Category Archives: U.S.A.

Pi is the guest star

Monument dedicated to Pi in Seatle
Source: Google Street View

One day by chance I discovered a Micajah Bienvenu’s sculptural work and I suppose that the picture that I had is due to the will of Destiny… I noticed that Pi was the great guest star of one of his sculptures! Reminding that discovery, I began to search in the net and I found that this Pi is or was installed at Harbor Steps in Seattle, near the corner of University St. and the 1st Avenue. However, I have not been able to discover whether the monument is still there or has been moved to another place because the author’s website says nothing about it.


I Saw the Figure 5 in Gold

I Saw the Figure 5 in Gold (1928)
Charles Demuth (1883–1935)
The Metropolitan Museum of Art (New York)

The American poet and pediatrician William Carlos William wrote the descriptive poem “The Great Figure” in 32 words in 13 verses:

Among the rain
and lights
I saw the figure 5
in gold
on a red
fire truck
to gong clangs
siren howls
and wheels rumbling
through the dark city.

The poem describes a red firetruck with the number 5 and it consists in one large sentence distributed in 13 short lines. Probably, William Carlos Williams saw this red firetruck with its great number 5 painted in it in a few seconds among the rain and this image inspired him to combine the words in this odd poem (there is a complete analysis of “The Great Figure” in this web).

Between 1005 and 1908, the versatile painter Charles Demuth met Williams in Philadelphia and twenty years later (1928), he dedicated one of his eight abstract portraits of his friends to Williams. This portrait entitled “I Saw the Figure 5 in Gold” pays homepage to William’s poem associating an accumulation of images related to him. For example, the names Bill and Carlos and the initials WCW are clearly distinguished. We can also see a big golden 5 in front of another 5 and red rectangles in the middle of a mess of lights and chaotic grey colors. I think that it’s easy to imagine the big 5 painted in the firetruck, isn’t it?

Location: The Metropolitan Museum of New York at New York (map)

Thales of Miletus in Nuremberg’s Chronicle

Portrait of Thales of Miletus
Nuremberg Chronicle (1493). Font: Wikimedia Commons

The Nuremberg Chronicle was written by Hartmann Schedel (1440-1514) in the XVth century to illustrate the human history from the Biblical passages up to the Middle Ages. It was printed in Nuremberg in 1493 and it became one of the best documented early printed books. The illustrations were drawn by Michael Wohlgemut (1434-1519) and his stepson Wilhelm Pleydenwurff (c. 1450-1494) and it’s not impossible that  Albrecht Durer (1471-1528) also collaborated in them.

The Chronicle also contains numerous genealogies and family trees and a lot of portraits of gods, kings, writers and philosophers and Schedel wrote about the great mathematician Thales of Miletus among them (folios LIX recto and XL verso):

Thales, the Asiatic philosopher and first among the Seven Sages of Greece, flourished in Athens at this time. The Seven Sages were named after him.

The Seven Wise Men of Greece, or the Seven Sages, as they are also called, were the authors of the celebrated mottoes inscribed in later days in the Temple of Apollo at Delphi: “Know yourself” by Solon of Athens, “Consider the end” by Chilon of Lacedaemon, “Know your opportunity” by Pittacus of Mitylene, “Most men are bad” by Bias of Priene, “Nothing is impossible to industry” by Periander of Corinth, “Avoid excesses” by Cleobulus of Lindus and “Certainty is the precursor of ruin” by Thales of Miletus.

The origin of the title “Seven Wise Men” was this: Some fishermen of Miletus sold a draught of fish to some bystanders before the net was drawn in. When the draught came in, there was also in the net a golden tripod. The fishermen claimed they had sold only the fish, while the buyer insisted he had bought the whole draught. To settle the dispute they referred the matter to the Oracle of Delphi. Being ordered to adjudge the tripod to the wisest man in Greece, they offered it to their fellow citizen Thales; but he modestly replied that there was a wiser man than he, and sent it to Bias. He also declined, and sent the tripod to another; and thus it passed through seven hands, and these seven were afterward called the “Seven Wise Men of Greece.” It was finally placed in the Temple of Apollo at Delphi. These seven men met together but twice—once at Delphi, and again at Corinth.

Thales was the first among the philosophers to practice astrology and to predict an eclipse of the sun. He acquired a knowledge of geometry from the Egyptians. He was also an excellent counselor in matters pertaining to civic customs. He had (as they say) no wife, and when asked why he did not take one, he replied that it was because of his love of children. He contended that water is the origin of all things, and stated that the world was associated with and born of the devil. It is said that he also invented the year, and divided it into 365 days. He wrote on the subject of astronomy, and his writings are comprehended in 200 verses. When a golden table (tripod) was accidentally found, and there was a misunderstanding as to whom it belonged, Apollo answered that it should be awarded to him who excelled all others in wisdom. So it was offered to Thales, but he gave it to Bias and Bias Pitachus. At last the table came to Solon, but he turned it over to Apollo, as a token of most renowned wisdom. Thales was poor and he devoted himself to the acquisition of wisdom. Item: By means of astronomy he was able to predict fruitfulness in future years. One night when he was led out of his house by an old woman to study the stars he fell into a hole. And the old woman said to him, If you cannot see what lies at your feet, how can you expect to recognize the things that are in the heavens? He died at 78 years of age. Thales of Miletus was born about 636 BCE, and according to the weight of authority he died about 546 at the age of 90; however, both dates are uncertain. Some say he was of Phoenician extraction, and this is probably the reason why the chronicler calls him an Asiatic philosopher. It is more probable, however, that he was born in Miletus. As a Greek natural philosopher his fame among the ancients was remarkable. He is said to have predicted the eclipse of the sun which occurred in the reign of the Lydian king Alyattes; to have diverted the course of the Halys, or Red River, the greatest stream of Asia Minor in the time of Croesus; and later, in order to unite the Ionians when threatened by the Persians, to have instituted a federal council in Teos, as the most central of the twelve cities. The application of wise man was conferred on him, not only for his political sagacity, but also for his scientific eminence. He became famous by his prediction of the eclipse that did actually take place during a battle between the Medes and the Lydians, and being total, caused a cessation of hostilities and led to a lasting peace between the contending nations. Thales was one of the founders of philosophy and mathematics in Greece. He maintained that water is the origin of all things. […]

Thales the Milesian, one of the Seven Sages, is considered famous. After the theologians and the poets, they were called ‘Wise’, that is, ‘Sages’. This Thales was the first who was able to predict an eclipse of the sun and moon (as Augustine says). The preceding folios make clear the accomplishments and words of these men […]

Talking about Pherecydes, Schedel says that he was master of Pythagoras and…

wrote many letters to Thales the natural philosopher, receiving many from him in return.

He also says that…

Anaximander, philosopher and celebrated scholar, was at first a disciple of Thales.

In his Commentary to the Book I of Euclid’s Elements, Proclus (412-485) says that Thales…

first went to Egypt and thence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attack being in some cases more general, in others more empirical.

Plutarch (c.46-120) says of him as one of the Seven Wise Men:

he was apparently the only one of these whose wisdom stepped, in speculation, beyond the limits of practical utility: the rest acquired the reputation of wisdom in politics.

[We can observe as a curiosity how the portrait of Plutarch is similar to the portrait of Thales!]

Portrait of Plutarch
Nuremberg Chronicle (1493). Font: Wikimedia Commons

I am going to look for another touristic reference to Thales to explain more things about him. He was the man who is supposed to introduce the geometry in Greece so I think he deserves another post.

Location: The Metropolitan Museum of Art at New York (map)

LocationV&A at London (map)

A Platonic Dalí in Washington

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.

And then…

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.

Location: National Gallery of Art in Washington (map)

Pythagoras’ Theorem in Ancient Mesopotamia

One of the more clear extant proofs of the knowledge of Pythagoras’ theorem by the ancient Mesopotamians is the clay tablet YBC 7289 (Yale Babylonian Collection). We can see in the tablet a square in which the two diagonals are drawn and three numbers are the clues to understand this representation. First of all, next to one of the four sides we can read the number 30 which is the measure of the side. The other two numbers are written in the middle of the square: 42;25,35 and 1;24,51,10 (in sexagesimal notation). The first one is the corresponding measure of the diagonal and it’s equal to the product 30 · 1;24,51,10 so it’s not difficult to deduce that 1;24,51,10 is the value of the corresponding square root of 2. This value is a very accurate approximation of the real value because we can compute:

1,24,51,102 = 1,59,59,59,38,1,40 

If we want to see the details of the tablet we can read page 27 of the book by A.Aaboe entitled Episodes from the Early History of Mathematics, Washington, D.C.: MAA, 1998 (originally published in 1964) which is possible to find very easy in internet:

Location: University of Yale (Yale Babylonian Collection)