Monument to Einstein in Ulm

Photography by Carlos Dorce

Photography by Carlos Dorce

This monument is located in the former Bahnhofstrasse B 135 (in 1880 renamed to Bahnhofstrasse 20) in Ulm where was placed the house where Albert Einstein was born in 1879.

The house was erected in 1871 and was destroyed in December 1944 in the bombardments of Ulm.

Photography by Carlos Dorce

Photography by Carlos Dorce

Einstein was born on March 14, 1879, and he lived in this house until the summer of 1880 when his fateher Hermann decided to move to Munich (on June 21, 1880, Hermann registered his family with Munich’s police).

This is a photography of Einstein’s birthplace before its destruction:

Photography by Carlos Dorce

Location: monument to Einstein in Ulm (map)

Ptolemy and Pythagoras in Ulm

Photography by Carlos Dorce

Photography by Carlos Dorce

The cathedral is one of the main attractions of Ulm because its tower is the tallest in the World (more than 161 metres!). However, one of the most interesting work of art which can be admired inside the church is the 15th century choir stalls crafted by Jörg Syrlin the Elder. There are 89 seats arranged in two rows with 90 busts of saints, Old Testament figures and classical philosophers and scholars as Ptolemy…

Photography by Carlos Dorce

Photography by Carlos Dorce

…and Pythagoras:

Photography by Carlos Dorce

Photography by Carlos Dorce

We must remember that Ptolemy represent the Astronomy in the Liberal Arts and Pythagoras usually represents the Arithmetic although his bust here is related with the Music.

Most of the people who visit the cathedral don’t know that this is one of the most wonderful medieval work of art which can be seen in Germany although this beautiful picture woul remain in their minds in a lot of years.

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Cathedral of Ulm (map)

Kepler’s last home in Regensburg

Photography by Carlos Dorce

Photography by Carlos Dorce

Kepler’s last home is this orange house located in Keplerstrasse 5 in Regensburg. Reading the famous Kepler’s biography written by Max Caspar:

[…] On November 2 [1630] he rode, tired, on a skinny nag, over The Stone Bridge into Regensburg. He took up quarters in Hillebrand Billj’s house in the street now named after him. This acquaintance was a tradesman and later an innkeeper.

Only a few days after his arrival Kepler came down with an acute illness. His body was weakened by much night study, by constant worry, and also by the long journey at a bad time of year. In the beginning he attributed no significance to his being taken ill. He had often before suffered from attacks of fever. He believed that his fever originated from “sacer ignis”, fire-pustules. As the illness became worse, an attempt was made to help him by bleeding. But soon he began to lose consciousness and became delirious. Several pastors visited him and “refreshed him with the vitalizing water of consolation”. It is not said anywhere that holy communion was afforded him. In the throes of death Pastor Christoph Sigmund Donauer rendered him aid. When, almost in the last moment of his life, he was asked on what he pinned his hope of salvation, he answered full of confidence: only and alone on the services of Jesus Christ; in Him is based, as he wanted to testify firmly and resolutely, all refuge, all his solace and welfare. At noon on November 15 this pious man breathed his last. […].

A plaque on the facade says that this is the house which I was looking for when I have arrived at Regensburg:

Photography by Carlos Dorce

Photography by Carlos Dorce

Bad luck! This small museum is only open in the weekends and it’s possible to rent a guided visit only for groups! I’ve not arrived here to give up! Finally, I’ve been able to visit it and the first thing that I’ve seen… the magnificent bust of the last owner of the house…

Photography by Carlos Dorce

Photography by Carlos Dorce

… over a plaque in German language where it’s possible to read a little part of this story:

Photography by Carlos Dorce

Photography by Carlos Dorce

The museum located in the house is very small and explains Kepler’s life and works focussing the interest in his astronomical discoveries and his three laws.

Photography by Carlos Dorce

Photography by Carlos Dorce

There is also a representation of the barrels whose volume was calculated precisely by Kepler in 1615:

Photography by Carlos Dorce

Photography by Carlos Dorce

Another bust representing the great mathematician is in the room of the first floor next to some information about his commemorative monument also in Regensburg.

Photography by Carlos Dorce

Photography by Carlos Dorce

There are a lot of Kepler’s works (which seem to be original) and this wonderful German edition of Napier’s logarithms (1631) which couldn’t be used by Kepler but exemplifies the great impact that this powerful calculator had in the beginning of the 17th century.

Photography by Carlos Dorce

Photography by Carlos Dorce

Of course, his Astronomia Nova, his Harmonices mundi,… and his Tabula Rudolphinae are also exhibited.

Photography by Carlos Dorce

Photography by Carlos Dorce

There also are explanation about his relation with Tycho Brahe and the Copernican system and a lot of astronomical instruments like sextants, globes, compasses,…

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, I want to say goodbye looking at this famous portrait. This man discovered the elliptical orbits of the palnets and his obsession with numbers let him find the second and the third law. Copernicus was right and Newton will be confirm all this theories. The World was explained (Wait Einstein, wait!).

Photography by Carlos Dorce

Photography by Carlos Dorce

Location: Kepler’s museum in Regensburg (map)

The mathematicians in the Walhalla

Photography by Carlos Dorce

Photography by Carlos Dorce

The Walhalla is a neo-classical hall of fame which honours the most important people in German history. It was conceived in 1807 by Ludwig I of Bavaria (king from 1825 to 1848) and its construction took place between 1830 and 1842 designed by Leo von Klenze.

Photography by Carlos Dorce

Photography by Carlos Dorce

The Walhalla was inaugurated on October 18, 1842 with 96 busts and 64 commemorative plaques for people with no available portrait and everything was presided by the great King Ludwig:

Photography by Carlos Dorce

Photography by Carlos Dorce

Among all these very famous people related with the German history there are some… of course… mathematicians who share this space with Bach, Göethe, Beethoven, Guttemberg,  Luther, Otto von Bismarck,…  First of all, Dürervis the great German painter from the Renaissance who applied a lot of perspective new techniques to his paintings:

Photography by Carlos Dorce

Photography by Carlos Dorce

The great astronomers are also here. Regiomontanus,…

Source: Wikimedia Commons

Source: Wikimedia Commons

Herschel,…

Photography by Carlos Dorce

Photography by Carlos Dorce

Copernicus,…

Photography by Carlos Dorce

Photography by Carlos Dorce

and Kepler:

Photography by Carlos Dorce

Photography by Carlos Dorce

The great Leibniz…

Photography by Carlos Dorce

Photography by Carlos Dorce

and the greatest Gauss (added in 2007), also have their busts in this hall of fame:

Photography by Carlos Dorce

Photography by Carlos Dorce

Finally, Albert Einstein’s bust was added in 1990:

Photography by Carlos Dorce

Photography by Carlos Dorce

I must say that the commemorative plaques also mention Alcuin of York, Albertus Magnus and the Venerable Bede, all ot them related with the wonderful Arithmetics!

Come to Regensburg to see this beautiful (and strange) place!

Location: Walhava in Donaustauf (map)

Another Kepler in Weil der Stadt

Photography by Carlos Dorce

Photography by Carlos Dorce

This statue of Johannes Kepler is located in a garden next to Kepler’s museum in Weil der Stadt. It seems to be a private monument in a garden but it’s possible to go inside.

Location: Monument in Weil der Stadt (map)

Kepler-Museum in Weil der Stadt

Photography by Carlos Dorce

Photography by Carlos Dorce

This building is the theorical Kepler’s bithplace in Weil der Stadt which hosts a very small museum about Kepler’s life and work:

At the age of six Kepler attends the German school. Continuing with Latin school he has to interrupt his attendance several times to help his parents with their work in the fields and at their inn. As a result he requires five years to complete the usual three school years.

The sickly child shows more enthusiasm at school than for hard work in the fields. His parents decide to send him to monastery school: First to the Adelberg monastery school (lower seminary) and then to Maulbronn (higher seminary).

His school comrades and teachers give him a hard time: At an early stage he starts to have his own ideas of church doctrine. His main struggle is with the meaning of Predestination and Communion.

Two celestial phenomena arouse his interest in astronomy: His mother shows him a comet, his father a lunar eclipse. Both phenomena remain in his mind for a long time. On the other hand, he never mentions his astronomy lessons in his written work.

Furthermore…

During the Age of Reformation the University of Tübingen, founded in 1477, forms the intellectual centre for Southern German Lutheran and for the Duchy of Württemberg. In 1536, Duke Ulrich orders the accomodation of poor students in Tübingen’s Stift. His aim is to ensure more graduates for loyal service in church and administration.

Coming from a humble background, Kepler wins a scholarship at the Stift. In 1589, he takes up his studies at the Faculty of Arts providing a general education, where the talented student receives many important stimuli. In particular, he studies the works of the Neoplatonists, whose ideas of a harmonically built creation make a deep impact on him.

However, his Professor of astronomy, Mästlin, influences him the most. Like a fatherly friend he familiarizes him with the ideas of Copernicus. Kepler sees an analogy in the central position of the sun to God’s omnipotence and consequently becomes a convinced advocate of the heliocentric view.

Kepler passes the baccalaureat exam at the Faculty of Arts as the second best in his class. […]. Before graduating, he accepts the position as provincial mathematician in Graz.

These were the first steps in Kepler’s life and the first thing that you see after entering the museum is the bust of this great mind:

Photography by Carlos Dorce

Photography by Carlos Dorce

Since 1594, as a provincial mathematician in Graz, Kepler…

[…] has to teach at the Lutheran seminary and write astrological calendars. His enthusiasm for astronomy inspires him to do his own research, and in 1596 he publishes his first work on astronomy Mysterium Cosmographicum.

He attempts to prove that a harmonic creation allows for only six planets. He regards the five regular Platonic polyhedra as elements of the planetary system, which, nested in the proper order, should determine the planet’s distance to the sun. As this approximately corresponds with the Copernican planetary distances, the work catches the attention of such important astronomers as Galileo Galieli and Tycho Brahe.

In spite of his fame, Kepler has to worry about his position in Graz. The Counter-Reformation creates great tension between the Lutheran inhabitants and the Catholic authorities. To recommend himself to the archduke Kepler dedicates a treatise to him on the solar eclipse of July 10th, 1600.

However, this does not prevent his expulsion from Graz one month later.

Photography by Carlos Dorce

Photography by Carlos Dorce

So these years in Graz were the period in which Kepler dreamt of his Mysterium cosmographicum and the possibility of the God’s design for the universe based in the regular polyhedra:

Photography by Carlos Dorce

Photography by Carlos Dorce

After Graz, Kepler became Tycho Brahe’s assistnat in Prague. After Tycho’s death, he assumed his position as imperial mathematician for Emperor Rudolph II:

[…] The quality of the [astronomical] data depends on the exactness of the particular orbit theory. Since all tables used around 1600 are inaccurate, Emperor Rudolph commissions Brahe and Kepler with the creation of the Tabulae Rudolphinae in 1601. When Brahe dies in the same year Kepler has to continue the work on his own.

It takes 22 years to complete the final version of the tables. Alone, the development of the elliptical orbits takes Kepler eight years. When he hears about the development of Napier’s logarithm, he integrates this into his tables and manages to simplify the calculation of orbital positions […]

Kepler discovered his first law and published it in his Astronomia nova (1609) and ten years later, he publishes his Harmonice mundi with the second and the third laws. Furthermore, Kepler had also time to wpork on infinitesimal calculus to compute the volume of some tonnels of wine:

Photography by Carlos Dorce

Photography by Carlos Dorce

I could follow explaining more things about Kepler’s life and works but this museum is very small so you must visit it. And Weil der Stadt is a very beautiful town!

Location: Kepler-Museum in Weil der Stadt (map)

Johannes Kepler Monument in Weil der Stadt

kepler0

Photography by Carlos Dorce

Weil der Stadt is located near Stuttgart. Johannes Kepler was born in this very veautiful town on December 27, 1571 and his memory is still there: this big statue is in the middle of the Market Square…

Photography by Carlos Dorce

Photography by Carlos Dorce

… and Copernicus, Mästlin, Tycho Brahe and Jobst Bürgi are with him in this monumental sculpture.

Photography by Carlos Dorce

Photography by Carlos Dorce

The four scientists are in the corners of the base of the statue and the words “Astronomia”, “Optica”, “Mathematica” and “Physica” are graved on each of the four sides.

Photography by Carlos Dorce

Photography by Carlos Dorce

I must say that here we have the first (imaginary) bust of Bürgi that I know. Bürgi was one of the originators of the logarithms because Kepler said that he had seen Bürgi using logarithms in astronomical calculus (Rudolphine Tables (1627)) before their “official” first occurrence in Napier’s  Mirifici Logarithmorum Canonis Descriptio (1614). Furthermore, Bürgi published his logarithms in his Aritmetische und Geometrische Progreß tabulen (1620) but his “red numbers” and “black numbers” couldn’t never win the “logarithms” which were the first calculator in all history.

Photography by Carlos Dorce

Photography by Carlos Dorce

Notice that this statue is not very similar to this other portrait from 1620:

Joost_buergi

J.Bürgi at the age of 67. Drawing by Aegidius Sadeler (1619)

The base of the statue also have four graved images representing moments in Kepler’s life like thispicture with Kepler in the middle explaining the Copernican system…

Photography by Carlos Dorce

Photography by Carlos Dorce

… with Hipparchus and Ptolemy watching how a central Sun brights in the middle of the universe.

kepler7

Can you imagine Kepler investigating about his elliptical orbits?

Photography by Carlos Dorce

Photography by Carlos Dorce

Next to Market Square there is his bithplace which hosts… no, no, no! Tomowwor will be another day!

Location: Weil der Stadt (map)

Abû al-Wafâ’ al-Buzjanî’s doodle

1075 aniversari del naixement d'Abu al-Wafa al-Buzjani

This beautiful doodle was published by google in the Persian and Arabic countries last 10th June because in 10th June 940 the great Abû al-Wafâ’ al-Buzjanî was born in Persia. Since 959, he worked in the Caliph’s court in Baghdad among other distinguished mathematicians and scientists who remained in the city after Sharâf al-Dawlâh became the new caliph in 983. He continued to support mathematics and astronomy and built a new observatory in the gardens of his palace in Baghdad (June 988) which included a quadrant over 6 metres long and a sextant of 18 metres.

Abû al-Wafâ’ wrote commentaries on works of Euclid, Ptolemy, Diophantus and al-Khwârizmî, and his works were very important in the developement of Trigonometry and Astronomy.

The conoids in Sagrada Família Schools

Photography by Àlvar Pineda

Photography by Àlvar Pineda

It is nothing new that Antoni Gaudi’s constructions are related to mathematics. In this post, we will focus on the conoid, a ruled surface that appears in Sagrada Família Schools (Escoles Provisionals de la Sagrada Família, in Catalan), near the Basílica i Temple Expiatori de la Sagrada Família in Barcelona.

First of all, we have to explain the concept of ruled surfaces: a surface S is ruled if through every point of S there is a straight line (called ruling) that lies on S. This implies that a ruled surface has a parametric definition of the form S(t,u) = P(t) + u Q(t).

Source: Wikimedia Commons

Source: Wikimedia Commons

As you can see, the roof of the schools is one ofthese surfaces which we call conoid: we can generate it by displacing a straight line above another straight line (the axis) and above a curve (often a sinusoid). Consequently, for every point on the conoid there is a straight line that passes trough that point and intersects de axis. If all of those straight lines are perpendicular to the axis, then the conoid is called right conoid. The conoid of this post is not in Sagrada Família but on the roof and the façade of the Escoles provisionals de la Sagrada Família. Antoni Gaudí designed that building on the commission of the entity that sponsored the project of the Sagrada Família, the Associació de Devots de Sant Josep (presided over by Josep Maria Bocabella (1815-1892)), and the school was for the children of the parish and also the children of the building workers of the temple. The building was divided in three classrooms, a hall and a chapel, and was constructed with brick. Its principal promoter was Gil Parés i Vilasau (1880-1936), the first parish priest of the Sagrada Família. He was also the school’s principal until 1930 and he used the Montessori method from 1915.

The building, inaugurated on November 15, 1909, has an amazing story of destructions and reconstructions. In fact, this peculiar school was intended to be demolished because Gaudí located it occupying land reserved for the construction of the Sagrada Família’s Passion façade. However, it was dismantled and rebuilt earlier than expected as a result of the several damages during the Spanish Civil War (1936-1939). Domènec Sugrañes i Gras (1878-1938) designed the restoration that finished in 1940, but the project had few funds and for this reason, in 1943 Fransesc Quintana (1892-1966) directed another refurbishment. Many years later, in 2002, the Passion façade was going to be built, so the building of the Escoles provisionals de la Sagrada Família was dismantled again and reconstructed in the corner between Sardenya and Mallorca streets, where the picture has been taken. In this regard, we can add that the building has become a small museum. It is important to note that the fact of being surfaces generated by straight lines makes the construction of the roof and the façade easier. Besides that, the profile of the roof is highly effective to drain off waterin a rainy day.

The contrast between the simplicity of the building (it was a very cheap and quickly erected structure) and its importance in twentieth century architecture is really remarkable.

This post has been written by Àlvar Pineda in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Sagrada Família Schools (map)

A Fractal garden in Barcelona

Photography by Àdel Alsati

Photography by Àdel Alsati

The Botanical garden of Barcelona, located in Montjuïc, has an extension of 14 hectares. It is specialized in the mediterranean climate and contains a wide range of plants from all over the world. Moreover, it is divided into the five main regions of the planet with this kind of weather, such as Chile, California, South-Africa, Australia and Southern Europe.

It was designed by the architects Carles Ferrater and Josep Lluís Canosa working in an interdisciplinary team whose two main priorities were, firstly, to distribute the plants so that they are placed together with the other ones of the same geographical region, and, in addition to that, that within every region, plants are disposed following their ecological affinities representing the different landscapes existing in those zones. Secondly, they didn’t want to do it making large earthworks.

Photography by Àdel Alsati

Photography by Àdel Alsati

They achieved the design of the park in a mathematical way since they designed the park following fractal structures: they split the land into triangles, so that every triangle contained the plants of a particular landscape, while each of the five regions was represented by a set of this triangles.

Map of the Botanical Garden

Map of the Botanical Garden

If we look at the zigzag shape of the path, and then at the trapezoidal pieces which constitute it, we can found a very good example of fractal geometry.

Photography by Àdel Alsati

Photography by Àdel Alsati

And… if we look more carefully, we’ll find it everywhere around us!

Photography by Àdel Alsati

Photography by Àdel Alsati

This post has been written by Àdel Alsati in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Botanical Garden in Barcelona (map)

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