold coins and the prayers of his wife who watches everything carefully. Perhaps the moneylender is Spain and the wife represents to Catalonia nowadays, or the painting can be compared with the European situation or… However, I am sure that the wife loves her husband and the prayers and the greed will find a common place to life together.
Perhaps this painting is not a purelly mathematical scene but nobody can say that counting money has a clear mathematical encouragement. If we switch on the TV or we read the newspapers we find decreases of the economy, interest rates,…
Location: Louvre Museum in Paris (map)