“MArs”, A WEB PORTAL ON “MATHEMATICS & ART” – A NEW APPROACH TO INTRODUCE MATHEMATICS THROUGH ART
Lorenzo FATIBENE1, Mauro FRANCAVIGLIA 1,2,3, Marcella Giulia LORENZI 3
Silvio MERCADANTE1, Pietro PANTANO 3,4
1 – Department of Mathematics, University of Torino,
Via C. Alberto 10, 10123, Torino, Italy; 2 – INFN, Section of Torino, Iniziativa Specifica NA12, Italy; 3 – ESG (Evolutionary Systems Group), University of Calabria, Via P. Bucci, Cubo 17b, 87037, Arcavacata di Rende CS, Italy; 4 – Department of Mathematics, University of Calabria,
Via P. Bucci, Cubo 30, 87037, Arcavacata di Rende CS, Italy.
Mathematics is the universal language of human abstract thought; it subtly pervades also Art in most of its forms, although sometimes as a hidden component. Art and Mathematics have evolved in parallel, alongwith changes in ways of conceiving, perceiving, experimenting and representing “reality”. This evolution can be paradigmatically divided into four steps. One refers to Classical Art, from the Antiquity to Middle Age (essentially based on the “rigid schemes” of Euclidean Geometry); another to Projective Geometry, i.e. the use of points at infinity as ordinary points, as a way to paint “what the eye sees” (Renaissance Perspective); a third one refers to the drastic changes of vision generated by non-Euclidean Geometry (with the introduction of curvature and higher dimensions as fundamental parts of “reality”); the fourth one is the “modern age”, with “Synthetic Geometry” and the deconstruction of rigid forms (on one side fractals and Topology and, on the other, the most modern and avant-garde forms of Art, often based on the latest achievements of Mathematics). In the Digital Era a skillful use of Web Technologies, Digital Art and Multimediality, offers new challenging ways to introduce Mathematics starting from a careful reading of artworks. We have thus elaborated and experimented an innovative “full teaching and visualization project”, explicitly aimed at attracting people to the beauty of Mathematics, based on the intersection of Mathematics, Computer Science and Digital Art. Relying on new technological possibilities offered to Communication, Visualization, Divulgation and Didactics of Mathematics, the project is articulated in four parts: 1) a formative path, based on a book on “Mathematics and Art” already adopted in PhD Courses and easily transferrable to e-learning platforms; 2) the Web Portal “MArs”, in which the deep interrelationship between Art and Mathematics is used to extract mathematical concepts out of classical and modern pieces of Art; 3) and 4) two scientific projects, “SCIENAR” and “ARTEMA”, respectively aimed at creating scientific scenarios based on mathematical models to produce artistic objects and at stimulating the virtual mobility of artists and cultural players interested to use mathematical models.
1. Communication and Didactics of Mathematics in the Digital Era
As we already discussed in the recent papers [1],[2],[3] the task of Divulgating, Communicating and Visualizing fundamental achievements of Mathematics, by no means a simple one, should be accompanied, in our era of innovative technologies, by efforts to make basic and advanced Mathematics teaching for young generations more attractive. Even if Mathematics is the language of abstract human thought and pervades most products of human experience, still many people consider it as a tedious “Science for Initiates”. Abstraction rather than intuition has been privileged in mathematical teaching of recent times, further contributing to the fall in appeal that Mathematics exerts on students. New tendencies aim, fortunately, to revitalize Mathematics without renouncing to rigour but evoking its more aesthetic and directly intuitive aspects; we have been recently working in these fascinating fields (the role of gestures in mathematical learning [4]; new ways to use Art in mathematical teaching [5],[6]; didactical proposals based on agents and multimedia [7],[8]).
The “Web Portal for Research” ( [9]) launched with the aim of reviewing intriguing national research initiatives, hosts among its “specials” an emblematic discussion about Communication of Science. Says S. Fantoni: “Scientists, nowadays more than before, are facing new challenges in the field of Communication. […] Communicating Science has become an absolute necessity […] The European Union considers the <Dialog between Science and Society> one of the fundamental steps towards the construction of a European space for research. […] The themes related with Communication of Science have therefore to become an essential part in the formation of young reserchers” [10a]. In another part of the “special” P. Greco describes [10b] the “Communicator of Science” as “an intellectual character who might be still unexisting. The new Society of Knowledge needs him more and more to tell herself her own fresh origin and her fast becoming. […] Communicating Science is becoming a more and more diffuse social need. […] We are therefore facing a two-fold necessity. Scientists have to communicate with non-experts, to share with them decisions that have a great impact on their own work. […] The system of Mass Communication is evolving and articulating in a such a way not to require and privilege rigid specializations, but rather requiring and privileging flexibility. […] These consequences define the unalienable characters of the Dwarf who is obliged to climb the shoulders of Giants to tell the story of the Society of Knowledge: an intellectual figure who is not lost in the complexity of Science, of Society and of their growing compenetration; someone who, at the same time, is able to <govern media>, i.e. to master the fastly evolving and diversifying means of Mass Communication”. As we discussed in [1],[2],[3] Mathematics is now facing the need of renewing its basic teaching and of promoting a “task force” of specialists in Communication who can help in promoting the dissemination of its specific culture by means of valid multimedial presentations and projects relying on Web Technologies. These are in fact best suited for achieving new forms of Didactics and Visualization in which high scientific quality is conjugated with enjoyable and “soft” realizations, smoothing out the difficulties usually related with the transmission of abstract scientific messages that are far from the common experience.
Among the recent experiences of ours aimed at using new technologies for Didactical and Communication purposes we first mention the field of Mathematical Physics, where we have recently realized a video and multimedia aimed at divulgating the fundamentals of Einstein’s Special Relativity [11]. Stimulated by the World Year of Physics (WYP) proclaimed by UNESCO [12] in 2005 to celebrate those papers by Einstein that in the “Annus Mirabilis” 1905 revolutioned Physics, a multimedia was realized for the “Pirelli Relativity Challenge” [13] with the aim of visualizing how and why that conceptual revolution eradicated the classical separate immutability of “Space” and “Time”, by replacing them with a new notion of “Spacetime” (see [14],[15],[16] for a description; [7],[8] for a discussion about the use of virtual agents for general educational purposes). Here we want further discuss an innovative approach to Mathematics in which the great impact of new technologies is used to help the understanding of mathematical structures through the “main door” of Art [17]. Scientific Communication and new didactical pathways can profitably use new media for the purpose, provided the “book paradigm” is replaced by a “laboratory paradigm”: in a laboratory one forecasts and investigates “knowledge”, allowing mental models to be confronted with “reality”. Science is born out of this relationship; in a sense Science is this relationship. Following the new pathway traced by UMI in the booklet “mattoncini” [18] we have recently launched an innovative teaching project based on a consistent use of the interrelationships between “Mathematics & Art”, already announced in several occasions (see [19],[20] and our earlier papers on this subject [21],[22]). We quote the following from [9]:“Mathematics is universally recognized as a powerful language to describe the World, to construct models, to calculate and forecast. Because of this it is considered to be a very useful tool, often indispensabile, for a lot of disciplines […] To reach the type of knowledge indicated it is not necessary to insist too much on formalism, since it often obscures the meaning of mathematical objects and puts a brake on the development of modelling abilities. The specific competences […] are better exploited through […] the following abilities: * reading and interpretation of texts; * writing and, more generally, Communication; * organization, storage and retrieving of knowledge […] also by means of informatic devices”. Along these lines we have first realized from 2003 to 2006 a PhD experience, tested on an audience including artists, within a Course on “Psychology of Programming and Artificial Intelligence” [23], accompanied by a text appearing in electronic format [24] on the Web in the Portal “MArs” [25]. The didactical path of [24], that in a sense constitutes the nerve of the structure of “MArs”, goes from Euclidean to Modern Geometry, peeling off the structures that enter our understanding of the sensibile World. Geometry is first introduced as the “art of measurement” and followed by Synthetic Geometry at different levels of abstraction; symmetry plays of course a dominant role in introducing Mathematics through Art. Mathematics and Painting have in fact developed in parallel, as other forms of Art (Architecture or Music) have also done. Mathematics, therefore, is not only an essential tool for Science and Technology but also for Humanities; and also out of Art - and not only from its “scientific applications” - it gained developments. As already stated in [1] Mathematics contributes to our way of conceiving and shaping the World we live in, while Art develops the means to harmonise, describe, represent aesthetically, transcend and transfigure the World of our sensations and perception. Because of this a new way of teaching, communicating and visualizing Mathematics using its interplay with Art should be envisaged. It is commonly accepted that Mathematics and Art share deep contacts, but usually “mathematical beauty” is introduced first at a theoretical level and only later shown to exist in artworks. This old-fashioned practise misses the scope of attracting non-specialists to Mathematics, since it belongs to the “standard frameworks” we have criticized in the Introduction. We strongly suggest instead a “reverse path”, where Art is the central theme out of which the existence of mathematical structures is first recognized and later grasped at a deeper level; Art should be considered as a mean to extract and understand structures, symmetries and broken symmetries out of their appearance within the structure, symmetry or apparently broken symmetry of a single piece of Art (see [26],[27]). There are in fact good aesthetic reasons to expect that a discussion of symmetry breaking might be valuable to the artists. Although some symmetry in an artwork might be appealing, too much is monotonous and uninteresting; indeed, the most symmetrical artwork would be an infinite blank canvas. A musical example of the aesthetic value of a small amount of broken symmetry is the preference for the rhythms of a human drummer over the relentless perfect repetition of a drum machine. In order to obtain such a different and completely new perspective on Mathematics one should then use all the modern tools that new technologies provide us to represent, communicate and visualize Science: Digital Art, Computer Graphics, Virtual and Augmented Reality, Multimedia and Web Technology. With the aim of helping to understand the central role that Mathematics plays in everyday’s life, starting from the emotional and aesthetic side of our perception rather that seeing Mathematics just as the “foundation of Science”. Not only to contribute to a better spreading of Mathematics among non-specialists, but also to promote a broader attention to the beautifulness of Mathematics; and even to help specialists to create new pathways for introducing deep mathematical concepts in an easier, more intuitive, more sensible and more palatable way [1].
3. The Portal “MArs” on “Mathematics and Art” and its Related Structures
Let us recall from our previous papers [1],[3],[5],[6] the historical perspective that – as we said in Section 2 – provides the “motivational skeleton” of “MArs”. Mathematics has developed in parallel with our way of perceiving, describing and representing the sensibile world. Our cultural history shows the closeness of the links between Mathematics – a means for discovering and describing reality – and Art, which aims to express or represent reality. The transition from Euclidean Geometry to the Geometry of Perspective in Renaissance [28], to non-Euclidean Geometry of XIX Century, up to the development of “topological forms” in XX Century, can and should be seen as a counterpart to the static paradigms of Arts and Architecture in the antiquities, to the conception of “beautiful painting” of Pier della Francesca, to the evolution of artistic shapes in the Divisionism, Expressionism and Impressionism, up to the complete destruction of symmetry in the contemporary and avant-garde forms of Art (Cubism, Fractal Painting). The theory of proportions was at the basis of Greek Geometry; in Renaissance the artist was a complete man: painter, sculptor, architect, mathematician and scientist (Piero della Francesca, Dürer, Brunelleschi, Alberti). The idea of an exactly Euclidean world and the need of representing faithfully the 3-dimensional world in two dimensions eventually led to Projective Geometry, while discussions about the “Fifth Postulate” of Euclidean Geometry arouse Hyperbolic Geometry that, in the artistic field, paved the way to Impressionism. Artists in XIX Century begun to represent what the eye actually sees rather than what the eye is “pretended to see” in a fully (but non-real) Euclidean world [29]; the XX Century has finally introducted time as a fourth “sensibile” dimension alongside with height, lenght and depth (in relation with Einstein’s Relativity): motion and curvature seen as a constitutive part of the World, not just embedded into it. Objects are so endowed with a dynamics impossibile to represent in lower dimensions (even if artists like Balla, Boccioni and Duchamp tried to provide pictorial representations of movement). Picasso painted in a “manifold-style” [30] while Dalì, designing a 4- dimensional hypercube opening up in 3-dimensional space, used dimensionality [31]. The new Mathematics of XX Century strongly relies on Curvature, Discreteness, Fractals and Chaos, as well as modern ways of making Art reflect these new ideas (fractals in Architecture [32] or in Pollock and Escher [33],[34]). In a sense modern Physics on one side (Quantum Mechanics) and Computers on the other have obliged us to rethink about the paradigms of a continuous world, giving new impetus to Discrete Mathematics; something that we discussed in [35], where an hypothetical path flows smoothly “from Menone to Heisenberg”.
According to the viewpoint that requires Art to act as a theme to develop a modern Communication, Divulgation and Teaching of Mathematics - at all possibile levels and for all possibile applications - we have projected and are currently developing an innovative “full teaching project”, associated with the Web Portal “MArs” dedicated to “Art & Mathematics” [25]. This basically new project points at allowing a progressive understanding of Mathematics, at progressive levels of deeper mathematical abstraction, in which Art comes, touches the emotions and eventually stimulates the need and the desire to penetrate more intimately into the structures which underly Art itself (often without revealing themselves). Art is therefore seen as a way to approach Mathematics, to enjoy the beautifulness of the structures existing in our representation of the World and eventually to reconstruct the theoretical tools necessary to understand and elaborate their true essence, as well as to stimulate further interdisciplinary applications to other fields (see [19],[20],[21]). The Web Portal “MArs” is a fundamental cornerstone of this project. A central Section of it aims in fact to present and propose “true” artistic objects first and use them later as a way to discover their mathematical richness, proposing interdisciplinary links and crossovers within Mathematics and Art. The Portal will of course act also as a container of multimedia, videos and presentations (or other products of Digital Art) that cover a broad range of interactive intersections between Mathematics, Physics and Art (both produced ad hoc for the Portal, or borrowed from elsewhere or finally just linked through Internet under permission). Just to quote a few examples, it will contain: i) a Section explicitly devoted to multimedia that accompany the contents of [24] and the related PhD Course the book was based on; ii) a Section devoted to present ESG products in fields of Generative Art (see [36]); iii) a Section devoted to present audiovisual products of the project “Più Veloce della Luce” (i.e., “Faster Than Light”) within the Law 6/2000 initiatives for the diffusion of Science (e.g., the multimedia mentioned above [11] and other didactical material for Mathematical Physics); iv) innovative didactical products for mathematical education at elementary levels (e.g., the movies quoted in [7],[8]); v) links to projects tending to emphasize the role of Mandalas in Mathematics and Art (see [37],[38]); vi) reports on Petry’s “Superstring Installations” where Mathematics, Physics and Generative Art find a stimulating intersection ([39] and ref.s quoted therein). The Portal “MArs” proposes also a selection of links to other WebSites (important for their contents and their complementary role with respect to “MArs”) as, e.g., the WebSites “Matematita” [40], “Connections in Space” [41] or “Atractor” [42] (many other are in fact present on “MArs”). Moreover, it envisages the presence of “more conventional” spaces, devoted to provide information about initiatives in the field of Art & Mathematics (Conferences, Exhibitions, Courses). The Portal intends also – in its future developments - to exhibit a collection of multimedia and software explicitly produced for the Visualization and the Didactics of Mathematics. In the future - also through two related projects described hereafter and presented in [3] - the Portal “MArs” will envisage an Electronic NewsLetter (in which both innovative and more traditional ways of communicating Mathematics through Art will find an appropriate forum), where Visual and Digital Art, Music, Architecture, Biology, Visualization and e-Learning will be seen in their interaction. The Portal will finally contain a less conventional space dedicated to “work in progress”: presentations of multimedia and/or innovative ideas in the field of Art and Mathematics; simulations and generative approaches to “Mathematical Art”; digital technologies to produce Art through Mathematics or to understand Mathematics through Art; new frontiers in Mathematics stimulated by Digital Art, Artificial Life and Virtual Reality. To summarize, the final task of “MArs” is four-folded. The Portal intends to perform the following actions: 1) to broadly cover a wide range of interests at the intersection of Art and Mathematics; 2) to open up a real dialectic between Mathematics and Humanities; 3) to stimulate interdisciplinary attention to innovative products that exulate the traditional competence of single frameworks and, as such, cannot be easily classified under traditional disciplinary classifications; 4) to give adequate space to new tendencies and new problems generated by Web-technology. This four-folded task shall be exploited by these explicit aims: A) to give a preferential attention to the most stimulating, innovative and interdisciplinary products (larger space devoted to critical discussions about the many relations existing between Art, Architecture, Mathematics and Humanities); B) to create an interactive dialogue between different “poles” of our present Culture, a strict actuality made possibile by Web-technologies; C) to strongly re-propose new ways of interaction between Art, Science and Technology, on the basis of the key role that “open minded” mathematicians can exert alongwith the role of “mathematically inclined” artists; D) to try to eventually re-create that fruitful “constructive and internal interaction” that existed between Mathematics and Art in Renaissance or in the Dutch painting of XVII Century; a liaison between the two disciplines that, as said in [1], has been interrupted and replaced by “external” interactions, only at a marginal, secondary and less promising level of efficiency.