Mathematicians, to prove their theorems, write on blackboards, paper, and parchment; as history tells us, even sand, like Archimedes. One uses clay.

Here we present the work of Robert Fathauer, a physicist/engineer/mathematician with a special skills for the arts, from clay modeling of mathematical forms to computer-coded tessellations. His recent book "Tessellations" summarizes years of science, research, and enjoyment.

**MM:** Your book is titled “Tessellations: Mathematics, Art, and Recreation.” What is a tessellation exactly?

**RF: **A tessellation is a collection of shapes that fits together without gaps or overlaps to cover the mathematical plane. It can be applied more generally to other surfaces like spheres, and real-world objects like baskets.

**MM:** By what (or whom) are you inspired to creating tessellations?

**RF: **I was mainly inspired by the tessellation art of M.C. Escher, who made drawing of birds, fish, and other creatures that fit together with no gaps between them.

**MM:** You are often referring to the importance of Escher’s work for your art and science. Do you feel more ‘yours’ the Escher of the impossible, of the labyrinthine, or the Escher of tessellations' perfect equilibrium?

**RF: **I’m drawn to all of Escher’s art, including his impossible structures, but it’s his tessellations that I have studied the most closely and used in my own work.

**MM:** “Mathematics, Art, and Recreation”: is one of them prevailing?

**RF: **No, I think it’s a balance. Some of my work is most art oriented, some emphasizes the math, and some emphasizes puzzles and games.

**MM:** Your book contains several models to build and modify tessellations. Is it a book thought of for a pedagogical framework, for kids, for math-passionate?

**RF: **It’s really for anyone who loves math and art. It’s not written for the professional mathematician or as a college textbook, but there is a lot of material that can be used by K-12 math teachers.

**MM:** Did pattern study and research change anything in your way to see the world?

**RF: **I do notice patterns and symmetry more than I used to, both in manmade things and in nature. I’ve become more aware of how the same types of patterns occur in many different places in nature.

**MM:** In mathematics textbooks, spiraling shells do often appear. In your book, you explain how to approximate them starting from adjacent triangles. Can you explain to the reader how to build them? While you are building them, does it happen to associate these figures with a movement, a rhythm?

**RF: **A lot of things can be arranged in spirals, particularly Archimedean-type spirals, ones in which the spacing between arms remains constant. These can be built with squares or equilateral triangles, for example. There are some artists who like to make spirals using rocks, leaves, and the like; e.g., James Brunt or Andy Goldsworthy. I don’t particularly associate spirals with particular movements or rhythms, but spirals in graphic designs do create a sense of motion, as well as evoking infinity.

**MM:** Knots of Figure 18.13 look like golden rings. How can we build them?

**RF: **If you want to physically build knots one approach is to bend copper tubing and join the two ends. Nat Friedman, one of the pioneers of the modern math-art movement, used to build knots this way. Knots can also be made from ceramics, but it’s not the easiest material for making knots. Other techniques that require a lot of skill and patience are wood and stone carving. A simple way to make models and understand knots better is just to bend a strip of paper.

**MM:** On page 176, the hyperbolic tessellation of Figure 10.2 makes us think of marine creatures, such as nudibranchs. How is it possible to build a hyperbolic tessellation?

**RF: **In a tessellation in the Euclidean plane, the angles of the tiles meeting at a point (a vertex) sum to 360°. In a hyperbolic tessellation they sum to more than 360°. Tessellations with angle sums greater than 360° can be built using construction toys like Polydron, in which polygons can be snapped together in different configurations. There are also techniques to make hyperbolic surfaces using paper. One approach is shown in this pdf by Chaim Goodman-Strauss: http://www.eugenesargent.com/g4g11/giftbag1.jpg

**MM:** Figure 20.10 shows one of your ceramics, “Radiolarian Form.” From D’Arcy Thompson to Haeckel, till Gaudí, radiolarians' symmetries have been exerting a strong appeal for artists and scientists. What are they for you?

**RF: **I find the geometric intricacy of radiolarians fascinating. It’s also interesting to me to compare them to different polyhedra, particularly the Goldberg polyhedra, which have large numbers of faces. Just the fact that nature creates such things is wonderful and amazing.

**MM:** Different from drawing, what does it mean to create a form with clay or ceramics?

**RF: **Thinking and visualizing in three dimensions is much more challenging than in two dimensions. I try to plan out my ceramic pieces so that they will look interesting from different angles. I think this is a key feature that gives sculpture something extra that two-dimensional art doesn’t possess. There is also the act of working clay with your hands that involves muscle memory, so that making mathematical forms relies on feeling as well as seeing.

**MM:** You have a background as an electrical engineer (Ph.D. at Cornell), with math and physics studies in college. You also have a past as a researcher at the Jet Laboratory Propulsion in California. Amongst physics, math, and engineering studies, what did mostly influence your view of nature, and for which reasons?

**RF: **I worked with developing new types of crystalline thin films for electronic device applications. Crystal growth and analysis of these materials was a big part of my work. That’s how I became aware of symmetry and lattices, including diffraction patterns and electron micrographs of structures too small to see with the naked eye. That work influenced my way of looking at nature and made me more aware of the mathematics underpinning natural forms.

**MM:** The book contains pictures and references from your travels, but also from the place where you live, Phoenix. Is the Arizona desert giving you unexpected insights, or, on the contrary, are your studies allowing you to enrich with meaning what you see?

**RF: **My mathematics and art explorations have made me more aware of tessellations, fractals, hyperbolic geometry, and polyhedra in the natural world. I look for these things in cactuses and other plants, rocks, and other landforms. Things I discover and observe in nature also inspire my art. Some of my ceramic sculptures start as polyhedra but then add organic features to become imaginary creatures not found in nature.

**MM:** At the end of the Preface, you are hoping that your work will ignite in the reader the same love for patterns you feel, “if not an obsession!” Do you have any student learning these techniques, or, better, this art of tessellation?

**RF: **I’m not a teacher and don’t have a position at a university, so I don’t have any students learning directly from me. However, a lot of people, some of which are students, follow my social media posts, so they are learning that way. Students sometimes contact me by email, and I always try to be helpful when they do.

**MM:** Thinking, creating, drawing, modeling, are characteristics of several genial personalities of the Italian Renaissance. To mention all but the most famous ones, we find Michelangelo, Leonardo, Raffaello. Which ones do you feel, or would like to feel closer to?

**RF: **I think my interests align better with those of Leonardo. Michelangelo and Raffaello were more pure artists, while Leonardo was interested in mathematical objects like polyhedra and knots, as well as in applying these concepts to design machines.

**MM:** Each one of the countless collective exhibitions of mathematical art you have organized must have resulted in engaging experiences. Would you like to share some of them with our readers?

**RF: **It’s really interesting to interact with people making mathematical art in other parts of the world. People have different ideas of how to use mathematics in art, and people from different cultures bring their own unique perspective and esthetic. Sometimes I’ll meet someone who has been working for years without being in touch with other people making mathematical art, and they’re always very excited to meet like-minded individuals and to have their work included in an exhibition.

Category Theory is a branch of mathematics born during the 1950's of last century to model several branches of mathematics with the aid of points, arrows, and diagrams. Categorical thinking has proven to be useful in other disciplines as well, and thus, *applied* categories have been developed. We talked about this in another post.

Books about category theory seem to be "abstract": we might wonder why there are so many arrows and points. But...

Some concepts of category theory can be applied to music as well as to the comparison between visual forms and musical forms. This post is part of my "virtual poster" for the conference Applied Category Theory (ACT) 2020 organized by MIT, that will be held online in July 2020.

Several scholars, including Jedrzejewski, Mazzola, Arias, Popoff, and Clark, have investigated the application of category theory to music.

I also tried to use some basic concepts and some simple connections. I use the language of categories to model musical structures and transformations and also to compare them with visual structures, forms from nature, and growth and transformation processes. In this short post, I will cite a few examples of this approach.

Too ambitious? Let's just smile about it.

Figure 1 shows two points and an arrow between them: it's a graphic representation of a morphism between two objects of a category. In the framework of music, the two points can represent two different values of intensity, such as a *piano* and a *forte*, and the arrow connecting them is a *crescendo*. (We'll see below why we can talk of a *category*).

Let's imagine a double "smile", to indicating two different ways to realize a crescendo, for example, slowly or quickly. One becomes the other thanks to an "arrow between arrows" (Figure 2).

Each point itself can contain points and arrows, creating in this way a nested structure (Figure 3).

We can thus obtain a sort of face, where the eyes themselves are also faces (graphs from this article [3]). These ideas can inspire graphic works, such a fragment of my drawing "Duality" (Figure 4).

Let's go back to music, which shows several nested structures. In the aforementioned example, a *crescendo* can be described as a variation from one intensity level to another one, e.g., from *piano* to *forte*. But there are several ways to do it, e.g., faster or slower. If each crescendo is represented by an arrow, the time variation between crescendi is represented by an arrow between arrows. Therefore, let the points be the intensities and the arrows be the intensity variations. The composition of two intensity variations yields an intensity again. The associative property is easy to verify, and the neutral element is given by the zero intensity variation.

Thus, we have a category with intensities as objects and intensity variations as morphisms. We can define a 2-cell where the objects are the intensity variations (1-arrow), and the morphisms are the intensity variations (2-arrows). The same musical passage can be performed by different musical instruments. We can thus define the 3-cells, and so on. See this article and this other one for details.

Morphisms between categories, called *functors*, can also be defined. Functors transform objects of a category into objects of another category and morphisms of the one into morphisms of the other one. These are basic concepts in category theory, but they nevertheless have a potential for generalization already. I applied these ideas to the form of trees (Figure 5).

Figure 5 shows, on the left, a young (top) and adult (bottom) *Butia capitata *tree*,* and, on the right, a young (top) and an adult (bottom) *Coccothrinax argentea *tree*.* Here, we are using the idea of the category to indicate a species. Morphisms within a category indicate the growth process; morphisms between categories indicate the comparison between the forms of individual of the same species ("form comparison" between objects), and the comparison between growth processes within different species ("growth comparison" between morphisms). This example comes from the book "Mathematics, Nature, Art" [1].

A suitable functor can turn trees' forms into music. Music develops in time, and thus it can represent the development of a form within the space of sounds. Several scholars see the plants' form as the result of a growth process through time. According to D’Arcy Thompson [6]:

Organic form itself is found, mathematically speaking, to be a function of time... We might call the form of an organism an event in space-time, and not merely a configuration in space.

And, as Francis Hallé [5] points out:

The idea of the form implicitly contains also the history of such a form.

The idea of functor can be applied to the forms of animals and their possible musical renditions as well; see a detailed post on this topic.

Another categorical structure is given by *natural transformations*, that, intuitively, compare the action of different functors. The same book [1] includes an example where the morphisms indicate the blossom of flowers in an inflorescence, a functor represents the "placement" of flowers within the spherical inflorescence, and another functor represents the placement of flowers into an ellipsoidal inflorescence. The spherical inflorescence is inspired by *Echinops ritro;* the ellipsoidal inflorescence is, however, invented here, see Figure 6. Flowers of *Echinops ritro* have five petals; here, they are simplified with three petals only.

In the same book, I defined a "sonification" functor that transforms single flowers into musical themes and inflorescences into distributions of themes on a sphere (in the space of sounds). In fact, it's possible to define nested categories and functors between functors.

The sonification technique I propose is grounded in the mathematical theory of musical gestures and on their "transferability" from one domain to another one.

Regarding gestures, I'd like to cite the definition of musical gesture given by Mazzola and Andreatta [4], see Figure 7, as a mapping from an oriented graph (with points and arrows) to a continuous path (that connects points in a space) within a topological space. Each vertex of the graph is mapped onto a point of the space, and each edge of the graph (arrow) onto a continuous arrow between points of the space, keeping the correspondence of original arrows' tails and heads.

In [2, 4], the oriented graph is Delta, and the set of points and continuous curves belongs to the space X. We can thus define gestures as mappings from Delta to X, and we can represent transformations between these gestures as 2-arrows, that can also be composed. In music, a non-characterized movement (gesture) can become a "piano" and then a "forte" movement (Figure 8).

We can also indicate as *gesture* the set of points in space and curves connecting them. Mazzola and Andreatta use the metaphor of a dancer's continuous movements, connecting discrete steps. I started from this metaphor to run a short study on categories applied to dance (Figure 9).

The orchestral conductor, wanting to conduct, let's say, a ternary time, thinks of a scheme with three points in the air touched according to a precise order; he/she will then join these points by performing some continuous movements in space and time (Figure 10).

We can consider a whole gesture as a point, defining arrows between gestures. The compound gestures are the *hypergestures *[4], that can be recursively defined. The composition of hypergestures (paths) is, as mathematicians would say, associative up to a path of paths. We can re-define hypergestures as an equivalence class of hypergestures, in order to formally have a 2-category. These ideas are discussed and proved in two theorems, see here [3].

We can take into account the cyclicity of conducting movements in correspondence with the cyclicity of musical time (e.g., the scheme 1-2-3 is repeated in a ternary time; see Figure 11).

If we consider the category having the points in space (and time) needed for orchestral conducting as objects, and the conductor baton's movements joining them (for sake of simplicity, we are only considering the conductor's right hand) as morphisms, the 2-morphisms are the variations of these movements --- speed changes, articulation changes, and so on. The formalism of 2-cells can be useful to analyze conductor's gestures, comparing them with the movements of orchestral performers. If, for example, the conductor points out aThe same formalism of Figure 12 can be applied to dance, see Figure 13.

We could investigate *similar* variations in the motion of a violinist's bow, or in a flutist's use of mouth and diaphragm. These concepts lead to gestural similarity [2]. Here is its heuristic version.

Conjecture A.1 (The heuristic conjecture) Two gestures, based on the same skeleton, are

similarif and only if they can be connected via a transformation:

(1) that homotopically transforms one gesture into the other, and

(2) that also leads to similar changes in their respective acoustical spectra.Homotopy is a necessary, but not sufficient, condition to get similar gestures.

When the conductor signals a *forte* for all performers, each performer will make different movements according to the parameter space for each instrument. However, the different movements will have some analogies (e.g., an increase of pressure of the bow for the violinist and of airstream pressure for the flutist) producing sound analogies (e.g., an intensity increase and a related timbre variation), which can be retrieved in spectrograms.

The idea of a functor can also be applied to the transformations from the visual symbols of the score to the sound of the performance. Natural transformations help formalize the comparison between different performances of the same composition. In this way, we can compare gestures by different musicians of different orchestras, their sounds, and so on.

Because the conductor's gesture does not directly produce a sound but instead only suggests it, the conductor's gesture appears to be more "abstract" with respect to the orchestral performers' movements. We might say that the conductor's gestures are ontologically different from performers' gestures.

As another topic, the listener and the conductor have opposite roles. Orchestral performance can be seen as a flux of arrows from the conductor (level 1) to the orchestral performers (level 2) to the listener (level 3). In category theory, the constructions which can be obtained by reversing all arrows are called *dual*. In fact, we can build an equivalent (but reversed) formalism, where the listener (level 1) pays attention to performers' movements and sound (level 2) and the performers pay attention to the conductor's gestures (level 3). In addition, the conductor's gestures synthesize the main points of the score and the essential information for performers. For these reasons, we can metaphorically be inspired by categorical constructions, such as limits and colimits, to model these interactions between different levels (Figure 14); find more detailed diagrams here [2].

A limit is the generalization of a product (i.e., simplifying, the arrows *start* from it); a colimit is the generalization of a direct sum (the arrows *converge* to it). As pointed out in the article, in [2] the *universal properties* are not described in detail, but some conceptual explanations are given, regarding why such a language could be adapted to a conductor/orchestra/listener.

The passage toward the abstraction, that we can metaphorically represent as a colimit in an extra-musical domain such as biology could be applied to species classification (taxonomy). Figure 15 shows, in the case of fish, the progressive abstraction, through arrow composition, from the single species, to the genera, and moving upward to reach an equivalent of the most abstract idea of "fish," represented here by a schematic drawing. (The book [1] discusses some differences from the Platonic idea [1]).

The gestural similarity conjecture can be extended to the interaction between music and images. Let's think of a collection of dots on a piece of paper and a sequence of *staccato* notes: staccato notes and dots can be considered as *similar* because they are produced by the same detached gesture, that is, as being drawn in the space of sounds, as being generated by the same creative gesture.

Clearly, there isn't any one-to-one correspondence because we can have infinite collections of dots to be associated with the given musical sequence. To such a musical sequence we wouldn't associate a continuous line, though. In the same way, we can sonify a given set of dots with different notes, all played as *staccato*. We can thus think of *equivalence classes* of possible musical renditions verifying gestural similarity.

In this way, music generated by an algorithm or "freely" composed can create the illusion of simultaneous production of sound and image; see this interview at Ca' Foscari University in Venice. Whatever the chosen technique for a form's musical rendition is, be it a free or an algorithmic one (exploiting Lindenmayer studies [5] in the case of plants), in my opinion, an effective rendition should verify gestural similarity. How to effectively translate a complex form into sound is a non-trivial problem, discussed in the article on "Quantum GestART".

To conclude, let's plunge into a mathematical ocean, where the forms are graphs of parametric equations (or rotation solids), and the music follows the gestural similarity criterion.

----

[1] Mannone, Maria. *Mathematics Nature Art.* (2019). Palermo: Palermo University Press.

[2] Mannone, Maria. (2018). "Introduction to Introduction to gestural similarity in music. An application of category theory to the orchestra". *Journal of Mathematics and Music*, 12(2): 63-87. https://www.tandfonline.com/doi/abs/10.1080/17459737.2018.1450902

[3] Mannone, Maria. (2018). "Knots, Music, and DNA". *Journal of Creative Music Systems, *2(2).* https://doi.org/10.5920/jcms.2018.02*

[4] Mazzola, Guerino, and Andreatta, Moreno. (2010). “Diagrams, Gestures and Formulae in Music.” *Journal of Mathematics and Music,* 1 (1): 23–46.

[5] Prusinkiewicz, Przemyslaw, and Lindenmayer, Aristid. (2004). *The Algorithmic Beauty of Plants*. New York: Springer.

[6] Thompson, D’Arcy Wentworth. (1966). *On Growth and Form. An Abridged Edition Edited by John Tyler Bonner*.Cambridge, Massachusetts: Cambridge University Press.

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We often hear about

youth lost in the world of alcohol and drugs, but less often of young people pleasantly immersed within geometry. And because geometry is a mathematical concept, and it is possible to investigate a bottle's curvature, as well as any liquids including alcohol which may be expressed meticulously via precise laws of physics and mathematical equations, it is very clear that getting immersed in mathematics is less dangerous than getting immersed in drugs. Also, in the world of mathematics, there is not any restriction on peddling or distributing new ideas, or any dangers of trafficking such things. Further, mathematics is ecologically sound as there is no trash production because it is always possible to suppose that the non-existence of trash is to stock it within an empty set.

Thus, music sometimes arrives at a university and guides students to write down notes inspired to given shapes and forms. And, as an even more unusual thing, mathematics arrives at Music Conservatory, takes possession of pianos, strings, screens, and becomes almost a way of thinking and connecting notes, musical instruments, objects, and even emotions.

Yes, emotions. Because even a "cold" geometric form can stimulate philosophical thinking and give birth to metaphors. And music.

In the following, I'm presenting a report of two math-musical experiences realized in an English university and in an Italian conservatory.

Last October I presented some topics from my math-visual-musical theory to Creative Coding students at the University of Greenwich in London, a modern and dynamic environment, open to new developments of mathematics and technology. I asked students to compose music starting from a schematic form of a flower, a fish, a cathedral. The form of the fish has been the most chosen one, and on this model, each student expressed himself or herself in graphic-musical terms, less or more complex according to their personal background.

Since November 2018 to March 2019 I gave a course, or, instead, a "Cycle of Seminars," at the Music Conservatory of Palermo, a glorious institution that, for a long time, has been named after Catania's composer Vincenzo Bellini, and that has been recently re-named after Palermo's composer Alessandro Scarlatti. The building is a historic one, and, from the windows of the upper floor, it's possible to enjoy a sight on the sea. In this place, entire generations of musicians grew up, and in particular, generations of composers and conductors. I crossed the threshold of this place for the first time twenty years ago.

The "Cycle of Seminars" I created and run, titled "Sound, Gestures, Diagrams," proposed by conductor Maestro Carmelo Caruso and duly authorized by Conservatory's staff, has been organized in ten lessons, open to Conducting and Composition students. It has been a course of my ideation, based on math-musical studies, and, of course, on the experience in London. It has been timely and assiduously attended by five students, at different levels of Composition courses.

The first four lessons were based on theory: I presented an excursus between mathematically-described musical gestures, communication between conducting baton and orchestra performers, music derived from plants' geometry, as a water lily or a *Ficus macrophylla* *s. columnaris*,* *a gigantic tree present in Palermo with remarkable specimens. Original examples have been alternated with nowadays classic ideas, such as Lindenmayer systems to formalize plants' growth. Ideas and references to “Flatland” and “On Growth and Form” appeared here and there.

Theoretical concepts led students towards a new way to envisage and connect different elements: from a complete form to its structures, and then to its "translation" from the domain of images to the codomain of music. And vice versa.

The fifth to ninth lessons have been organized in a workshop-lab style. I gave all students a unique theme, that each one of them interpreted according to his own sensibility and imagination: the "Klein bottle," from the name of German mathematician Felix Klein (1849-1925).

Klein's surface (usually called as "bottle" for a translation mistake from German) is a non-orientable surface: moving across it, we find ourselves inside the form, and then, keep moving across the surface, we are again outside of it. Conversely, surfaces such as the sphere have a boundary that separated the inside from the outside, and it is not possible to move from the one to the other without making a hole on the boundary. If cut into half, from the Klein bottle we obtain two Möbius strips. A Möbius strip is a mathematical object that inspired artistic thinking and scientific production, also in the field of mathematical music theory. The Klein bottle is a surface that lives in four-dimensional space, and, if represented in three dimensions, it intersects itself.

My interest towards such an extraordinary form was ignited in 2018 in San Diego, in California, during the Joint Mathematics Meetings, when I met the mathematician, astronomer, and artist Cliff Stoll, and I saw his stand full of fragile Klein-shaped glass bottles, that were attracting even most the abstract thinkers attending the conference.

If Klein bottle has recently been used as a study model in the field of mathematical music theory and of music and storytelling, to the best of my knowledge the works completed by my students are the first truly creative musical applications of this shape. The very original composition of (in alphabetic order) Marco Barilà, Mattia Camuti, Biagio Genco, Massimiliano Seggio, and Massimiliano Vizzini, even with their variety, take into account both the musical rendition of theoretical topics and the music taste of their authors. On the other hand, even Xenakis himself made use of abstract models and of his musical ear at the same time!

The final works range from counterpoint interpretations of different paths on Klein's surface, to a quadraphony that wraps the listener and makes himself or herself immersed within the surface, where he or she is "listening" to the fourth dimension through the spatialization of moving sound; to the Shepard tone to give an idea of vertiginous rising and descending on the surface of the "bottle"; to the reference to Möbius strips; to the ancestral ouroboros meant as an "eternal return"; to singable melodies that quietly end after a journey where mathematical surface almost becomes an existential metaphor. Fundamental musical structures emerge from a geometric shape, and, vice versa, the geometric structure suggests musical sequences now melodious, now dramatic, but always expressive. Behind each composition there are deep creative ideation, thinking, choices, months of intense work, pages and pages of notes and schemes. All of them -- it must be highlighted -- lived by students with great enthusiasm and in addition to their commitments and their study load.

The tenth and conclusive meeting of the cycle of seminars happens, as Tosca was singing, by "caso o fortuna" (by fortune or fate), in coincidence with the World Day of Mathematics: Pi Day! It has been the ideal opportunity to present a "Prima della Prima" (Preview of the Première) of the final event: "Klein Concert -- Five Compositions on a Geometric Shape."

In the opening, I presented the course's structure and organization; then, each student-composer described his composition, whose I performed some passages at the piano, synthesizing string score. Each step has been integrated by a sequence of slides. Closing section involved the integral listening of two compositions for electronic music on the given theme.

For the Music Conservatory it has been a new experiment, and was considered truly successful. In fact, as I was expecting, given a specific extra-musical theme, all works, independently from their diversification, presented a common structure, that re-constituted essential lines of Klein bottle, that is, of the proposed model. All compositions are, in fact, within the same equivalence class given by the initial geometric theme, and thus they present a unity within the variety. Variety is an object of artistic interest, and unity, of scientific one. However, because varieties belong themselves to mathematics, art and science seem once more to merge together as the two superposed Möbius strips.

Joining Klein to Palermo wasn't an entirely random choice. Felix Klein belonged to the Circolo Matematico di Palermo, founded by Giuseppe Guccia in 1884. Its golden era ended with the death of its founder, in 1914, in coincidence with the beginning of the "Great War" that also destroyed communications between scientists of different parts of the world. The symbol of the Circolo was Trinacria or Triskelion (a very ancient symbol and emblem of Sicily), so familiar to mathematicians of that time, to be cited in a monograph on Simmetria by Hermann Weyl.

The Circolo included renown mathematicians such as Hilbert, Poincaré, and Noether.

If in 2018 Palermo was the Italian Capital of Culture, it is thus worth of attention reminding that it once was a sort of world capital of mathematics.

During the aforementioned tenth meeting, the participation of the orchestra had been planned, but it was unexpectedly postponed *in extremis* for technical and bureaucratic reasons. However, as *inter omnes constat*, if we can solve differential equations, even changing boundary conditions we can still reach the solution, even if it is different from the expected one.

The event caught the attention of media, including newspapers "Il Giornale di Sicilia," "La Repubblica" (video here and here, update June 2019) and the online newspaper "Il Sito di Sicilia," as well as curiosity and enthusiasm between math-musicians.

The highly-expected "Klein Concert" happened in May (with the orchestra!) at the Music Conservatory of Palermo. I shared the podium with two conducting students: Salvatore Barberi and Antonino Sfar. Here, the video of the concert.

STEAM is moving forward, mathematics enchants, music world let itself be pleasantly inspired!

]]>Why do we need theoretical research in physics? What did it mean to be a young physicist in the Soviet Union? And finally, which characteristics should a good student of physics have? Dirac medalist and member of the National Academy of Sciences, Prof. Mikhail Shifman answers these questions and more for *Math is in the Air*. Mikhail (Misha) Shifman, a Russian theoretical physicist based in the US, at the University of Minnesota, is one of the leading scholars in quantum chromodynamics and quantum field theory.

MM: Professor, can you tell us something about your scientific journey, from Russia to the US?

MS: My scientific career started in the early 1970s. I was a graduate student in Moscow, at the Institute of Theoretical and Experimental Physics (ITEP), from ’72 till ’76. That was a fortunate time for young people in theoretical high energy physics: this was the time of great experimental discoveries. Theorists were guided by breakthroughs in experiments. I was a little bit late: my life would have been completely different if I had started a year earlier, because I became a graduate student in 1972, at the time when Yang-Mills theory was proven to be renormalizable and the Weinberg-Salam model appeared. Shortly after, in the spring of 1973, asymptotic freedom of Yang-Mills theory was discovered. This was the beginning of Quantum Chromodynamics (QCD), the theory of strong interactions.

So, I couldn’t participate at the initial stage, but I delved into the area very early, especially after the November revolution of 1974, when particles with charmed quarks were discovered. I lived through these years of general enthusiasm. This was a very exciting time. Young people should work on new things! It is much easier to build a career in this way. There is less competition if discoveries come almost weekly.

So, I participated at very early stages of QCD. Another fortunate event for me was supersymmetry –the first serious expansion of our space–time after Einstein. It was invented in 1971, in Moscow, first by Golfand and Likhtman and two-three years later at CERN by Wess and Zumino.

At first, I didn’t pay that much attention to supersymmetry. There was no evidence of empiric support for it. In the early 1980s I got interested in a completely different aspect of supersymmetry -- non perturbative phenomena. These phenomena occur at strong coupling. In 1982, Valentin Zakharov and Arkady Vainshtein suggested me to consider strong coupling effects in supersymmetric Yang-Mills theory. This paved the way to a novel direction of research which is extensively used even now.

About my personal life. Starting from the early ‘80s I was receiving invitations to give talks at international conferences, review talks or lectures. Exchange of ideas, fresh ideas from outside, are very important in theoretical physics. However, I was not allowed to leave the USSR. At least, not until Gorbachov came to power, which happened in 1985. And then, with Perestroika, in 1989, I was invited for a year to lecture in Switzerland, at Bern University. This course was supposed to last two semesters, so, it was a long-term journey. I made an attempt to obtain permission from authorities, and a miracle happened: they allowed me to go with my family. Previously, this would have been completely unconceivable, but this time, after a number of adventures, finally I got permission to go to Bern. But without my older daughter, who had to stay in Moscow as a “hostage.” I started to lecture there, and then, our older daughter got permission to join us. Another miracle. I visited many major labs to talk physics with my colleagues. In the spring of 1990 the USSR became unstable, which made me nervous. On the other hand, I was sort of happy. The day of the Soviet invasion in Czechoslovakia in 1968 was an eye opener. I understood that ideology of communism is inhuman.

The Soviet Union collapsed in a year or two. All obstacles were eliminated. In spring of 1990, my daughter who had been studying at a University near Moscow, obtained an exit visa to join us, and exactly at that time, I got a letter that I did not expect.

Larry McLerran, who is a theorist at the borderline between nuclear and high energy physics, sent a letter informing me that his newly opened institute at the University of Minnesota had some senior vacancies and invited me to come. I was invited to visit Minnesota. We came there, all my family. I got offers also from a couple other places, but we decided to accept McLerran’s offer because there were a few people in his institute whom I knew very well from Russia. Also, Minneapolis is a nice city, with lots of cultural events. And we were not that much afraid of the cold winters, they reminded us winters in Moscow as they used to be 30 years ago.

We moved in August/September of 1990, that is, 29 years ago! We settled there and I never had any regrets. It’s a good place. I have done a lot of good physics here, and my children grew up here. I think that our Institute is very nice..

In Russia, our group at ITEP was one of the best in the USSR –this was my salvation from the insanities of the external world around me. ITEP was very isolated. Journals arrived with a long delay. It was difficult to report achievements. But that was what we had there, the so-called developed socialism. Isolation bothered me a lot. This is one of the reasons why I decided to not go back to Moscow, at least, not immediately. I got a couple of offers from Germany and France, but still decided to accept the offer from Minnesota. I love Europe, this is my culture, but to start a new life from scratch is easier in the US.

MM: During your career, you received several awards, including the Dirac medal. Which one do you consider as the most important one, and which one is the dearest to you?

MS: In 1999 I got my first recognition, this was very encouraging! And then, a number of other awards. In particular, the Lilienfeld Prize in 2006 meant two things: it is awarded for theoretical physics achievements and the ability to explain them to the general public. Communicating with the general public is my hobby.

I wrote a few books on history of quantum physics in the ’30s and ‘40s — the greatest achievements — that occurred despite the mad world in which physicists lived, in Europe and the USSR at that time. The work required many weekends. I was working on that just for personal satisfaction.

Quantum mechanics, quantum field theory, and nuclear physics are my favorites from the school days in the the Soviet Union. I learned lessons of Landau, using his famous eight-volume course of theoretical physics.

Recently, I published a book based on the letters between Sir Rudolf Peierls and his Jewish-Russian wife Genia Kannegiser.

Their life was full of adventures and misadventures, full of love and physics. This book is also based on archival materials, memoirs of their friends and relatives, and conversations with their daughters. Genia Kannegiser came from a huge family of the Mandelshtams which is nearly extinct now. It gave Russia Osip Mandelshtam, arguably the greatest poet of the 20-th century. He perished in a Gulag camp.

After the Lilienfeld prize, I received, as an award, the Blaise Pascal chair of the CNRS in France. A year I spent there was a very happy time. I was surrounded by good people.

And, finally, in 2016 came the Dirac Prize…

MM: Dirac medal: can you describe us, in a nutshell, the research that led you to this prestigious award?

MS: I was awarded with the Dirac medal and prize in 2016. It is especially important to me for two reasons. First it is given only for significant achievements in theoretical physics. Second, they have a great company there: Witten, Zeldovich, Polyakov, Zumino, Gross, Green, Schwarz, Wilczek, Parisi – all my good friends and people whom I admire.

I received the Dirac Medal with Nathan Seiberg and Arkady Vainshtein. Arkady is my collaborator from 1973-74, we wrote lots and lots of paper together. In fact, he was my informal teacher. Basically, most of what I learnt, I learnt from from him when I was a student in Moscow. He’s recently retired, but we are still in touch.

Nathan Seiberg is a younger theoretical physicist from Israel, who has been since many years in Princeton. He was doing great work on topics overlapping with our earlier research, such as strong coupling and supersymmetry. Together with Witten he immensely advanced research in this area. This field is still very much alive and thriving these days. Nati is an exceptionally good physicist.

I had never met him in person before I came to the US. In Moscow, I used to read almost all papers he authored on arrival. Also Witten’s papers. They were inspirational for me.

MM: We are now used to emails and fast internet communication. How was it doing research and communicating in the 70s in the Soviet Union?

MS: Well, once in a while, somebody from the West would come to our institute. I remember James Bjorken used to come pretty often, as well as Marshall Baker. These visits to a small extent broke our isolation. People sometimes came to conferences in Moscow or elsewhere in the USSR. But these were rare occasions. Besides, to be able to attend such a conference, one needed a special permission. I remember I was denied such a permission for Neutrino-75.

We could write letters. Letters addressed outside the USSR required another special permission. Sometimes it could takes weeks or months before the letter to a Western addressee would arrive. We could send our papers to European journals. And guess what… again a number of permissions were need. It was hard to communicate with the Western colleagues in the Soviet Union. The official doctrine was that all of them are enemies by default.

I wrote a short article on that. From it you can understand the conditions of isolation and how we survived.

MM: But isolation is not always a bad thing. Oppositely to ‘publish or perish,’ Prof. Shifman writes that:

“Now I would like to mention one more aspect which concerns me at present, a very strong pressure existing in our community, to stay in the “mainstream”, to work only on fashionable directions and problems which, currently, are under investigation in dozens of other laboratories . This pressure is especially damaging for young people who have little alternative. Of course, a certain amount of cohesion is needed, but the scale of the phenomenon we are witnessing now is unhealthy, beyond any doubt. The isolation of the ITEP theory group had a positive side effect. Everybody, including the youngest members, could afford working on problems not belonging to the fashion of the day, without publishing a single line for a year or two. Who cared about what we were doing there anyway? This was okay.”

About admissions at ITEP, Professor writes: “ITEP was more than an institute. It was our refuge where the insanity of the surrounding reality was, if not eliminated, was reduced to a bearable level. Doing physics there was something which gave a meaning to our lives, making it interesting and even happy.” Admissions were difficult, and “Even extremely bright students, who were too mathematically oriented, like, say, Vadim Knizhnik, were having problems in passing these examinations.”

Vadim Knizhnik lived only 25 years, but his contribution to mathematical physics was relevant; it includes developments in string theory and algebraic geometry, as concisely described in the obituary published by the Circolo Matematico di Palermo.

According to Prof. Shifman: “We had a wonderful feeling of stability in our small brotherhood. A feeling so rare in the Western laboratories where a whirlpool of postdocs, visitors, sabbatical years come and go, there are a lot of new faces, and a lot of people about whom you do not care so much. [At ITEP], The rules of survival were quite strict. First, seminars – what is now known worldwide as the famous Russian-style seminars. The primary goal of the speaker was to explain to the audience his or her results, not merely to advertise them. And if the results were non-trivial, or questionable or just unclear points would surface in the course of the seminar, the standard two hours were not enough to wind up. Then the seminar could last for three or even four hours, until either everything was clear or complete exhaustion, whichever came first.”

“Scientific reports of the few chosen to travel abroad for a conference or just to collaborate for a while with Western physicists, were an unquestionable element of the seminar routine. Attending an international conference by A or B by no means was considered as a personal matter of A and B. Rather, these rare lucky guys were believed to be our ambassadors, and were supposed to represent the whole group. In practical terms, this meant that once you made your way to a conference, you could be asked to present important results of other members of the group. Moreover, you were supposed to attend as many talks as physically possible, including those which did not exactly belong to your field, make extensive notes and then, after returning, deliver an exhaustive report of all new developments discussed, all interesting questions raised, rumors, etc. The scientific rumors, as well as non-scientific impressions, were like an exotic dessert, usually served after nine. I remember that, after his first visit to the Netherlands, Simonov mentioned that he was very surprised to see a lot of people on the streets just smiling. He said he could not understand why they looked so relaxed. And then he added that he finally figured out why: <<... because they were not concerned with building communism...>> This remark almost immediately became known to “Big Brother” who was obviously watching us this evening, as usual, and it cost Simonov a few years of sudden “unexplainable allergy” to any Western exposure.”

MS: My first paper was published in 1972, that is, 47 years ago! A few years ago I counted that I had 64 different collaborators from all over the world. This number is growing. I always liked working with people, not only after I moved to the US, but also in the Soviet Union. From 1990, since I’m in the US, I’ve been collaborating with the Americans, French, Italian, Swiss, German, Russian, English, Israelis… wherever people do theoretical physics.

MM: Why do we need abstract research in theoretical physics in today’s world?

MS: There are many reasons. Technology comes after theory. In the last thirty years, we had the Internet, Google, social networks, smartphones, Whatsapp, GPS, and many other things which changed the world, made all people better off. And not only in Europe and North America: in India, Africa, South America – everywhere – people use computers and learn a lot, speak with their friends and relatives thousand kilometers away, buy books and clothing and so on. And all of this is the consequence of the discovery of transistors in 1947 by Bardeen, Brattain and Shockle seventy years ago. Fundamental research is an investment in the future. This is costly investment, but without it the future of our children and grandchildren will not be better. Especially expensive is research in high energy physics. I hasten to add that the World Wide Web, a byproduct of research at CERN, payed off ten-fold or maybe hundred-fold. Another example is MRI. No modern clinic can function without it. Many years ago magnetic resonance in atoms was a part of fundamental research. In the history of humankind people always have been doing things which hardly seemed practical for society. Take Galileo: who could think that his experiments in Pisa would lay the foundation of every single machine operating today? At his time Galileo’s throwing balls from the Tower of Pisa was really obscure. But he opened up physics for us, everything is based on that.

Experimental and theoretical physics grew into a huge tree now. Everything physicists do needs money, resources. Of course, there are limitations because society is not infinitely rich and after all it is society as a whole which pays for fundamental research. Therefore priorities should be set. Some projects could require five years, others ten, still others 20. Funding comes from public funds. Thus, the public has a say. Educated people can have some knowledge in the subject; their opinion is relevant and important.

Some people will always be against costly scientific projects such as accelerators: this is the case of Sabine Hossenfelder, a well known physicist and writer, interested in foundations of physics. She authored a recent book titled *Lost in Math: How Beauty Leads Physics Astray*. She is an ardent opponent of future accelerators. I understand her motivations. To a certain extent I agree with ideas expressed in this book. But I disagree with her conclusion. If you stop developing a certain area, people who work there will go elsewhere. The scientific and technological culture resides in people. It is like the Olympic flame, transferred from hand to hand. In Germany, Hitler forced hundred or two of their best scientists to leave the country. Not too many, right? But they were the best. Now, 70 years later, German research is far from global dominance – the level it had in the 1910s-30s. Why? The chain was broken. The generation which should have replaced Born and Einstein was not there. One missed generation was enough to lose all what had been accumulated earlier. Germany invests now a lot of money in science, but I will risk to say it will never achieve the status it had before Hitler. If we start building accelerators in ten or 15 years from now (when hopefully people decide to renew experiments in high energy physics) there will be nobody who could do it or could teach young people. Nobody, who could pass the flame to students. If a generation is destroyed, it is very hard to restore this connection.

Therefore, unlike Sabine, I believe we need at least some projects in high-energy physics to keep the scientific culture alive.

MM: This is very intriguing: science relies on people, and, in particular, professors and students. This reminds me of art creating and art schools, where teacher-to-student transmission of knowledge is crucial.

Can you tell us something about your relationship with the world of the arts, and in particular, with music and with the visual arts?

MS: I’m a big fan on painting! I’ve strong opinions about what I like or dislike. I try to support young Russian artists by buying their paintings. Usually they are affordable. I have a small gallery at home, perhaps, ten paintings or so. As far as music is concerned, this is a problem — I don’t have a good ear. My grandchildren have perfect pitch, and they play piano (three of them) and violin (one of them). Maybe they inherited their music talent from my wife, who sings very well.

MM: Which characteristics should a good student of physics possess?

MS: A good student must be very curious. Also he or she must make some compromises in his or her life, sacrifice something: physics takes so much effort and time, a lot of traveling, less friends… Work ethics is also important. Every problem, when you start thinking on it, seems very hard. It is important to be able to overcome initial hesitations. A good student has to read a lot, attend courses of good lecturers, talk to peers, and look through new books and papers. It is very good to find a group of thinkers close to you. If you are alone, it is extremely difficult, almost impossible to be a good student. Science is so broad now, one can easily get lost.

You cannot learn only from books, but also from people: you need other people around with whom to discuss projects. Having around just five or six people interested in a topic is a blessing. There is an interesting article on this issue, written by a Nobel prize in Physics, Gerard ’t Hooft.

MM: Thank you Professor!

Italian translation available here:

http://www.mathisintheair.org/wp/2019/05/perche-la-fisica-e-importante-intervista-a-mikhail-shifman

John Baez is an American mathematical physicist, and a professor of mathematics at the University of California Riverside, and an activist for the environment. I have been in touch with him via email and through his online course on category theory. Recently, I had the pleasure to met him in person in London, during a conference about Physics and Philosophy dedicated to Emmy Noether. In the last few days, I had the honor to interview Prof. Baez for the blog Math is in the air.

**MM.** You are one of the pioneers in using internet and blogging for scientific education, with ‘This week’s finds.’ Which words would you use to feed the enthusiasm of young minds towards abstract mathematics?

**JB.** It seems only certain people are drawn to mathematics, and that's fine: there are many wonderful things in life and there's no need for everyone explore all of them. Mathematics seems to attract people who enjoy patterns, who enjoy precision, and who don't want to remember lists of arbitrary facts, like the names of all 206 bones in the human body. In math everything has a reason and you can understand it, so you don't really need to remember much. At first it may seem like there's a lot to remember - for examples, lists of trig identities. But as you go deeper into math, and understand more, everything becomes simpler. These days I don't bother to remember more than a couple of trigonometric identities; if I ever need them I can figure them out.

But the really surprising thing is that as you go deeper and deeper into mathematics, it keeps revealing more beauty, and more mysteries. You enter new worlds full of profound questions that are quite hard to explain to nonmathematicians. As the Fields medalist Maryam Mirzakhani said, "The beauty of mathematics only shows itself to more patient followers."

**MM.** I love the reference to patterns, and the beauty to find. Thus, we can say that mathematical beauty is not ‘all out there’ as the beauty of a flower can be. Or, that some beautiful geometry present in nature can give a hint or can embody some mathematical beauty, but people have to work hard to find more of it — at least they have to learn how to look at things, and thus, how to mathematically think of them.

In the common opinion, a rose, or a water lily is beautiful (and it is!), but a bone is not ‘beautiful’ per se. Personally, each time I find patterns, regularities, hierarchical structures, I get excited and things seem to be at least mathematically interesting. I would like to ask you how would you relate the beauty in the natural world, both visible and ‘to discover,’ and the beauty of math.

I’m wondering if they should be considered as two separate sets with occasional, random intersections, or as two displays of a generalized ‘beauty,’ as two different perspectives. Or, maybe, if the first can guide our search into math, or if math can teach us ‘how to look at things and finding beauty.’

**JB.** I think all forms of beauty are closely connected, and I think almost anything is beautiful if it's not the result of someone being heedless to their environment or deliberately hurtful.

It's not surprising that flowers are very easy to find beautiful, since they evolved precisely to be attractive. Not to humans, at first, but to pollinators like birds and bees. It's imaginable that what attracts those animals would not be attractive to us. But in fact there's enough commonality that we enjoy flowers too! And then we bred them to please us even more; many of them are now symbiotic with us.

Something like a bone only becomes beautiful if you examine it carefully and think about how complex it is and how admirably it carries out its function.

Bones are initially scary or 'disgusting' because when they're doing their job they are hidden: we usually see them only when an animal is seriously injured or dead. So, you have to go past that instinctive reaction - which by the way serves a useful purpose - to see the beauty in a bone.

Mathematics is somewhere between a rose and a bone. Underlying all of nature there are mathematical patterns - but normally they are hidden from view, like bones in a body. Perhaps to some people they seem harsh or even disgusting when first revealed, but in fact they are extremely elegant. Even those who love mathematics find its patterns austere at first - but as we explore it more deeply, we see they connect in complicated delicate patterns that put the petals of a rose to shame.

**MM:** Thus, there seems to be an intimate dialogue between nature, both visible and hidden, and mathematical thinking. About nature and environment: in your Twitter image, there is a sketch of you as a superhero saving the planet, with the mathematical symbol ‘There is one and only one’ applied to our planet Earth.

Can you tell the readers something about the way you combine your research in mathematics with your engagement for the environment?

Also, it is often said that beauty will save the world. Do you think that mathematical beauty can save the world?

**JB:** I mainly think of beauty - in all its forms - as a reason why the world is worth saving. But we are very primitive when it comes to the economics of beauty. Paintings can sell for hundreds of millions of dollars, and we have a market for them. But nobody attaches any value to this critically endangered frog, *Atelopus varius*.

To my mind it's more beautiful and precious than any painting. Not the individual, of course, but the species, which has taken millions of years to evolve. We are busy destroying species like this as if they were worthless trash. Our descendants, if we have any, will probably think we were barbaric idiots.

But I digress! I switched from pure mathematics and highly theoretical physics to more practical concerns around 2010, when I spent two years at the Centre for Quantum Technologies, in Singapore. I was very lucky that the director encouraged me to think about whatever I wanted. I was wanting a change in direction, and I soon realized that mathematicians, like everyone else, need to think about global warming and what we can do about it: it's the crisis of our time. I spent some time learning the basics of climate science and working on some projects connected to that. It became clear that to *do *anything about global warming we need new ideas in politics and economics. Unfortunately, I'm not especially good at those things. So I decided to do something I can actually do, namely to get mathematicians to turn their attention from math inspired by the physics of the microworld - for example string theory - toward math inspired by the visible world around us: biology, ecology, engineering, economics and the like. I'm hoping that mathematicians can solve some problems by thinking more abstractly than anyone else can.

So to finally answer your last question: I'm not sure the beauty of mathematics can save the world, but its beauty is closely connected to clear thinking, and we really need clear thinking.

**MM:** Yes, in a certain sense, despite culture, technology, and thousands of years of human history, people are quite primitive when it comes to evaluating beauty as detached from the economy.

You brought up an important point: the research focus of mathematicians. This is a tricky point because young researchers are kind of split between following new ideas and projects, and the search for funds, that often leads them to join existing projects or just well-funded areas and to put aside their more ‘visionary’ ideas. What would be your suggestion to find a balance?

**JB:** I don't know if I can give advice here: I've never needed to search for funds, I get paid to teach calculus and other courses, so I always just do the best research I can. That's already quite hard - I could talk all day about that!

I suppose if you're struggling for funds you have to fight to remember your dreams, and try to work your way into a situation where you can pursue these dreams. I imagine this is also true for any entrepreneur with a visionary idea. Academics struggling to get grants really aren't all that different from executives in a large corporation trying to get funding for their projects.

**MM:** My last question is about the theme of peace, very important to the Baez family:

Many innovations are related to the military. Do you think that the needed clear thinking you mentioned, can first of all come from times, themes, and ideas of peace?

**JB:** We are currently in a struggle that's much bigger, and more inspiring, than any war between human tribes. We're struggling to come to terms with the Anthropocene: the epoch where the Earth's ecosystems and even geology are being transformed by humans. We are used to treating our impact on nature as negligible. This is no longer true! The Arctic is rapidly melting:

And since 1970, the abundance of many vertebrate species worldwide has dropped 60%. You can see it in this chart prepared by the Worldwide Wildlife Fund:

If this were a war, and these were humans dying, this would be the worst war the world has ever seen! But these changes will not merely affect other species; they are starting to hit us too. We need to wake up. We will either deliberately change our civilization, quite quickly, or we will watch as our cities burn and drown. Isn't it better to use that intelligence we humans love to boast about, and take action?

**MM:** Thank you Professor, I hope these words will enlighten many people.

We have all seen, at least once in our life, a juggler tossing balls in the air. Why is that so impressing at our eyes?

Despite having just two hands, any respectable juggler can **juggle** three balls at the same time. Considering for simplicity that one can handle one ball for each hand, how is that possible?

Let's try to analyze Animation 1. We can see that each ball is tossed by one hand to the other: the right hand tosses the balls to the left hand and vice versa. Just as the floating ball floating is about to fall down, the juggler tosses another ball up to free his hand and catch the falling one. Juggling three or more balls is possible only by iterating this principle.

The **pattern** represented in Animation 1 is known as three-ball *cascade*. Let's analyze now Animation 2 and compare it with Animation 1.

In this case we immediately note that the number of balls is still 3, but the pattern is different. Indeed, by observing it carefully, we see that the juggler tosses the three balls at three different heights.

As you can easily imagine, there is a wide variety of patterns and, if we were to assign a name to each pattern (as in the case of the *cascade*), we would have to make a prohibitive effort of memory.

For this reason Paul Klimek and Don Hatch, at the beginning of the 80s, independently invented a notation system to describe and name juggling tricks nowadays called **siteswap**. Afterwards, this system has been developed and extended by other jugglers, like Bruce Tiemann, Jack Boyce and Ben Beever.

Siteswap is able to describe (and name) all juggling patterns with any number of jugglers and balls, covering both the case of *synchronous* and *asynchronous* throws. In the following two animations we can see the same pattern done in both the asynchronous and synchronous versions.

(NOTE: some patterns can be only asynchronous while others can be only synchronous).

For simplicity, we will describe the so-called **Vanilla siteswap.** This siteswap notation allows us to describe all the patterns where the balls are tossed asynchronously by a single juggler using both hands.

Before going through the description of this notation method, we must underline that siteswap has a limitation. Let's observe the following two animations.

We have seen already the left-side animation: the three-ball cascade. The right-side pattern, known as three-ball *Mill's Mess*, is still a cascade but it's done by crossing and switching the hands' position alternatively. Even though the two patterns look very different, they have the same siteswap notation, i.e. they are identical. Indeed, if we focus on the trajectories of the balls with respect to the positions of the hands, we see that *Mill's Mess* is identical to the normal cascade.

Therefore, siteswap is able to describe juggling patterns by considering the height and the direction in which the balls are tossed (a ball can be tossed to the same or to the other hand) but without considering "how" the pattern is executed.

After this quick introduction, we will now describe how siteswap works. The basic idea is very simple: we assign a positive integer number to each throw that corresponds to the number of **beats** (soon we will deepen this concept) that the ball takes to complete his trajectory. We use odd numbers (**1**, **3**, **5**, ...) for throws from one hand to the other hand and even numbers (**2**, **4**, **6**, ...) for throws from one hand to itself. The number zero (**0**) is used to indicate when one hand is not holding balls during a beat.

In other words:

- A
**0**means a beat when the hand is empty. - A
**1**means a direct throw from one hand to the other, during which there is no time to catch or throw other balls, i.e. it is executed in one beat. - A
**2**means a very small throw (almost imperceptible) of a ball to the same hand. While the ball is completing its trajectory, the hand who tossed it has no time to do anything else while the other one has a beat to catch and throw another ball. - A
**3**means a throw from one hand to the other during which both have a beat to juggle a ball each (so there is time to juggle two other balls). - A
**4**means a throw from one hand to the same hand during which the tossing hand can juggle another ball while the other hand can juggle two balls (so there is time to juggle other 3 balls). - ...

Therefore, the numbers indicate the height at which the balls are tossed relatively to the execution speed of the throws. Indeed, it is possible to toss a **5** with top height under our head if we juggle quickly, or over 3 meters if we juggle slowly. What really matters are the beats left to juggle other balls during the trajectory of the toss. This depends, of course, by the speed of the juggler.

Furthermore, as it is easy to guess from the animations above, the patterns are repeated cyclically. In other words, there is a **period** after which the pattern is repeated (identically or symmetrically). With the siteswap notation we only write the throws that identify the period of the pattern. For example, the period of **531531531** is **531**. We refer to it as **531** by removing the redundant part and without loosing any information.

Once the concepts detailed above are clear, we can try to recognize some patterns:

Once we are familiar with the concept of siteswap we can go through a little bit of theory. Let's try to imagine the pattern **432**. First, say with the right hand, we toss a **4**, i.e. the ball will falls in the same had. Then we toss a **3** with the left hand, i.e. the ball will fall in to the right hand. While the two balls are still completing their trajectory, the right hand executes a **2**, in other words it performs a small toss to itself. What will happen is that the right hand will find itself with three balls falling on it at the same time. In siteswap jargon this event is called *collision*, and the pattern is impossible to repeat. Indeed, the sequence **432** is not executable.

How can we distinguish an executable sequence from a non executable one? Fortunately maths comes to the rescue! Indeed, there is a theorem that characterizes siteswaps and gives us a condition such that there are no collisions.

**Characterization theorem of siteswaps**

A finite sequence of non-negative numbers (where is the number of digits) is executable if

for each .

Here, the operator returns the remainder of the division

Let's come back to the previous example and verify, using the theorem, that the sequence **432** is not valid:

In this case we get 2 for every digit of the sequence and, according to the theorem, this is not a valid siteswap. We now try to apply the theorem to a valid siteswap that can be obtained by switching the last two digits of the sequence above: **423**

It is clear that this siteswap respects the condition imposed by the theorem (and you can actually find it in one of the animations above).

Suppose now to have a valid siteswap, for example **534**, How many balls do we need in order to execute it? Again we have another nice and helpful theorem used by the jugglers from all over the world.

**Theorem on the number of balls**

If is a valid siteswap (where is the number of digits), then we have that

Let's try how many balls we need for the **534 **pattern:

The answer is 4 balls!

Do you think that those patterns are science-fiction? Try to watch the following video by Ofek Snir, a great juggler that executes (among other stuff) some very hard siteswap with 7 balls.

As already specified above, this article only talks about the *Vanilla siteswap*. Actually there are also siteswap notations to represent other categories of patterns such as synchronous, the patterns where one hand can hold and toss more than one ball at time (in jargon **multiplex**) and the ones executed by more than one juggler (in jargon **passing**). Here are some examples

PANGOLIN (Ground Pangolin, manis temminckii)

A pangolin curled up in the defensive position:

When the pangolin is frightened, it curls up, becoming a sort of armored ball that the predatory animals are not able to open, but easy to be caught by poachers.

The pangolin is characterized by a strong scale armor that makes it look almost like a small dinosaur. The pangolin is too little to run and too big to hide, thus nature gave it the armor. The particular armor of the pangolin can guide us to the discovery of some basic concepts of category theory.

Let us consider one scale.

Then, two scales.

The repetition of two scales is defined by a transformation that we call g:

The scales are our “objects,” and the arrows g are our “morphisms.” If we compose several g-arrows, we again obtain scales, as the following image shows.

If we don’t make any repetition, thus, if we move from a scale and we apply the arrow “1” that gives us again the same scale, we just defined what mathematicians call an “identity.” Thus, already the very first observation allows us to define the category “scale,” with its objects, arrows, identity-arrows, and arrow composition.

The whole image of the pangolin in its closed-defensive position is much more complicated. We can try to add new “transformations” to build such a whole image.

After the “horizontal” composition, we add a vertical composition, given by the “vertical” repetition of scales, through the arrow h.

In this way, by combining g and h, we can build several rows of scales, the ones partially superposed to the others. However, by looking at the real image of the pangolin’s armor, we can see that the rows are offset as if we introduced a little shift in the even rows. Let us indicate such a transformation with Sh (where Sh stands for shift and not for ‘s composed with h’):

We obtain all the scales by repeating N-times the action of g and h. To obtain a whole image that is more realistic, we can modify the “general shape” of the image via the arrow I:

Finally, we “close” the obtained figure, to imitate the image of the defensive-position curled-up pangolin’s back. Here, we use letter L which stands for Loop.

Remaining within the category “scales,” we reconstructed a complex image through a sequence of progressive transformations.

If we keep working with a category-typical operation, we can take such a collection of objects (points) and morphisms (arrows), and we can apply this to something else. It is like… to have some bridges, from an initial to a final point (to be opportunely shifted into a new initial and new final point), and we can transform each bridge into another bridge: we would create a “bridge of bridges”! The concept of “transformation between transformations” is, in fact, the “primum movens” that gave birth to Category Theory in the Forties of last Century.

Now, let us try to apply all of that to music. We can move from the category “images” to the category “musical fragments.”

There is an endless number of ways (mappings) to transfer a visual shape into a musical structure (in general, a set of non-sound data into a sound data). We are in the field of “sonification.”

In our analysis, we can choose a melody that imitates the upper contour of the scale, with a raising and lowering movement. Is that a case that, in English, the term “scale” applies both to music as well as to a part of the armor?

Now, let us apply to music, one by one, each transformation we could apply to images. Our priority is to opportunely translate each arrow from the visual field into the sound domain.

The horizontal repetition g could be ‘translated’ in a time repetition (g’) of a musical fragment; the vertical repetition h becomes a simultaneous repetition of the same melodic fragment at different pitches (h’); the spatial shift of even rows becomes a time shift of even melodies.

There are several ways to musically render the transformations I and L. A possible choice consists in the assignation of an intensity change in correspondence of the change of shape (I’), and a cycle of repetition of fragments and melodies with L’. We get a series of musical fragments (objects) connected by transformations (the arrows g’, h’, Sh’, I’, L’). The composition of two or more arrows still gives musical fragments, and an arrow that does not change anything and returns a musical fragment that is identical to the initial one is the identity. In this way, we build up the category of musical fragments.

We obtain the same result if we separately sonify each image, and also if we sonify the initial one and we successively apply the musical transformations. For such an invariance of results, we obtained diagrams that are called “commutative.”

The transformation that brings each image into a musical fragment (red arrows), and each visual transformation into a musical transformation (green arrows), is the “sonification functor.” In category theory, a functor is a mathematical entity that transfers objects and morphisms of a category into objects and morphisms of another category. It is a sort of… hyper-bridge between bridges and islands!

Let us call S_{1} the functor we just defined: we have g’ = S_{1}(g), h’ = S_{1}(h), …, and so on.

Let us now suppose to have another sonification functor, S_{2}. Category theory also allows here to define what changes between the melodic fragments we produced with S_{1} and the melodic fragments we produced with S_{2}, if we restrict the analysis to the symbolic point of view. These transformations are called “natural”: they are transformations α defined from S_{1} to S_{2}. All said above can be summarized in the following image.

Via the tools given by category theory is then possible to create music, and also to analyze written music—by “decomposing” the structures as we did for the pangolin’s armor—as well as to analyze basic and advanced elements of musical practice. An example for all: a crescendo at the piano, from ‘piano’ to ‘forte’ can be described through an arrow; a slower crescendo and a faster one can be represented as two arrows between the same points, connected by a temporal transformation (an arrow between arrows.)

Summarizing, the particular example we considered can also be interpreted as a structure to compose, program, and also as a scheme to improvise music. The following link leads us to an improvisation whose unique “score” is constituted not by notes to play, but by transformations to apply.

(Idea, study, schemes, and drawings by Maria Mannone)

**Selected References.** A very clear textbook to start the study of category theory is “Conceptual Mathematics” by Lawvere and Schanuel. For a deeper reading, we suggest the classic textbook “Categories for the Working Mathematician” by Sounders Mac Lane, and, for an interdisciplinary view on the topic, the reader can examine “Categories for the Sciences” by David Spivak. The recent literature on mathematics and music about the applications of category theory to music includes the works by Guerino Mazzola, Franck Jedrzejewski, and Alexandre Popoff. The Italian book “Le Figure della Musica” by the composer Salvatore Sciarrino, even if not explicitly talking about category theory, highlights the importance of concepts of elementary mathematics to compare structures and transformations between music and visual arts. The works by Lawrence Zbikowski concerning the relationships between music and movement from a cognitive point of view can find in category theory a formal explanation, that graphically describes the “transformations of transformations.” The philosopher Charles Alunni uses diagrammatic thinking in his works, and the physicist John Baez, an expert of the topic, administrates an interdisciplinary blog, considered as one of the most important references for category theory. My studies are about applications of category theory to the orchestra, structures in composition, and relationships between music and images.

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I've always been amazed by the beauty of nature and its wonderful patterns: symmetries, spirals, meanders, waves, cracks or stripes.

At the very beginning I started creating artworks with basic geometry and fractals, but later I discovered the possibilities of using randomness, physics, autonomous systems, data or interaction to get more expressive and meaningful artwork.

My goal as a generative artist and experimental animator is to create autonomous systems that make the essence of nature's beauty emerge by modeling not only its appearance but its behavior.

Mathematics, physics and computation are essential tools to implement a set of rules that an autonomous system must follow to simulate nature's appearance and behavior.

When I started this project I had in mind so many different ideas related to generative art and creative processes, and I was willing to explore them. But I had no idea what path to follow. "*Spaghetti coding*" is a pejorative phrase in programming world to refer to a piece of code that has a complex and tangled control structure. So basically I called the website "*Spaghetti Coder*" because it just reflected my state of mind at the beginning of the project.

"Spaghetti coder" is a creative and experimental art project inspired by generative artists from the past century such as Ellsworth Kelly, Sol LeWitt or François Morellet, but using not only chance as a main generator but physics, agents, data or interaction.

I got inspired by the essence of pioneers in Generative Art and I'm trying to evolve that conceptual vision using nowadays tools (programming, media, data, IOT) and exploring how it suits in experimental animation world.

Depending on the nature of the artwork I'm developing at the moment, I use many different mathematical tools: trigonometry, vector spaces and calculus, operators and matrices, densities and distributions, combinatorics, graph theory ...

And I usually write my code using programming languages such as Java, Processing, C++, openFrameworks and GLSL to generate visual products (drawings, animations, 3d graphics), or SuperCollider and Chuck to compose algorithmic audio (soundtracks, audio effects).

When I use different programs that work together or there's some kind of user interaction through hardware (MIDI controller, instrument or IOT device), a communication protocol is required. Then I frequently use OSC, MIDI or DMX protocols.

AAAC (An Autonomous Agent Choreo). from Spaghetti Coder on Vimeo.

I think I was able to make some kind of rhythm emerge through the motion and sound of +5.000 autonomous agents and in the end it suggested many interesting variations in the global perception of basic elements of design (color, shape and texture). I'm also really happy with people's feedback and how AAAC worked in animation festivals. It has been screened at many of them: CutOut Fest, SIMULTAN, Tasmanian International Video Art Festival, Seoul International Cartoon & Animation Festival, Azores Fringe Festival, Bogotá Experimental Film Festival, Punto y Raya Festival...

The only experience that is worse than the annual condo meeting or queuing up at the post office is, probably, the parent-teacher conference.

It's an ordeal for the parents, forced to wait a long time. It's cause of panic for the students, who are afraid that their parents may be mad at them. But, I assure you, it's a terrible experience for the teachers too.

Obviously, since bad marks in maths are widespread in Italian schools (but the same can be said of schools all over the World), queues to speak to the maths teachers are never-ending, exactly like queues at the post office. In such situations, the long waits result in the best and worse behaviours on could possibly imagine.

As an experienced teacher, I tend to split parents into three categories: the strikers, the compliant and the flatterers.

The strikers are the funniest ones. They sit down, ask you how their child is doing and, before you have even finished to speak, they charge. They do it recklessly, without thinking, coming up with completely made up arguments like:

- "Last year, with the previous teacher, my son had good marks" (Actually he had E, with me at least he's got to C)
- "But, in elementary school he was great" (Sure, but that was 12 years ago, now we're studying limits, you may agree that that is somewhat more difficult)
- "I am an Engineer and I know what I'm talking about. My son told me that the results were correct. I don't understand why you gave him a D" (that's correct, too bad that he copied the results without showing how he got them. How did he get them? Was it some sort of godsend?)
- "They guy who gives private lessons to my son says that, with him, he always solves the exercises." (OK, what should the poor guy say other than your son can solve the exercises... with his help?)

The flatteres are, instead, the most dangerous ones. They always start with something like:

- "My daughter said that, thanks to you, she finally gets maths. She couldn't get it with last year's teacher." You would like to say: "Look, you have probably said the same thing to the other teacher when you were talking about your other child. Also, too bad that last year's teacher was me.", but you just shut up and pretend to be playing the game. The parent, in the meantime, scrutinizes you, trying to understand whether he fooled that idiot of a teacher or not.

- "Your explanations are excellent", they go on, "and my son is working really hard. It's a shame that last exam didn't go so well. But I'm sure that if you could help out with the grades, he would get back on track. It would be too bad if he lost motivation given that you're such a great teacher." And then the may speak ill of the other teachers, expecting that you'd support them against the guy who teaches philosophy or the one who teaches sciences.

With this lot, I usually keep a low profile. I don't expose myself to their trap and let them talk. Not much, but I do let them talk. Luckily the fact that there's a queue of people waiting to speak with me is a good excuse.

The compliant parents, instead, are the ones you try to plead for compassion.

"My son's going through may health problems, you know. He finds it hard to eat and has been so weak lately." And you say: "Are you talking about Alessandro, madam? The 6'4" tall guy who plays football four times a week with the semi-pro team?"

Then, after explaining the most improbable health and family problems, and saying that their child have experienced many different teachers across their school years, the compliant parent plays what they think is the ace up their sleeve:

"Anyway, sir, we can't expect much from my son. Nobody in the family really understands a thing about maths.

Now you would like to tell them:

*"Luckily, neither stupidity nor not understanding maths are hereditary. Actually, your son has many chances to get better."* But clearly you don't say that and try to encourage them reminding them that it takes time to catch up with maths.

In the end, this is the problem: we live in a society where it's not a problem to say:

"I never understood maths."

Many are even proud of this. How do they not understand that by saying that openly, what they're actually saying to their children is:

"Give up. Maths is useless. I never studied it and yet here I am."

The problem is with the parents (and in some of the humanities teachers) who candidly say that they do not understand maths. **They don't realize that they are indirectly giving permission to their childred to not care about maths!**

When this happens, a teacher has almost powerless. As long as family and society will take it as granted that one can live with no maths, and that maths is not part of a good citizen's cultural baggage, this is a hard battle to fight.

You will always meet students who say "Sir, I never got maths" and they will feel justified by the society.

But the parent-teacher conference come to an end too, sooner or later. At the end of the day, the maths teacher goes back home a little beaten up, a little dizzy because of all the words that have been spoken. He thinks with some melancholy about those "normal" parents (there are some obviously) with "normal" children (what a blessing!).

Next day, you start again your classes hoping that you'll convince your students to change their minds, they who are so sure they don't get maths. You start again your classes, trying to help them love as much as they can the topic that you yourself love (thanks to some other good teacher).

P.S.

I am sure that in a few years, when I will attend the parent-teacher conference as a parent, I will fall in one of the categories I've been talking about. There's nothing you can do about it, being a parent is really hard... harder even than mathematics!

(translated by Stefano)

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We can consider, for now onwards, a metric space. By my previously post it is clear that a metric space is the only one in which we can define a distance. If we use an Euclidean metric is simple to prove, but also to understand in an intuitive way, that the shortest distance between two points is a line that directly goes from the first point to the second one.

Now we try to repeat the same with a different metric. We can use a taxi driver metric; with this metric we can draw the minimum length curve between two points. In the following picture the minimum length curve is the line crossing the minor number of blocks.

The first important concept we can appreciate from the picture is that the minimum length distance is not univocal in some metrics. In some metric spaces more than a single minimum length curve is allowed.

Everyone can draw a straight line across a paper, probably it depends on how good you are in drawing. Given two points we can draw the line from one point to other one. Our brain is used to think the outstanding space in an Euclidean metric so we can figure out a segment connecting two points as a piece of straight line. It is so easy.

In some metric spaces a straight line is not a straight line in the sense we are usually to think about it. In some other metric spaces a straight line doesn't exist. In math, in this strange and fantastic world, a straight line has a definition. If in a metric spaces exist a minimum length line connecting two points and if this line is unique it represents a straight line for that metric space.

According to the above definition in the taxi driver metric space there isn't a straight line (the minimum length line is not unique) and in the Euclidean space it is the so common straight line we are used to draw.

In the end we can figure out a particular space made by only 4 lines. The man inside the space can see only the four lines and can move only along one of them. No other paths or lines or space or points are accessible by him.

In this case the red line represent the minimum distance path between the man and the final point. For the man the red line represent a straight line between the two points. Surely it is not a straight line from our point of view but it is due to a better space concept and to the possibility of observing the space from the outside.

In the end, we can conclude saying we are in the same condition of the man described above. We are cheated by our limited space perception. To trick us is more easy than we can think.

You are walking on train rails, straight on; in front of you the way and you walk in the same direction. Are you moving along a straight line? I let you think for few seconds... 1... 2... 3...

No, you are not moving on a straight line. We are walking along a minimum distance path between two points on the Earth surface.

This is possible because we perceive the earth as flat. We are like points in a 2D space. Our height is really small if compared with the earth radius so we are unable to perceive the earth curvature. All we perceive is a bidimensional space around us.

When we move along earth surface we don't perceive the fact we are moving along a curve path. Why? Simply becouse the path we are following is the only path we can see. Surely we can see other paths between two points but they are longer than the one we are moving on, so this path is effectively the straight line between two points for us.

The true concept of Euclidean distance isn't applicable to the path we follow on earth surface. When we move along the earth surface we can measure the purple path as distance between two points. The real Euclidean distance is represented by the red line (see figure above).

The difference between the two lines is as greater as the points are far away. If we move from one end to another of a football stadium the distance we travel is small if compared with earth radius. In this case the purple distance and the red one are really close each other and we can consider them as the same distance.

If the difference between the distance on earth surface and the Euclidean distance is neglectable for small distances it is not in the case of a flygth from Rome to New York.

Looking at a planisphere we can see Rome and New York are almost on the same parallel. Intuitively we can think the shortest path between the two cities is following the parallel on which they are located. Another time the perception of the space tricks us so we are wrong again.

On earth surface the shortest distance between two points is called geodesic. The geodesic is obtained intersecting the earth with a plane passing for the two points and the earth center. The consequence of this is the parallel (except the equator) aren't geodesics. The meridians are geodesics instead. An airplane flying from Rome to New York don't follow the parallel but the geodesics that pass over the two cities (see figure above).

The concept of geodesics was introduced by Georg Friedrich Bernhard Riemann.

When in its books of geometry Euclide try to set up the bases of modern geometry, he fixed some definitions. These are indemostrable but are so intuitive that they don't deserve a proof. They are called postulates or axioms. They give us a basic definition basic geometric entities (points, lines, planes, circles, angles).

The most interesting postulate is the fifth. It say, more or less, that given a line and a point out of it, then these exists only one line passing for the point and parallel to the first line, for this reason it is also called postulate of parallel lines. For a lot of time some mathematics tried to deduce the fifth postulate from the other four because it did not seem very intuitive.

For a moment we can think about the straight lines are in the euclidean definition. The shortest line between two points. On the earth surface these lines are the geodesics. We can then try to apply the fifth postulate to the earth surface. We can draw a geodesics, keep a point on earth surface, not on geodesics, and draw a geodesics passing for this point. The two geodesics we have drawn intersect each other in two points. If we repeat the procedure for all points on earth we can find that given a line and a point not on it, there is no line passing for the point and parallel to the first line. The fifth postulate is wrong.

The hypothesis of the falsity of the fifth postulate give us another geometry. The elliptical one. In this geometry the first four postulate of Euclide are ture, the fifth isn't. Every theorems and consequences of the first fourth postulates is true again but aren't the theorems or consequences of the fifth postulate. We have a strange and not intuitive geometry but a valid geometry. The elliptic geometry is really relevant in all problems in which the distances over the earth surface are not neglectable respect to the radius such as navigation, orientation, positioning and long distance travelling.

In the end there exists another way in which the fifth postulate can fail. Given a line and a point there exists infinite lines passing from the point and parallel to first line. Also in this case we obtain a different geometry. It is consistent and valid and is called hyperbolic geometry. It was postulated by Lobachevskij. This type of geometry with the add of the time give the Minkowski geometry that is the base fo relativity theory of Einstein and is used to describe the behavior of the Universe over large distances.

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