Tag Archives: Sphere

Pi Day’s doodle

Today is Pi Day!

Google published this doodle four years ago. We can see some formulas in it related with circles, spheres, trigonometry and the Archimedian value of pi.

Advertisements

The Museum of the History of Science (III)

Photography by Carlos Dorce

Photography by Carlos Dorce

The mathematicalinstruments are also part of the collection. For example, there is a 17th-century box with some wooden polyhedra and some models for the study of Spherical Trigonometry:

Photography by Carlos Dorce

Photography by Carlos Dorce

And more wooden models in this mathematical box:

Photography by Carlos Dorce

Photography by Carlos Dorce

John Rowley was one of the leading London instrument makers in the late 17th and early 18th centuries and there are some mathematical compasses and instruments made by him in the collection:

Photography by Carlos Dorce

Photography by Carlos Dorce

Object number 1 is a proportional compass meanwhile number 5 is a ruler with pencil and dividers and number 6 is a slide rule.

Of course, if we are in a museum where the History of the Mathematics is exhibited, Napier’s rods must be here:

Photography by Carlos Dorce

Photography by Carlos Dorce

Unsigned, English, c. 1679

Photography by Carlos Dorce

Photography by Carlos Dorce

Unsigned, English, 17th century?

Photography by Carlos Dorce

Photography by Carlos Dorce

Charles Cotterel’s Arithmetical Compendium, Unsigned, English, c.1670

As in the Pitt Rivers Museum, the abacus also have their space in the showcases:

Oriental abacuses use beads on rods to represent numbers. Addition and substraction can be quickly performed by flicking the beads to and fro. Rather than ten beads in each column, the Chinese abacus uses five ‘unit’ beads and two ‘five’ beads (1 and 2). The Japanese abacus has just four ‘unit’ beads and one ‘five’ in each column (3).

Photography by Carlos Dorce

Photography by Carlos Dorce

The next Arithmetical instrument was made in the 18th century for counting. Addition was performed by turning the brass discs but since there isn’t no mechanism it was up to the user to carry tens:

Photography by Carlos Dorce

Photography by Carlos Dorce

I am going to finish this post with this reproduction of the Measurers by the Baroque painter Van Balen (1575 – 17 July 1632) which can be seen upstairs:

The Measurers. Hendrick van Balen

Location: The Museum of the History of Science (map)

Fiona Banner Full Stop Courier 2003

Photography by Carlos Dorce

Photography by Carlos Dorce

Today has been our last day in London and we’ve decided to go for a walk along the Thames and we’ve discovered this set of ‘mathematical’ sculptures. Probably they aren’t too important but I’ve liked a lot.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: map

Millenium Square in Bristol

Photography by Carlos Dorce

Photography by Carlos Dorce

The Millenium Square in Bristol is a very interesting square next to Bristol’s old port. Its main building is At-Bristol which is a science centre with this big “sphere” in the left of the facade. There are fountains, sculptures ad there also is a very big sundial in the middle of the square:

Source: Google Maps

Source: Google Maps

Furthermore, the ack facade of At-Bristol has a corious cone in front of it:

Photography by Carlos Dorce

Photography by Carlos Dorce

I’m sure that it’s possible to discover more mathematical objects in this square. If you find them, please let me know!

Location: the Millenium Square in Bristol (map)

The wonderful vault of a Royal Library

Library of the monastery
Source: Wikimedia Commons

The King Philip II of Spain decided in 1550’s that he wanted to have a great library near his court in Madrid and he chose the new Monastery of San Lorenzo de El Escorial to place it in spite of other bigger villages. He didn’t want that the new library was a regular room inside a monastery so it had to be a very important place. Therefore, the library was placed on the second floor of the monastery just above his royal chambers but never above the basilica. Between 1565 and 1576, the king bought almost 5.000 books and manuscripts and the library became one of the most important libraries in all Europe.

Imaginary portrait of Juan de Herrera (1791) from
the book “Retratos de Españoles ilustres
Source: Wikimedia Commons

The mathematician and architect Juan de Herrera (1530-1597) designed a large room (54 m. long x 9 m. wide x 10 m. high) with big windows in both sides under a great barrel vault. This vault had to be decorated by an important painter and Philip II decided that Peregrino Tibaldi (1527–1596) had to be the right artist to do the work. Philip II was advised by Juan de Herrera and other humanists and he decided that the main subject of the paintings of the vault had to be the Liberal Arts. Furthermore, the seven arts would be together with the Philosophy and the Theology on both ends of the room. The Philosophy represented the compendium of the Human knowledge and she is accompanied by Aristotle, Plato, Seneca and Socrates:

The Philosophy

The Philosophy

The Theology is on the side next to the convent and she represented the Divine knowledge. Therefore the vault represented the way from the Human Philosophy to the Divine knowledge through the seven Liberal Arts: the Arithmetic, the Geometry, the Astronomy (Astrology), the Music, the Rhetoric, the Grammar and the Dialectic. We can see a mathematical detail on the fresco below the Philosophy: it represents the School of Athens and there is a discussion between the Academics leaded by Socrates and the Stoics leaded by Zeno of Elea.

The School of Athens

The School of Athens

The scholars aren’t listening to the speakers because each of them is “playing” with something different. We can see at the lower left corner a man measuring something with a compass and two books, a sphere and an armilar sphere, a dodecahedron and a compass in the middle of the picture:

Detail of the School of Athens

Detail of the School of Athens

Going from the Philosophy to the Theology, we arrive at the Arithmetic after admiring the Grammar, the Rhetoric and the Dialectic. The Arithmetic is a woman turned to a table with simple mathematical operations rounded by muscled young men with tablets with arithmetical operations ans counting with their fingers:

The Arithmetic

The Arithmetic

There is also a representation of the Queen of Saba talking with King Solomon  According to the Book of the Kings (I,10,1), the Queen of Saba went to meet Solomon to ask some enigmas to him so we can see a ruler, a balance and a tablet with some numbers written on it. In the red tablecloth we can read “Everything has number, weight and measure” in Hebrew:

King Salomon and the Queen of Saba talking about numbers

King Salomon and the Queen of Saba talking about numbers

The other panel next to the Arithmetic represents the school of the Gymnosophists who lived near the Nile and thought their philosophical theories from the numerical computations. In the middle of the picture we can see one of the gymnosophist with a compass looking at a triangle with the word “Anima” and the arithmetic progression 1, 2, 3 and 4 and the geometric 1, 3, 9 and 27 written on it. The other gymnosophists are computing with numbers written on the sand:

The Gymnosophists

The Gymnosophists

Finally, at both sides of the Arithmetic on the roof we find four people related with this subject: Archytas of Tarentum (c.428–c.347 BC) and Boethius (c.480-c.525) in one side and the Platonic Xenocrates (c.396/5 – 314/3 BC) and Jordan in the other. They are writing numbers in their tablets.

There is the Music after the Arithmetic and we find the Geometry after it:

The Geometry

The Geometry

She has a compass in one of her hands and the young men around her have different geometrical instruments. The two scenes which are on the corresponding walls next to her are dedicated to some Egyptian monks drawing geometrical figures on the sand…

Egyptian monks measuring the lands

Egyptian monks measuring the lands

and Archimedes’ death:

Archimedes' death

Archimedes’ death

Notice that Archimedes is drawing the demonstration of the Theorem of Pythagoras made by Euclid!

Finally, the four chosen figures are the Astronomer Aristarchus of Samos (IIIrd c. BC) and the Persian astrologer Abd del Aziz also known as Alcabitius (Xth century) in one side and Archimedes (c.287-212 BC) and Regiomontanus (1436-1476)  in the other. Aristarchus is measuring angles and has a dodecahedron at his feet, Alcabitius has a carpenter’s square, Archimedes has a compass and a sphere to measure the Earth and Regiomontanus is pointing at a dodecahedron.

The last Liberal Art is the Astrology. She is backed on a terrestrial globe and her eyes are looking at the sky. She has a compass in one of her hands and the little boys around her have an armilar sphere and some astronomical books:

The Astrology

The Astrology

In one of the two panels on the walls we can see Dionysius the Areopagite observing a solar eclipse the day of Jesuschrist’s death in Athens (Luke, 23,45) We can notice a quadrant and an astrolabe in the hands of the amazed men!

Dionysius the Areopagite observing a solar eclipse

Dionysius the Areopagite observing a solar eclipse

The other fresco represents King Ezekiel resting in bed and looking how time is delayed 15 years by God because of the repentance of his sins:

astronomia2

Ezekiel resting in bed

The four famous men are Euclid, Ptolemy, Alfonso X and Johannes of Sacrobosco. Euclid is represented here meaning the relationship between Astrology and Geometry. He has drawn three geometrical schemes. One is a triangle and a square inscribed in a circle and another square. Another scheme seems to be two overlaid squares partially hidden by Euclid’s name. In the middle of both pictures there is a man measuring the stars. Johannes of Sacrobosco has a quadrant in his right hand.

Euclid and Johannes of Sacrobosco

Euclid and Johannes of Sacrobosco

King Alfonso X of Castile (XIIIth. c) is the author of the Libros del Saber de Astronomía (“Books of the Astronomical knowledge”) and on the tablet which he has in his hands we notice a compass and the Ursa Maior (the compass is anachronistic!). His left hand has an open book with a horoscope

Alfonso X

Alfonso X

So you can see that this wonderful vault is an open mathematical book designed by Tibaldi and Juan de Herrera. I’ve been twice in the library and now I am waiting for the next time that I could enjoy this artistic part of the monastery of San Lorenzo de El Escorial.

Location: San Lorenzo de El Escorial (map)

A celestial globe in the Singer House

Celestial globe in Singer HousePhotography by Carlos Dorce

Celestial globe in Singer House
Photography by Carlos Dorce

The Singer House was designed for the Russian branch of the Singer Sewing Machine Company. We fix our attention in it because of the glass globe crowned by a celestial globe of the corner. When the Singer House was built it was forbidden o build a house higher than the Tzar’s Winter Palace (23,5 m). Then the architect Pavel Suzor decided to construct this tower in the corner of the building so it made an impression of high rise. Its architect was Amandus Adamson. Today it’s the biggest book shop of the city.

The history of the building is quoted in the St. Petersburg encyclopedia:

THE HOUSE OF BOOKS (28 Nevsky Prospect) is the biggest book department store in St. Petersburg. It was housed in the former building of the Zinger company: the seven-storied building with a high corner tower crowned with a glass sphere is a shining example of art nouveau architecture (1902-04, architect P. Y. Suzor, sculptors A. G. Adamson, A. L. Ober). The Consulate of the USA was housed here in the 1910s. The building was given to the book storehouse of the Petrograd State Publishing House in 1919. Soon the House of Books became the main book store of Leningrad, at the beginning of the 1930s it passed to the jurisdiction of LenKOGIZ. In the years of the siege it continued to work intermittently up to the end of November 1942. It was opened anew, after the repairs, in November 1948. The trade in the House of Books is carried on the ground floor and on the first floor; approximately 20,000 books of various spheres of knowledge were in assortment in 2003. Departments of inquiry and bibliography and the department Book by Correspondence operate. There are subsidiaries on 30, Liteiny Avenue, 4, Bolshoy Avenue of Vasilievsky Island, 9, Shevchenko Street, 20, Lenina Street etc. Editorial offices of the journals Kniga i Revolyutsiya (The Book and the Revolution), Literaturnye Shtudii (Literary Studies), Zvezda (the Star), Leningrad Departments of the publishing houses the Children’s Publishing House, the Soviet Writer, the Art, the Enlightment, Fiction, the union Leningrad Book etc. were housed in the House of Books in different years. Book trade was transferred stage-by-stage to 62 Nevsky Prospect due to the reconstruction of the building on 28 Nevsky Prospect.

Source: Wikimedia Commons

Location: Singer House (map)

Finnish “Peace in the World”

Monument to the Peace in the World (1989)Photography by Carlos Dorce

Monument to the Peace in the World (1989)
Photography by Carlos Dorce

This monument is in the Kallio district of Helsinki (next to the sea). It was a present from Moscow to the city of Helsinki made by O. S. Kirjuhin. It represents the Peace in the World and people is holding a terrestrial globe surrounded by a garland of laurel. There are represented the equator and eight meridians on the globe.

Location:  Halamiemenranta (Kallio district)

The Sphere protects

Presentation of the exhibition “The Sphere protects”
Cosmocaixa (Barcelona)
Photography by Carlos Dorce

We can define the three-dimensional sphere as the set S2 = {(x,y,z) ∈ ℝ: x2 + y2 + z2 = R2}. It’s one of the most famous shapes studied in schools and we find it in the nature very easily. In the Cosmocaixa of Barcelona there is a special exhibition about shapes and the sphere is one of the main protagonists of it. The sphere protects and the reason hat we can read in one of the explanations is:

In the inert world, the sphere emerges easily in isotropic conditions, that is, when no one direction in space has priority over another. That is why stars and planets are round. That is why, too, if we blow air into a liquid, the bubble adopts this form, the minimum that contains a given volume. Natural selection favours circular symmetry in living beings (eggs, medusas, sea urchins, fruit, seeds, etc.) for two reasons: a) because it occupies the minimum surface, the sphere is the frontier with the exterior that loses heat most slowly and b) because, not having any edges, it is the most difficult shape to catch or bite. In both cases, then, the sphere protects.

We can also see different examples of this fact like these four examples: 1) These radial aggregates of pyrite crystals from Peru:

Radial aggregates of pyrite crystals
Cosmocaixa (Barcelona)
Photography by Carlos Dorce

2) These Marcasite nodules (FeS2):

Marcasite nodules
Cosmocaixa (Barcelona)
Photography by Carlos Dorce

3) These Dinosaur eggs (Titanosauridae) from China:

Dinosaur eggs (70 million B.C.)
Cosmocaixa (Barcelona)
Photography by Carlos Dorce

4) This colonial coral (Scleractinia) from the Pacific Ocean:

Colonial coral
Cosmocaixa (Barcelona)
Photography by Carlos Dorce

The exhibition is very interesting and it tries to answer the question “Why are some forms more common than others?”:

What do a star like our Sun, a planet, a fish egg, an orange, a bubble and the point of a pen have in common? All these objects, inert, living or manmade, share the same form: they are spheres. Why are there so many spheres, circles and circumferences? Does the fact that something is spherical help in some way? Spheres are found more frequently than other forms. What are the forms that we are most likely to find in nature? Does being circular, spiral, hexagonal or fractal in shape serve any prpose? Do objects with the same form share anything else apart from their shape? Perhaps they share the same function, that is to say, the property that helps the object in question, wether inert, living or manmade, to persist in nature.

As you can imagine, I must talk about spirals, hexagons,… but I’ll do it in next posts!

LocationCosmoCaixa at Barcelona (map)