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A strong message that mathematics revealed in the late nineteenth and the early twentieth centuries is that geometry and space can be the realm of freedom and imagination, abstraction and rigor. An example of this message lies in the infinite variety of forms of mathematical inspiration that the Mexican sculptor Sebastian invented, rediscovered and used throughout all his artistic activity.

The author’s father, Luciano Emmer, was an Italian filmmaker who made feature movies and documentaries on art from the 1930s through 2008, one year before his death. Although the author’s interest in films inspired him to write many books and articles on cinema, he knew he would be a mathematician from a young age. After graduating in 1970 and fortuitously working on minimal surfaces—soap bubbles—he had the idea of making a film. It was the start of a film series on art and mathematics, produced by his father and Italian state television. This article tells of the author’s professional life as a mathematician and a filmmaker.

On March 16, 2019, a large exhibition on soap bubbles in art and science opened in Palazzo dei Priori, home of the National Gallery of Umbria in Perugia, Italy. Marco Pierini, the director of the Galleria Nazionale dell’Umbria, curator together with Michele Emmer of the exhibition, wrote in the Introduction to the catalogue [1, p. 7]

I was a professor at the Ca' Foscari University in Venice for 7 years. I had my studio in the Ca' Dolfin palace, famous for the 10 canvases by Gianbattista Tiepolo which are currently at the Metropolitan Museum in New York, the Hermitage in St. Petersburg and the Kunsthistorisches Museum in Vienna. I know the city of Venice very well and in 1991 I wrote a book on the geometries of the city La Venezia perfetta (The Perfect Venice) [1] and a revised and reduced version of the book was published in English with the title Venetian Geometry, or the Perfect Venice in one of the three volumes dedicated to the city by Alain Vircondelet Venice Art and Architecture published in 2006 by Flammarion [2] and partially in the magazine on architecture and Math Nexus [3]. A second edition of my book with a new cover and introduction was published in 2019 [4]. In 2013 I was elected a member of the Instituto Veneto di Scienze, Lettere e Arti (IVSLA) founded in Venice by Napoleon. Since then I have organized my yearly conferences on mathematics and culture at the Institute which is based in Palazzo Franchetti on the Canal Grande, one of the most famous Venetian palaces, unique for its history and its architectural features (Fig. 1).

In a 2016 article I wrote a short history of the recent films on mathematicians until 2015. The paper was published by the title Cinema and Mathematics: Recent Developments [M. Emmer (Imagine Math 5. IVSLA & UMI, 2016)]. Since then there have been interesting developments.

This text comments the surprising Eloge des Mathematiques, contained in Les Chants de Maldoror, written from 1868–1869 by the Comte de Lautreamont.

This chapter describes the influence of topology and of topological thinking on the work of the great architect Zaha Hadid.

Matematica significa teoremi, inventare teorie, metodi, algoritmi. E’ un’attività creatrice a tutti gli effetti. Poco comprensibile, però. Ecco allora il raccontare, raccontarsi, anche per chiarire a se stessi, che cosa avaviene nel momento in cui appare davanti agli occhi di un ricercatore la rivelazione della avvenuta dimostrazione di un risultato che nessuno ha mai ottenuto prima.

La matematica come fonte di ispirazione per raccontare, per visualizzare, immaginare altri mondi. E di cosa tratta il cinema sin dai suoi esordi ? L’idea è uella di mostrare quali sono i ruoli privilegiati dei matematici nel cinema, dalla storia alla investigazione e di serial killer a quelle di matematici realmente esistiti. Alle volte pieni di problemi che sconfinano nella pazzia alle volte introversi ed isolati, ma in ogni caso sempre geniali e imprevedibili.

Leonardo da Vinci probably did not consider the possibility of realizing images with real movement. Many centuries later, however, the author's father, Luciano Emmer, had the idea of reinterpreting the images of the famous artist and scientist using the technique of cinema.

The Dutch graphic artist Escher was very much interested in plane symmetry. He was fascinated by the problem of infinity and the representation of infinity on a sheet of paper. An exchange of ideas with Coxeter was the origin of the series of works called "Circle Limit". This article also describes the animation technique, which by adding time and movement to the images of Escher try to contribute to Escher's dream of reaching infinity.

The hall of the 1986 Venice Biennale dedicated to Art and Science opened with works by Lucio Saffaro (the artist whose works are featured on all the covers of books in the Imagine Math book series), along with Felice Ragazzo’s reconstruction of Kepler’s model of how the universe works, when Kepler still believed that the orbits of the planets were circular and circumscribed by the regular solids first described by Plato [27, 5, 28, 6, 8, 9].

Plateau’s problem consists in taking a generic curve in three-space and finding a surface with the least possible area bounded by that curve. The empirical solution may be obtained by dipping a tridimensional model of the curve into soapy water, resulting in a form called a minimal surface. When a soap bubble is blown, the soapy surface stretches; when blowing ceases, the film tends toward equilibrium. Frei Otto used soap film models to design his tensile-structures, developing a technique to obtain a precise photogrammetric evaluation of soap film models and another method to simulate peaks in a membrane of soap films. H.A. Schwarz solved Plateau’s problem for a non-plane boundary by developing the periodic minimal surface. Infinite periodic minimal surfaces, combinations of saddle polygons or surfaces, are more stable. Such a surface has been adapted as a play sculpture in the Brooklyn Museum, where children can actually enter into the labyrinthic structure of periodic minimal surfaces.

During the Desert Storm, the Gulf war, it was possible to read in the newspapers words such as: “Inmathematical terms, was is becoming more and more electronically controlled and, as a result, it is moving away from the battlefield. Then, when war comes down to earth, it becomes bloody, it loses its mathematical asceticism” Reading the newspapers in those days, one had the impression that modern warfare is based on mathematics, as if it were not men but computers that decided where to carry out “surgical operations”. By contrast, the volume published a few years before the Gulf war conceived as a didactic unit to be used in schools with a guide for the teacher with the titleLa matematica della guerra (The Mathematics of War) published by Gruppo Abele in Turin begins with the words “Mathematics, like any other discipline, lends itself to building several paths towards education for peace”. The volume, written by a group of teachers belonging to an anti-violence organisation forming part of the “education for peace” project, highlights the power or ambiguitiy of mathematical models used to simulate war or conflict situations and demonstrates that in some cases the use of mathematics leads to a better understanding of the situation, but in other cases, the mathematical model itself can lead to conclusions which are either wrong or morally unacceptable.

The teaching of mathematics, knowledge of mathematics, knowledge of the sciences are some of the parameters used to evaluate the ability of a country to improve its capacity for production and creativity. The spread of scientific knowledge, and especially of mathematics, is an essential factor. It is precisely those countries that were once defined as ‘emerging’ and are now among the most industrialised and powerful on earth that have invested in basic training, in excellent scientific training, and in mathematics in particular. Obviously these include China, India and Korea. Hence the importance of teaching, and of mathematics teaching.

Michele Emmer discusses the relations between soap films, arts, mathematics, visual arts, architecture with very recent examples. It’s because I don’t do anything, I chatter a lot, you see, it’s already a month that I’ve got into the habit of talking a lot, sitting for days on end in a corner with my brain chasing after fancies. It is perhaps something serious? No, it’s nothing serious. They are soap bubbles, pure chimeras that attract my imagination. Fedor Dostoevsky, Crime and Punishment

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, as Housman insisted, the importance of ideas in poetry is habitually exaggerated: ‘I cannot satisfy myself that there are any such things as poetical ideas…. Poetry is not the thing said but a way of saying it.’

Let us imagine that we have been travelling on a great journey to some far-off world. We shall call this world Tor’ Bled-Nam. Our remote sensing device has picked up a signal which is now displayed on a screen in front of us. The image comes into focus and we see. [...] What can it be? Can it be some strange looking insect? Or could it be some vast and oddly shaped alien city, with roads going off in various directions to small towns and villages nearby? Maybe it is an island — and then let us try to find whether there is a nearby continent with which it is associated. This we can do by ‘backing away’, reducing the magnification of our sensing device by a linear factor of about fifteen. Lo and behold, the entire world springs into view. [...] We may explore this extraordinary world of Tor’ Bled-Nam as long as we wish, tuning our sensing device to higher and higher degrees of magnification. We find an endless variety: no two regions are precisely alike — yet there is a general flavour that we soon become accustomed to. [...] What is this strange, varied and most wonderfully intricate land that we have stumbled upon? No doubt many readers will already know. But some will not. This world is nothing but a piece of abstract mathematics — the set known as the Mandelbrot set [1, p. 74–79].

Art and Mathematics are inextricably linked. One of the main linkages is the idea of space and its transformations. There is no doubt that throughout the history of Western culture, there were moments when the relationship between mathematics and art has remained hidden, if not absent, and others in which it appeared with great vividness. And one of the privileged periods is obviously the Renaissance.

It is the 1960s, and Brigitte Bardot is at the height of her success. She appears as herself in a film entitled simply Dear Brigitte [1]. By simply calling the film that everyone immediately knew what is was about. She was the dream girl of the son of the character played by James Stewart. Stewart’s character, Robert Leaf, is a poet and professor of English literature in an American university. He is perennially in conflict with the scientists at his university, holding science to be dry, and scientific training, particularly mathematics, to be of little use. One day tragedy strikes at home (a sort of tragedy, of course, since the film is a family comedy). Leaf’s son, who goes to elementary school, is a mathematical genius. Or better, the boy has a gift for mental calculation. In ordinary language, and thus also in movies, we often find ‘mathematical genius’ used for those who can perform rapid calculations in their heads. His teacher discovered this by chance, and quite pleased, goes to tell the boy’s parents. When he hears that his son is a mathematics whiz (at elementary mathematics, obviously), the father becomes white as a sheet and puts a hand on the mother’s shoulder to comfort her. Then, when the teacher leaves, he begins to talk to the boy, begging him not to tell anyone about this skill he has, since it will lead to no end of grief, especially when people begin to shout, when they pass him on the street, ‘He is a mathematician!’, a phrase Stewart pronounces with disgust, saying, ‘And we don’t want that, do we?’, ‘No, Sir’, ‘Of course not.’

Computer graphics work not only as pure visualization of almost all well‐known phenomena, but also as new ways of studying mathematical and geometrical problems. A new branch of mathematics has been developing in the last few years, called visual mathematics. Visual computers and computer graphics could be a future unifying language between art and science. Interesting results can be expected for both scientific and artistic fields.