# How are math and art related?

While there are innumerable pictorial and plastic representations of certain mathematicians and, as we saw above, quite a few representations of mathematical objects or concepts in the sense of generalized descriptive geometry, the number of such art objects is something about mathematics as a whole without reference to a particular person as a social phenomenon or their position in the world, rather small.

Woodcut by Reisch | The school of Athens |

As a first example we show the woodcut from the book "Margarita philosophica" by mathematicians, which is very well known **Gregor Reisch** (1504), who allegorically describes the current contest between the abacus adopted by the Romans (i.e. calculating with calculating stones on a lined calculating table, board or cloth or with balls that are moved on wires) and the algorithmic advancing from the Orient Arithmetic shows. On the woodcut, the followers of the abacus calculation are (by mistake) through **Pythagoras** (around 580-around 500 BC) and the supporters of the more advanced algorithmic computing (equally wrong) by the Romans **Boetius** (480? -525?) Represents. They are also named with the appropriate names. Arithmetic personified stands as the referee in the background, and the facial expressions of the opponents clearly show how the competition ended. In that the woodcut illustrates in a drastic and humorous way the subordination of the old calculation method, it represents a central problem of everyday mathematics at the time, which affected everyone who had to calculate. Involuntarily one wonders whether and how one could artistically implement the current situation of that part of mathematics with which not only professional mathematicians come into contact and their position in the world in a similarly convincing manner. The best way to do this is perhaps with a certain type of collages and photo montages that have come into fashion, in which classic images are combined with set pieces from computer technology, including the poster for our exhibition.

To the right you can see the famous mural "School of Athens" (1508-11) by **Raffael**located in the Stanza della Segnatura of the Vatican. With the "school" is obviously meant the Academy of Plato, which, however - typical of the Renaissance - has been moved to a building from the time of Raphael. The people visible in the picture represent famous scholars of antiquity such as Euclid, Archimedes, Ptolemy, but bear the features of Raphael's contemporaries: Leonardo da Vinci, Michelangelo, Bramante, ...

Euclid on the Beautiful fountain (nuremberg) | Frontispiece to Opera Omnia |

The figure of Euclid on the beautiful fountain in Nuremberg could of course also be included in the mathematician's portraits. Apart from the fact that portrait resemblance cannot be expected in the subsequent depiction of ancient people, their function here is the personification of geometry as one of the seven liberal arts of medieval university studies. Together with personifications of arithmetic, astronomy, harmony (the four mathematical sciences of the quadrivium at that time) and rhetoric, grammar and dialectic (the trivium) as well as the philosophy personified by Aristotle, they form the bottom floor of this 19 meter high fountain, which was completed around 1596 with a total of 40 figures on the Nuremberg main market [Zintl, 1993]. So that the observer does not have any doubts about the figures and their meaning, Euclid, like all other figures, bears his name on a banner, as well as a compass and a right angle in his hand.

To the right of this we see the frontispiece to the von **David Gregory** Euclid's "Opera omnia" published in Oxford in 1703. The text below shows the picture as an illustration to an interesting place in the "Ten Books on Architecture" by **Vitruvius** (1st century) from: The Socratic student Aristippos is stranded with some companions on the shore of an unknown island (it is Rhodes). When he saw a geometric figure carved into the sand, he said, "Friends, we can draw hope to be close to civilized people and therefore out of danger." Does that mean geometry as a symptom of a humanistic outlook? As much as we mathematicians would like to see ourselves in such a reputation, from today's perspective one has to think of the many (and in some cases already practiced) possibilities of misusing mathematics to the detriment of humanity.

Hand from the Musei Capitolini |

Occasionally the relation of a work of art to mathematics is so special that special knowledge is required to discover it. We owe Heinz Lüneburg (University of Kaiserslautern) the following example. He photographed the stone hand shown here in the Musei Capitolini in Rome and kindly allowed us to use his picture. The layman will only notice the naturalistic rendering of a hand down to the last detail, and he will possibly interpret its gesture as "raised in warning". However, anyone who knows the method, which was widespread throughout the Roman Empire and still in the Middle Ages, of "dynamically" storing numbers between 1 and 9999 by positioning both hands during intermediate calculations (in the words of H. Lüneburg) will recognize the position of the hand the number 2900. It is important that it is the right hand: The left hand stores the numbers from 1 to 99, the right hand with the same signs the number multiplied by 100. The detailed explanation of the finger numbers and their use can one can read in Lüneburg's book Leonardi Pisani Liber Abbaci or reading pleasure of a mathematician, 2nd edition, Mannheim-Leipzig-Vienna-Zurich, 1993, p. 50f or in Karl Menninger: number and number, 2nd edition, Göttingen 1958.

Antique bust of Archimedes | Archimedes' sculpture by Gerhard Thieme |

Now we come to the mathematician portraits and monuments. There are thousands of them, and we can only use a few examples from different epochs to show typical or remarkable things. The oldest such art objects include portrait busts of ancient scholars such as Plato, Aristotle and Ptolemy from Roman times, which can certainly never claim portraits, since they were created long after the death of the person concerned. Nevertheless, through their frequent reproduction in the course of time, they have caused one to recognize them and involuntarily introduce oneself to the person concerned. As an example we show the ancient bust of the **Archimedes of Syracuse** (287-212 BC) (Museo di Capodimonte, Naples). While the connection between bust and person is only established through art-historical knowledge, in the contemporary Archimedes sculpture by the sculptor Gerhard Thieme the reference without an inscription is clear to every viewer who knows something about Archimedes and especially about the legend of his death. This sculpture was originally commissioned for the Würzburg university campus. In 1977/78 the artist created four non-identical variants, which were set up in front of Sankt Marien in Güstrow, in front of the library of the University of Magdeburg, in the park of the Archenhold observatory in Berlin-Treptow and in the courtyard of a school in Ellrich (southern Harz) . We show the variants from Treptow and Güstrow.

Archimedes sculpture in Güstrow | Archimedes' death |

The Roman mosaic, which includes the invading mercenary, also reveals without further explanation that it is about the death of Archimedes. Numerous other works of art were dedicated to Archimedes, including baroque fantasy portraits by **Domenico Fetti** (1589-1624) and **Jusepe de Ribera** (1591-1652) and a painting (existing in two versions) "Seneca discovers the tomb of Archimedes".

Book illustration with Nicole de Oresme |

From medieval art in the narrower sense, we are showing a book illustration depicting the Bishop of Normandy **Nicole de Oresme** (around 1323-1382). Oresme embedded mathematical and natural philosophical studies in his theological work in a manner typical of his time. Mathematics owes him, among other things, the first consideration why the sum 1 + 1/2 + 1/3 + 1/4 + ... of all parent fractions exceeds every finite bound, as well as the introduction of fractional exponents and rules for calculating with powers. In the picture, an armillary sphere (a simple model of the geocentric astronomical system), also typical of the time, identifies him as a scholar inclined to the mathematical sciences.

St. Stephen's Cathedral in Vienna |

While Nicole de Oresme is, so to speak, a representative of the theoretical mathematics of the Middle Ages, we can also show a representative of the practical mathematics of the time: the sculptor and cathedral builder **Anton Pilgram** (around 1450 - around 1515). Like some of his colleagues, he portrayed himself in the churches they created. The picture shown is from St. Stephen's Cathedral in Vienna. There is even a second self-portrait of Pilgram there, known as the "window-gazer". In both works, Pilgram, who was probably born in Brno, is characterized by the geometric construction instruments in his hands. More about the cathedral construction huts and their traditions can be found in the booklet by F. Kaderavek.

Portrait of Euclid |

The fantasy portrait of Euclid is one of a series of 28 pictures by famous scientists and poets by the Dutch painter **Joos van Wassenhove** (also known as Justus van Gent, around 1435 - after 1480) painted on behalf of Duke Federico da Montefeltro of Urbino. We meet the same duke as the patron of Luca Pacioli and presumably sponsor of the famous picture below, which probably shows him himself in the company of Luca Pacioli. While Euclid is almost always portrayed as an old and ethnically neutral man, here he is apparently still in his prime and obviously of Mediterranean blood. Today we are surprised that a picture of all things that is supposed to represent Euclid was painted in such a geometrically unsatisfactory way and was apparently also accepted by the science-loving Duke: The compass, as shown in the picture, and probably also the board, should be sent to the geometer immediately fall out of your hands.

Pacioli with Guidobaldo da Montefeltro |

One of the most famous mathematicians portraits of the Renaissance is the picture of the in Naples **Luca Pacioli** (1445-1517) with his student and patron, the **Duke Guidobaldo da Montefeltro** (who probably also commissioned and paid for the painting.) It was made by **Jacob Welch** (in Italy as Jacopo de 'Barbari, 1440-1515), who also worked temporarily in Nuremberg and was in contact with Dürer. The turn things have taken in the meantime can be seen in the additions to the pictures: the armillary sphere or the circle are by no means the only attributes with which the artist can identify a mathematician as such. Incidentally, the archimedean semi-regular body hanging down from above was supposed to give the impetus in the 20th century to the surprising discovery of a previously overlooked such body: if you twist the upper or lower of the three layers from which it is made up by 45 degrees, you get one A body that satisfies the classic definition of the Archimedean polyhedra, but its corners are not indistinguishable in pairs.

One of the most important and versatile mathematicians was **Pierre de Fermat** (1608^{1}-1665). Of the various works of art dedicated to Fermat, we show the lesser-known sculpture (1898) by Théophile Eugéne Victor Barrau (1848-1913) in the Toulouse City Hall. The lady on Fermat's lap is said to be the muse who inspires him. At the turn of the 20th century, you always needed an excuse to portray well-built naked women. There can be no doubt that Fermat must have had a muse: he was one of the founders of four major mathematical disciplines, differential calculus, probability calculus, coordinate geometry and number theory. One of his number theoretic claims that for *n> 2* those relating to the Pythagorean theorem *a ^{2} + b^{2} = c^{2}* analog equation no longer with whole numbers

*a*,

*b*,

*c*can be fulfilled, has withstood all attempts at proof of the most famous mathematicians for more than 330 years and was only approved by

**Andrew Wiles**finally proven. The popular book by [Singh, 1998] is recommended for those interested.

Newton by W. Blake | Bust of Rysbrack |

Tomb design |

Pictures and busts can give posterity the impression of a "hero" that never existed. When the first memorial for famous British people, the temple of the "British Worthies", was erected in Stowe near Buckingham (England) in 1735, next to the busts of Elizabeth I, Francis Bacon, John Locke and Shakespeare there was also a Newton, the no one who has seen a picture of the living Newton would identify as Newton, perhaps comparable to the heroic pose of Vigeland's Abel. The creator of this bust was **Michael Rysbrack**. The French architect **Etienne-Louis Boullée** (1728-1799) designed a bombastic spherical tomb for Newton in 1784 and wrote: "Oh Newton! With the rank of your intelligence and the sublime nature of your genius you have determined the shape of the earth. I imagine you with your discovery to envelop. " Perhaps it was a reaction to this exaggerated heroic cult, mixed with the tradition of Archimedes' drawing in the sand, which the scandal-ridden, early-romantic English painter and poet **William Blake** (1757-1827) to his peculiar Newton picture (1795).

Portrait of René Descartes | Inverted copy |

The painted or drawn portraits of mathematicians from the time before the invention of photography are important props in the history of science, and they are always in demand when it comes to a famous scholar, e.g. on the occasion of a milestone anniversary, through a biography, a memorial, a conference or a Honor postage stamp. Although there are very many such pictures, it has rarely happened that the artist who created the picture is as famous today as the person portrayed. The portrait on the right is an exception **Frans Hals** (1580? -1666) by the mathematician and philosopher around September 1649 **René Descartes** (1596-1650), who, alongside Fermat, was the second father of the coordinate method. It has a strange story: After Descartes' sudden death at the Swedish royal court in February 1650, so many people wanted a picture of the man who was already very famous and valued at the time, several copies were made in the 17th century, some of them reversed (see the attached small picture) and with clearly different facial expressions. The original is owned by the Danish Carlsberg Foundation and is unfortunately in very poor condition [Slive, 1989]. In the mathematical literature, images of different copies are widespread, mostly without precise information on the origin.

Descartes' youth image | Descartes portrait by Weenix | Sims figure in the Louvre |

The picture of the youth is by an unknown hand and is probably not authentic. The second was made in 1647 by the Dutch portrait painter **Jan Baptist Weenix** (1621-1660) painted. Both are completely unknown in the mathematical history literature.

The Descartes case raises the question of how far one can trust historical scientist portraits or assume authenticity. We mention two more shocking examples:

Portrait of Christoph Clavius |

- The portrait shown on the right exists in two practically identical versions in Rome and Graz. According to the signature, however, the picture in Rome shows the Jesuit mathematician
**Christoph Clavius**(1537 - 1612), the picture in Graz the equally well-known and important Jesuit mathematician**Paul Guldin**(1577-1643). According to Dr. Andreas Pechtl (Mainz) an engraving by**Francesco Villamena**(1566 - 1624) of good quality and probably authentic. The portrait of Guldin, since it names the year of Guldin's death, was only painted in oil by an unknown hand after 1642, apparently based on a copy of the above-mentioned copperplate engraving.Romanian

StampHungarian

Stamp - In 1960, the Hungarian mathematician's 100th year of death
**Janos v. Bolyai**, whose home is now part of Romania, the Romanian Ministry of Post asked the local authorities for a picture of the famous man in order to issue a commemorative stamp. Under the circumstances at the time, people apparently did not dare to answer that such a picture does not exist, and the random picture of a contemporary of Bolyai's (presumably a member of the Habsburg family) was sent. In the same year the Hungarian Post copied this picture on a Hungarian commemorative stamp for Bolyai. Since then, trusting in authenticity, these pseudo-portraits have served as a template for illustrations of articles on Bolyai and non-Euclidean geometry in numerous books and encyclopedias.

Portrait of Felix Klein | Vigelands Abel |

With a big leap in time we come to a second case in which a famous painter portrayed a famous mathematician: **Max Liebermann** (1847-1935) painted Felix Klein, who has already been mentioned several times.The picture hangs in the Mathematical Institute of the University of Göttingen. For his part, Klein, whose sense of modernity was evidently not developed to the same extent in art as in mathematics, has shown himself in his book *Lectures on the development of mathematics in the 19th century* (posthumously 1926) was very critical of the work of a sculptor who was very famous, at least in his Norwegian homeland: **Gustaf Vigeland** (1869-1943) created the monument to the Norwegian mathematician **Nils Henrik Abel** (1802-1829), which was erected in front of the Oslo Castle in 1908. Klein wrote about it:

I cannot fail to take this opportunity to remind you of the very different kind of monument which was erected instead for Abel in Kristiania and which must seriously disappoint anyone who knows its nature. On a towering, steep granite block, a young Byronian athlete strides over two grayish victims. If the hero can at best be understood as a symbol of the human spirit, then one asks in vain about the deeper meaning of these monsters. Is it the vanquished fifth-degree equations or the elliptical functions? Or the sorrows and worries of everyday life? |

Abel monument in Oslo-Blind | Abel monument in Froland Verk |

Vigelands Abel was created as a result of a competition announced in 1902 on the occasion of Abel's 100th birthday. At that time the first prize was awarded to a design by **Ingebriks Vik** (1867-1927), which was only realized around 1965 and has been on the new Oslo-Blinder campus ever since. Felix Klein would probably have liked him better. The second prize went to a design by the sculptor **Gustav Lærum**which is now through **Per year** and **Roald Kluge** was expanded and executed. The inauguration of the monument in Froland Verk was on August 5th, 2002.

Bernard Bolzano |

The name of the Prague-born mathematician, logician and moral theologian **Bernard Bolzano** (1781-1849) is still associated with several important terms and phrases that every math student learns in the first year. However, since he was politically intolerable under the Metternich regime, Bolzano spent the last 25 years of his life isolated, living on the charity of friends, without any means of action and hardly able to publish anything. His extensive work has only recently been published in full (120 volumes planned!). The portrait that the Viennese painter **Heinrich Hollpein** (1814-1853) by Bolzano around 1839, is in the Czech National Museum in Prague. This painting also impressively demonstrates that a masterfully painted portrait can reveal more about a person than a photo.

Copernicus Monument in Torun |

Since the invention of photography, there has been little demand for the previously indispensable realistic portrait (although it can still reveal more about a person than a snapshot). Artists who are commissioned for a memorial are more than ever challenged to make a symbolic statement about the nature and achievement of the person honored. A particularly successful example is the monument in Torun for the great son of this city **Nicolaus Copernicus** (1473-1543), created by **Jozef Kopczynski**. The name Copernicus does not appear here, instead the inscription *Sol omnia regit - the sun rules everything*. We might as well have discussed this monument in the broader chapter on descriptive geometry. Should a visitor to this exhibition now ask whether or why Copernicus was a mathematician, there are two answers: First, in the understanding of the 15th and 16th centuries. Jhs. astronomy is a mathematical science (see the comments on the quadrivium). Second, the greater part of the main work *De revolutionibus orbium coelestium* (1543) of Copernicus is purely mathematical from today's point of view. In particular, it played an important historical role in the development of trigonometry.

And yet it moves (Galileo Galilei) |

Also the (again existing in two casts) Galileo sculpture by **Fritz Cremer** (1906-1993) is not naturalistic. Basically, it seems to relate more to Bert Brecht and his Galileo drama than to Galileo himself. As with Copernicus, in the case of Galileo the layman's question is justified as to whether Galileo is a mathematician. The obvious answer that Galileo with his laws of throwing and falling opened the door to the mathematical formulation of physical laws (according to Galileo, mathematics is the language in which the book of nature is written), that he raised many fruitful questions, e.g. about the mathematical description the cycloids or the so-called chain line, one can add that Galileo was apparently the first, around 200 years before Bolzano and 270 years before Georg Cantor, to draw attention to the "paradoxes of the infinite" apparently represent only a small part of the natural numbers, reversibly assign the natural numbers. So here, in contradiction to Euclid's 8th axiom, is the whole equal to one of its real parts?

David Hilbert |

The question asked above about the artistic value of portrait-like pictures since the invention of photography can be answered without words by the picture of the famous mathematician, which is also hanging in the Göttingen Mathematical Institute (like Liebermann's Felix Klein) **David Hilbert** (1862-1943) answer.

We prefer the woodcut with which we close our gallery.

Felix Hausdorff | "Poeta laureatus" |

The famous mathematician **Felix Hausdorff** (1868 - 1942), who taught at the University of Greifswald from 1913 to 1921, frequented artistic circles during his Leipzig years and wrote philosophical and fiction texts and a play himself under the pseudonym Paul Mongré. The Hausdorff caricaturing woodcut shown here entitled "Poeta laureatus" was created by Hanns Alexander Müller in 1910 based on a drawing by Walter Tiemann. For a better assessment, we put a portrait photograph of Hausdorff next to our visitors.

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