Denmark and Sweden have been two expensive countries for my students. So in our first day in Copenhagen we had to manage for finding a cheap restaurant to have lunch… and we got it! We noticed it in an advertisement and we decided to try it. The restaurant was the Greek restaurant called Samos located in Skindergade 29. The restaurant was very homely and the food is so good. The buffet costed 49 DKR plus the beverages.
I think that I will come back if I visit Copenhagen again but not only for its nice Greek food. There is a representation of Pythagoras in the napkins! So if you want to eat Greek food being observed by Pythagoras you must try the restaurant Samos in Copenhagen!
Doge’s Palace in Piazza San Marco is one of the most touristic attractions of Venice. The palace (XIVth c.) is very beaytiful and there is a hidden mathematical secret in one of the capitals of its columns. The capitals of the columns of the palace are dedicated to some biblical passages, quotidian Medieval scenes and… there is one capital dedicated to the Liberal Arts! So we can find here our famous three most representative figures of the Arithmetic, the Geometry and the Astronomy:
Pythagoras is counting money and next to his coins we can read the number 1399 and Euclid has a compass in one of his hands. The column is the first one next to the corner in front of the sea:
You must see this column in Venice!
Location: Piazza San Marco (map)
In room 56B of the Museo del Prado we can admire Giovanni dal Ponte’s Seven Liberal Arts. There are some masterpieces in the same room 56B as Fra Angelico’s Annuntiation and therefore people don’t use to stop in front of this mathematical panel. In the web of the museum we can read:
This decoration of the front of a chest depicts the seven Liberal Arts, accompanied by an equal number of figures that represent the most relevant personages in each discipline. All are being crowned with laurel wreaths by small angels.
Astronomy presides over the composition, carrying the heavenly sphere, with Ptolemy (first and second centuries A.D.) sitting at his feet and reading one of the thirteen volumes in which he surveyed the history of Greek astronomy. To the right, Geometry holds an angle iron and a compass, walking hand-in-hand with Euclid (fourth and third centuries B.C.). He is followed by Arithmetic, who carries a counting tablet and is accompanied by Pythagoras (sixth century B.C.). At the right end of the composition, Music bears an organ, followed by its inventor, Tubalcain. To the left of Astronomy, Rhetoric carries a scroll and is accompanied by Cicero (first century B.C.), who carries one of his texts. Then comes Dialectics, who carries an olive branch as a symbol of agreement among the Arts, and a scorpion, whose pincers represent the opposing positions of dialectical thought. He is accompanied by Aristotle. At the left end of the composition is Grammar, with its disciplines, preceeded by two children and accompanied by Donatus (fourth century A.D.) or Priscian (fifth and sixth centuries A.D.).
This work exemplifies the coexistence in the arts of that period between the late Gothic heritage —visible in the use of gold and lineal calligraphy— and the new Renaissance style, which is clear in the solid and monumental definition of the figures, recalling works by Masaccio (1401-1428)
Our Ptolemy, Pythagoras and Euclid are the guest stars again and we have here a mathematical reason to visit room 56B. For example, Euclid is following the Geometry who is wearing a ruler and a compass:
After Euclid, Pythagoras is following the Arithmetic who holds a counting tablet:
Finally, Ptolemy is below the Astronomy:
De la Hire also painted the allegories of the Geometry and the Astronomy. The Geometry is a young woman with a paper in her left hand in which we can see some geometrical constructions as the famous Euclid’s demonstration of the theorem of Pythagoras:
In her left hand, she also holds a right angle edge and a compass. We can also see that there is a sphinx and an Egyptian background in the right which represents the Egyptian origin of the Geometry. Proclus stated that:
Since, then, we have to consider the beginnings of the arts and sciences with reference to the particular cycle [of the series postulated by Aristotle] through which the universe is at present passing, we say that, according to most accounts, geometry was first discovered in Egypt, having had its origin in the measurement of areas. For this was a necessity for the Egyptians owing to the rising of the Nile which effaced the proper boundaries of everybody’s lands.
Herodotus says that Ramses II distributed the land among the Egyptians in equal rectangular plots on which he levied an annual tax. When therefore the river swept away a portion of a plot and the owner applied for a corresponding reduction in the tax, surveyors had to sent down to certify what the reduction of the area had been.
The Geometry is next to a globe which nods to the science devoted o measuring the Earth (Geo + metry = Earth + measurement). There is also a snake representing the ancient goddess of the Earth.
Today is March 18, 2013 and Philippe de la Hire was born on March 18, 1640. Philippe de la Hire was a French mathematician who worked on astronomy and the conic sections. He provided an exposition of the properties of the conic sections and he applied the analytic geometry to some indeterminate problems about intersection of curves.
Philippe de la Hire’s father was the painter Laurent de la Hire (1606-1656) who got a lot of commissions from distinguished politicians, the Church and rich Parisian who wanted to have a portrait in their houses. But Laurent de la Hire had also time to paint the Allegories of the Liberal Arts and the Arithmetic can be enjoyed in the Walters Arts Museum of Baltimore.
The Arithmetic holds a book in which we can read the name of Pythagoras (c. 570 aC – c. 495 aC) and there is a paper on the book with the words “par” (“even”) and “impar” (“odd”) and an addition, difference and a multiplication. Can you imagine the young Philippe watching his father painting this picture? Maybe Philippe found his way to Mathematics in that moment!
The Spanish Chapel is one of the most wonderful chapels which can be enjoyed in Santa maria Novella. My students didn’t want to visit it but the teacher could convince most of them to enter the church and they weren’t disappointed.
The fresco entitled The Triumph of St. Thomas Aquinas was painted by Andrea di Bonaiuto (1365-1367) and it was dedicated to…
the great Dominican Doctor of the Church who, illuminated by the spirit of Wisdom, as described in the book lying open in his hands, and supported by the Theological and Cardinal Virtues and the study of the biblical writers of both the Old and New Testaments, defeats heresy, personified by Nestor, Arius and Averroes, and dominates the sciences. These are represented by fourteen allegorical female figures, alluding in part to the Sacred Sciences (left) and in part to the Liberal Arts (right). Each of these is accompanied by a historical personage, famous for having distinguished himself in that articular discipline.
The Theological and Cardinal Virtues are the Charity (over St. Thomas), the Faith and the Hope (at her respectively left and right sides), the Prudence (below the Faith), the Temperance (at the left side of the Prudence), The Justice (below the Hope) and the Fortitude (at her right). On the left of St. Thomas, there are (from left to right) the Biblical authors Job, David, St. Paul, St. Mark and St. Matthew, and on his right (from left to right), St. John the Evangelist, St. Luke, Moses, Isaiah and Solomon. Below St. Thomas, we find Nestor, Arius and Averroes:
The fourteen allegorical women and the corresponding eminent men are (from left to right): the Civil Law with Justinian, the Canonical Law with Clement V, the Philosophy with Aristotle, the Holy Scriptures with St. Jerome, the Theology with St. John of Damascus, the Contemplation with St. Dionysius the Areopagite, the Preaching with St. Augustine, the Arithmetic with Pythagoras, the Geometry with Euclid, the Astronomy with Ptolemy, the Music with Tubalcain, the Dialectics with Pietro Ispano (?), the Rhetoric with Cicero and the Grammar with Priscian (?):
Finally, here you are my privileged students who enjoyed the wonderful Spanish Chapel:
Another of the important equations which stands out in the main entrance of Cosmocaixa in Barcelona is Fermat’s last theorem:
Fermat was reading Diophantus’ Arithmetic about the pythagorean triples x2 + y2 = z2 in 1637 when he noticed immediately that n = 2 was the only case which satisfies the equation xn + yn = zn. Then, he wrote in the margin of his edition of Diophantus’ work:
it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain
Probably, Fermat was only able to prove the new theorem for n = 4 using his proof by infinite descent: Fermat proved that there are not three numbers x, y, z such that x4 + y4 = z2 . First of all, we can suppose that x, y, z are co-prime from (x2)2 + (y2)2 = z2. If <x,y,z> = d, then we can divide all the equation by d2 and we’ll have another equation (x2)2 + (y2)2 = z2 with <x,y,z> = 1.
Now, from the solution of the Pythagorean triples, there are p, q such that x2 = 2pq, y2 = p2 – q2 and z = p2 + q2.
We can observe that y2 = p2 – q2 implies that p2 = q2 – q2 is another Pythagorean triple so there are u, v co-prime such that q = 2uv, y = u2 – v2 and p = u2 + v2.
So, x2 = 2pq = 2(u2 + v2)(2uv) = 4(uv)(u2 + v2). Since <p,q> = 1, we know that <uv,u2 + v2> = 1 and since their product by 4 is a perfect square, they must be perfect squares too. So there exists a < p such that a2= u2 + v2 = p.
Finally, if we had (x2)2 + (y2)2 = z2 , we have obtain p and q co-prime with z = p2 + q2. Then it is possible to obtain another pair u and v co-prime with p = u2 + v2 and p < z, u < p and v < q. So we can iterate this algorithmic procedure so we’ll obtain a set of pairs of natural numbers each of them lower than the previous and this fact is false, since we’d have an infinite decreasing succession of natural numbers without end!
Nobody accepts that Fermat could have known any other case apart of n = 4! In 1753, Leonhard Euler wrote a letter to Christian Goldbach telling him that he had proved the case n = 3 but the demonstration that he published in his Algebra (1770) was wrong. He tried to find integers p, q, z such that (p2 + 3q2) = z3 and he found that:
p = a3 – 9ab2, q = 3(a2b – b3) ⇒ p2 + 3q2 = (a2 + 3b2)3
He worked with numbers of the form a + b√-3 to find two new numbers a and b less than p and q such that p2 + 3q2 = cube and then he applied the method of infinite descent. Unluckily, he made some mistakes working with the new complex numbers a + b√-3.
This story is so exciting but, almost quoting Fermat, this post is too short to contain everything. If you are interested in, you must read Simon Singh’s Fermat last theorem which is as interesting as the theorem itself.
When you enter the Cosmocaixa building in Barcelona (the museum of science), you notice that the main door is surrounded by a very big panels full of Mathematical, Chemical and Physical equations:
The reason of this decoration is:
Not everything imaginable happens in reality. Objects and facts have inviolable restrictions: the laws of nature. Its knowledge helps to fulfill an old dream of living beings: to anticipate the uncertainty. The laws are written with mathematical equations, a relationship between relevant quantities (mass, energy, charge …) and their changes over time and space. An equation is rises its fundamental law as greater is its scope. A great equation of a law is like a poem: a concentrate of intelligibility and beauty.
Curiously, one equation that stands out is:
The demonstration of this formula is very easy. If we suppose a, b > 0, then we can square both terms of the inequality to obtain:
4 a b ≤ (a + b)2 = a2 + b2 + 2 a b ⇒ 0 ≤ a2 + b2 – 2 a b ⇒ 0 ≤ (a – b)2
which is obviously right.
The geometric and arithmetic means between two numbers was known in time of Pythagoras (and also the harmonic mean: a, b and c are in harmonic progression if a(b-c) = c(a-b)). Archytas (c.430 BC-c.360 BC) defined the three means in his On Music . Iamblicus (IIIrd c.) relates that Pythagoras knew them and Archytas and Hippasus adopted the name of harmonic mean instead of subcontrary mean. Heath (A History of Greek Mathematics, 1981) says that:
Nichomachus [of Gerasa] too says that the name ‘harmonic mean’ was adopted in accordance with the view of Philolaus about the ‘geometrical harmony’, a name applied to the cube because it has 12 edges, 8 angles and 6 faces, and 8 is the mean between 12 and 6 according to the theory of harmonics.
The arithmetic mean of 6 and 12 is 9 and their geometrical mean is 8,4852…
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.
Salvador Dalí (1904-1989) is one of the most important surrealistic painters. This eccentric Catalan painter finished in 1955 the Sacrament of the Last Supper after nine months working in it and this painting is mathematically remarkable because of the dodecahedric form over Christ and the twelve apostles. The thirteen sacred figures are having supper when it’s not at night and we can see a beautiful Catalan landscape behind the main scene. If Dalí wanted to capture this Biblical moment, why the dodecahedron was painted over it? The dodecahedron and its twelve sides may be related to the twelve apostles. Furthermore, a philosophic point of view can show to us a platonic idea for this famous painting. Plato (c.428 BC-c.347 BC) wrote in the IVth century BC the Timaeus where he associated the four basic elements to fours regular polyhedra: tetrahedron was the fire, hexahedron was the earth, octahedron was the air and icosahedron was the water. Thus, the universe was in perfect harmony with Mathematics!
Firstly, Plato described two kinds of triangles as the base of all the nature:
In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes ; and every plane rectilinear figure is composed of triangles ; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles ; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies
With these two triangles, Plato built the squares and triangles which had to be the sides of the regular polyhedra:
Now is the time to explain what was before obscurely said : there was an error in imagining that all the four elements might be generated by and into one another ; this, I say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds: three from the one which has the sides unequal ; the fourth alone is framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. […]. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles ; and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles ; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side] having generated these figures, generated no more ; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles ; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases.
There was yet a fifth combination which God used in the delineation of the universe.
Great Plato! Which is this fifth combination? The answer is now so easy: the dodecahedron! The platonic solid which was used by God to draw our universe. It’s easy to understand the reason why this harmonious figure is presiding the scene.