Claude Bruter, Dmitri Kozlov
European Society for Mathematics and the Art (ESMA)
In the middle of the XVIII century, Leonhard Euler said that mathematics is “the basis and the key-stone of all human knowledge”. Today these words sound more relevant than ever. The foundations of the natural sciences such as physics, chemistry, and biology, formulate in the language of mathematics and are not accessible without knowledge of this language.
Economic success in the modern world depends on the development of applied sciences, and their development, in turn, depends on the level of the fundamental science, the language of which is mathematics. It is obvious that in the near future, only those countries will succeed in economics that will be able to make the language of mathematics understandable to the majority of their citizens.
There is no doubt that mathematics forms intellect and stimulates the capacity for logical thinking and synthetic perception. Mathematics awakens the imagination, develops power of observation, and trains attention. Due to its universality, mathematics is a true means of cognition of the world, used both in fundamental sciences and in their technical applications.
Today the majority of economically developed countries that want to remain so in the future try to evoke interest in mathematics among the young. Many cities of the world have museums of science in which children form their ideas about various scientific fields, playing with interactive exhibits, many of which based on mathematical principles. The world’s first museum dedicated to mathematics opened in New York City at the end of 2012, and since then it has always enjoyed great success with the public.
Nevertheless, a museum, even if it is equipped with the most advanced interactive exhibits, has a number of significant limitations in its capabilities to influence people. The very concept of museum as a form of popularization of mathematics addresses only to the intellectual side of the human being. The tasks of any museum are mainly informational and cognitive, but a human being cannot be reduced only to an intellectual function.
Since ancient times, three components of the human being: physical, emotional and intellectual were distinguished, manifesting respectively as body, soul and spirit. People learn the world around them with their whole beings, in harmony of all three of these components. One or another form of cognitive activity usually requires the predominance of one of these components, but this does not mean the complete exclusion of the other two. Their mission consists in supporting the main acting component, in maintaining its connection with the integral human essence.
In mathematics, as in other fields of science, the path to intellectual knowledge lies through feelings and emotional experiences. A person before studying anything seriously should become interested in this, form a positive internal attitude. In turn, feelings and emotions closely relate to physical sensations, to the general physical state of the human body, and its readiness for activity.
Unfortunately, many people on their way to mathematics sometimes face obstacles that can be psychological, emotional and intellectual. They can also arise because of imperfections in education, which often leads to incorrectly laid foundations in this area of knowledge. They adversely affect the quality of future work and leads to a significant reduction in highly qualified specialists and teachers in this discipline.
It is necessary to recognize that true culture is universal in its nature: it is not only artistic or literary, but it is also scientific. Increasing the intellectual level of society can only be true if it is also accompanied by an increase in its cultural level, which in turn initiates the development of intellect. Since mathematics occupies a central place among other sciences, it is necessary to give access to this form of culture for all people.
All these considerations formed the basis for the project of a special landscape and architectural complex, called the “Mathematical Park” [1]. The aim of the project is to visualize fundamental mathematical ideas by means of architecture, sculpture, landscape and decorative art, making them clear and accessible to the public.
The project includes a system of small architectural structures namely park pavilions or follies, each of which represents a section or a field of mathematics. External forms, interiors and filling of the pavilions serve to allow visitors of the Park to see and feel the images of the basic ideas of mathematics. The layout of the Park, the terrain landscape, and the nature of the surroundings reflect the connection of individual sections of mathematics with each other. Park sculpture and small architectural forms emphasize the purpose of the main pavilions and establish their connection with the landscape.
The main purpose of the Mathematical Park is to overcome the psychological barriers that often arise in some people on their way to mathematics. This goal can be achieved by means of the aesthetic qualities of many mathematical objects that evoke in every viewer a sense of beauty. Just a simple contemplation of this beauty can arouse in the audience feelings of admiration, surprise and curiosity, which, in turn, should encourage them to make their own intellectual efforts and try to understand the mathematical significance of what they saw. After that, they will be able to try to go further, getting deeper into the knowledge of mathematical ideas, which stimulate their interest and positive feelings.
No special knowledge is required to feel the aesthetics of the forms of the pavilions, sculptures and landscape of the Mathematical Park, to enjoy the prospects suddenly opening up among the green spaces. The perception of them is as natural as contemplating any other architectural structures, sculptures or landscapes. As a result, the public of all ages will be able to enter into an almost tangible contact with fundamental mathematical principles, feel and understand their meanings and as a result to penetrate into the magical world of classical and modern mathematics without any additional effort.
The Mathematical Park is a system of interconnected architectural and landscape designs organically integrated into a single ensemble. The main concept of the project is to express the basic mathematical ideas in the form of memorable architectural, sculptural and landscape images in the natural environment. The mission of the ensemble consists in creating a general imaginative representation of mathematics for visitors. In the process of their acquaintance with the pavilions dedicated to individual sections of mathematics, this general image will gradually specify.
Functional zoning of the Mathematical Park includes zones typical for any park (zones of entrance, parking, trade, and food, administrative zone, zone of rental means of transportation in the park, etc.), as well as special thematic ones, which include: sightseeing , exhibition, cognitive-entertaining and information-communication zones [2]. The general layout of the Park subordinates to this zoning, and the designs of all zones reflect the mathematical principles.
Functional purpose of the excursion zone is to conduct scientific and educational activities in order to popularize mathematics. It is supposed to organize the reception of visitors, direct thematic excursions, and organize scientific and practical seminars and conferences. This component of the project includes thematic park pavilions and their internal filling, park sculpture, small architectural forms and landscape of the Park territory. The purpose of these objects is to express visually the ideas of mathematics with the help of various kinds of arts such as architecture, landscape design, sculpture, painting, music, and through modern technologies such as computer graphics and animation, laser and holographic projections, transformable structures, etc.
The exhibition area of the Mathematical Park accommodates permanent and replaceable expositions of artworks devoted to mathematics, as well as illustrative examples of the application of mathematics in science and technology. The task of this part of the project is to demonstrate the connection of mathematics with nature and with the life of each person. The exhibition opportunities of the Mathematical Park can also be of interest to educational, research and production organizations whose activities depend on mathematics.
The cognitive-entertaining zone is a zone of experimental activity intended for games, creative knowledge and training of the Park visitors. It hosts interactive exhibits intended primarily for the gaming activity for children with the purpose to form in their consciousness basic mathematical concepts. The widest use of mathematically based computer games, toy construction sets, transformers and puzzles proposes to install in this zone of the Park. Also in this zone, it is supposed to create special cognitive-entertaining laboratories and creative groups for practicing arts related to mathematics.
The information and communication zone of the Mathematical Park is intended to spread information about the purposes of the Project and the scientific content of the park pavilions, as well as for activities in the virtual space. Unlike the other three zones, this zone distributes itself throughout the Park. In particular, printed information products will be available to visitors directly in the pavilions of the Park to let them find out which mathematical ideas and facts are associated with each of the pavilions. Thanks to this information, the people will be able after their visit to the Park to refresh all that they have seen and share the impressions with their friends. In order to fix the results of the impact on the spectators of the pavilions and landscapes of the Park this zone provides visitors with special series of films and computer animations. In addition, the functional task of the distributed information and communication zone of the Park consists in providing access to the interfaces of the Internet portal of the Mathematical Park and all information resources of the International Network of Mathematical Parks.
The “Mathematical Park” project includes at least ten pavilions described below in the order of the historical development of mathematics [3]. The names and architectural forms of the pavilions create vivid and memorable images of mathematical facts in order to make them visible and accessible to the public. The sizes of the pavilions can vary depending on the total size of the Park, the nature of its landscape and the buildings adjacent to it.
1. “The Apollonius Headdress” – a pavilion dedicated to classical geometry from Euclid to Descartes, the generalization of which is projective geometry. The architectural form of the pavilion represents the cone geometry. Its outer surface serves as a screen on which moving light rays project the shapes of quadric curves and visually demonstrate geometric optical phenomena associated with them.
Figure 2. Pavilion “The Apollonius Headdress”. General view.
Figure 3. Pavilion “The Apollonius Headdress”. Angle view.
2. “The Horn of Plenty” – a pavilion dedicated to classical analysis, which deals with the concept of limit and ideas of approximation and convergence and describes the processes of growth and evolution. The external form of the pavilion expresses the concept of approximation by the example of a continuous function approximated by a step function.
The concept of convergence presents in the interior of this building designed on the rule of the golden section, which characterizes the recurrent sequence of Fibonacci numbers. Images and projections on the walls of the pavilion demonstrate the recurrent properties of fractal shapes and certain limit curves. Computer programs and models illustrate the convergence phenomena of different mathematical objects and sequences.
3. “The House of Numbers” – a pavilion dedicated to number theory. The shape of this building reflects the properties of some well-known numbers, such as e and π. The exterior part of the building is in the rays of bright light, while its interior stays in darkness, symbolizing the fact that the world of numbers is still full of mysteries. The walls of the building provide screen surfaces for special computer projections to generate different versions of number representations, for example, in the form of continued fractions, or examples of their use, in particular, for encoding messages.
4. “The Gauss Observatory” – a pavilion dedicated to differential geometry, and its form expresses the idea of curvature, both geometric objects and the space itself. The base of the building has the shape of minimal surface and supports a cylindrical tower with a spiral staircase in the form of a helicoid inside of it. The tower covered with a spherical dome that has partially transparent surface. A flat, movable screen, tangential to the surface of the dome, covers the transparent part of the dome. A moving beam of light emitted from a hidden source inside of the dome and reflected by mirrors of a special shape draws quadric curves on the screen. The opaque part of the dome provides for demonstration of the Gaussian curvature concept. In the interior of the building, special computer animations and projections visually represent the laws of spherical geometry, the properties of soap films and minimal surfaces.
5. “The Whitney Umbrella” – a pavilion dedicated to algebraic geometry and complex analysis, which have many connections between them. The form of the pavilion is the Whitney umbrella – a geometric surface, the fact of existence of which is associated with four fundamental mathematical concepts: singularities, stability, bifurcations and bundles.
6. “The Euler Bridges” are not a pavilion, but a landscape composition dedicated to the ideas of topology, which emerges from the study of the properties of figures independent of their metric properties. In 1736, Leonhard Euler proved that it is impossible to walk along the seven bridges of the city of Konigsberg only once and return to the original point. This proof illustrates one of the foundations of topology – the concept of connectivity. A small lake with an island in the middle and seven bridges thrown across it carefully inscribed into landscape of the Mathematical Park for a visual representation of the Konigsberg problem. On the island, a group of sculptures with the forms of the main topological surfaces illustrates the concept of the Euler characteristic.
Figure 4. Landscape composition “L. Euler Bridges”. Top view.
7. “The Seventh Temple” – a pavilion dedicated to algebra and group theory, which plays a very important role, both in mathematics and in physics. It visually represents the concept of group by means of space filling with polygons and polyhedra. Polyhedra locate on the hyperbolic inner surface of the convex covering of the building and on the upper frieze of the outer walls. Convex and star-shaped polyhedra locate outside the building on the playground for young visitors of the Park. Both of them revolve around one of their axes of symmetry and represent groups of motion of polyhedra. Decor of this building consists of flat ornaments on the walls and friezes representing the fundamental symmetries and their organization in groups. In the interior of the building, a number of computer animations and projections visualize the actions of group theory.
Figure 5. Pavillion “Symmetry Temple”.
8. “The Knotted Stained Glass” – a pavilion dedicated to differential topology. The form of this building is a one-sided closed surface known as the Boy surface, which is a real model of the projective plane [4]. The entire surface of the pavilion is completely made of colored and transparent stained glass. In the architecture of this building, there are many examples of structures with knotted and interlaced forms found in living nature. Computer animations projected onto the glass surface of the building, illustrate how to construct some of these forms. Special models of transformed knotted structures demonstrate other topological operations [5].
Figure 6. Pavilion “The Knotted Stained Glass”.
9. “The Poincaré Surprises” – a pavilion designed to represent the ideas of movement and change, which played an important role in the formation of modern mathematics. The inner part of the building and the content of projections and animations illustrate these ideas.
10. “The Luminous Torus” – a pavilion dedicated to applications of mathematics in the field of optics. This building is in the form of a torus, made partly of glass. Visitors inside the torus can observe the effects produced by light rays directed at transparent objects with different refractive indices.
Figure 7. Pavilion “The Luminous Torus”.
In addition to the ten main pavilions, the Project also includes two auxiliary buildings, which forms and imagines are designed to support and supplement the general architectural and landscape composition of the Mathematical Park.
1. “The Bourbaki Amphitheater” is a meeting room in the form of an amphitheater with a transforming dome cover, which allows to use it as an open or closed space depending on the season and time of day. This building intends for theater and music performances, exhibitions, conferences and seminars.
2. “The Unfinished Building of Mathematics” is the administrative and informational center of the Mathematical Park. The architecture of this building combines various shapes and characteristics of known geometric and numerical constructions, symbolically reflecting the idea of incompleteness and potential growth of the majestic building of mathematics itself. The administration office and the rooms of the Park’s staff locate in one part of this building, and the hall serves as an exhibition space for changeable expositions. In the basement of the building is a library dedicated to the interrelations between mathematics and art. This building is also a place for a united computer center of the Park that manages the work of the computers of all its pavilions and the functioning of the internal and external networks of the Project.
In addition to the main and auxiliary pavilions, the Project provides for various small architectural forms on the territory of the Park, specially designed to express certain mathematical principles and harmoniously combine with the neighboring pavilions and landscape elements.
The park can also become a place for temporary exhibitions of sculptural works depicting mathematical objects, as well as a place for creating a variety of land art installations, reflecting mathematical ideas.
The planning and spatial organization of the Park reflects the interconnection of individual sections of mathematics. The location of the main pavilions, the network of roads and paths between them, and the landscape of the surrounding spaces emphasize this idea. Landscape design of the territory of the Park consists of garden and park compositions combined thematically with the pavilions.
The compositional links between the elements of Park are not only aesthetic and allegorical, but also symbolize the logical relations between them. If the sections of mathematics to which the pavilions are dedicated are closely related, the pavilions place in a visual connection between each other. If the pavilions belong to quite different fields of mathematics or the connection between the corresponding sections of mathematics is not so obvious, the space between them opens gradually through a series of intermediate landscape objects.
The similarity of natural forms and mathematical laws plays an important role in the landscape ensemble of the Park. One of the aims of the Project is to demonstrate visually to the visitors the possibilities of describing natural phenomena with the help of mathematics.
“The Projective Park” arranges near “the Apollonius Headdress” pavilion. The forms of its layout express the principles of projective geometry and linear perspective.
“The Phyllotaxis Glade” surrounds “the Horn of Plenty” pavilion. Its plan repeats the characteristic drawing of two families of spirals twisted towards each other like a basket of sunflowers. The visitors of the park can visually comprehend the strict mathematical patterns underlying the formation of the living nature by walking along the paths-spirals.
Islets of grass and flowers named “The Euler Flowerbeds” surrounds “the Euler Bridges”. The paths of various shapes and lengths line between the islets forming intersections with different number of branches. The flowerbeds and paths form respectively the faces and edges of a flat graph and the crossroads form its vertices. The visitors of the park can practically verify that the Euler characteristic of a flat graph does not depend on the shape and number of its elements.
Figure 8. Topologocal shapes alley.
“The Garden of Symmetry” joins to “the Seventh Temple” In the figures of its regular plan the visitors can discover all 17 possible types of flat ornaments depicted on 17 facets of the pavilion.
“The Knotted Forest” is a natural extension of “the Knotted Stained Glass” pavilion. In this forest, the curved and intertwined trunks and branches of special trees serve as illustrations of knot theory, the intricate and rapidly growing new field of mathematics.
In general, the landscape and spatial organization of the Park follows scenario principle. The individual scenarios of walks in the park reflect different degrees and speeds of immersion in the world of mathematics for visitors with different mathematical backgrounds.
The Project proposes to build a network of mathematical parks in different countries of the world. Each of them will reflect national specifics of architecture and landscape as well as local historical features of evolution of mathematics. Scientific and artistic conceptions of the pavilions and their composition will highlight local traditions and modern achievements in mathematics and in the fields of its practical applications. These Parks will not copy each other, but at the same time, they will have much in common because of the fundamental unity of mathematical science. Each of them will form one of the vertices of the global network of mathematical parks.
In parallel with building of the global network of mathematical parks in the real world, the Project also aims to create a network of virtual mathematical parks in the Internet. This network will be a system of interactive Web Portals that allows the user to make a virtual journey through any of the real mathematical parks and visit any of their pavilions, both already functioning and still being under construction or just in project.
Schools, universities, libraries and scientific organizations of the whole world will be able to connect to the Web Portals of the Parks. Each of these organizations can create its own virtual mathematical room – a local portal, directly connected with the global network of mathematical parks. As a result, the latest world achievements in the field of popularization of mathematics will become available for all educational institutions connected to the Global Network.
Comprehensive courses of lectures on various sections of mathematics, on-line seminars, master classes of the world’s leading specialists, etc. will be available on the Web Portals. The informational resources of the network will constantly update and improve by the collective efforts of their creators and users. The resources of the network of virtual mathematical parks at their basic level should be freely available to any Internet user.
Users of the Web Portals will be able to communicate with professional mathematicians and share their own achievements in mathematics for discussion with the broad mathematical community. To achieve this aim, modern mathematical and software methods of image processing and data transmission will be used. As a result, a virtual network of mathematical parks can become the center of concentration of mathematical knowledge of all humanity and an intellectual resource of a planetary scale. To implement this concept, the Project envisages utilization of existing Internet channels and creation of new, just emerging data transmission systems and the next generation Internet, the fragments of which either exist or are under investigation.
Figure 9. Global network of Mathematical Parks.
The global network resource in the form of a unified system of virtual mathematical parks, accessible from any point of the Earth will function as a communication tool for the people of the whole world who are interested in the ideas of popularizing mathematics. In addition, this system will serve as the means of collective creative activity organized on the principle of feedback, resulting in the formation of a unique information base, necessary for the further development of the concept of the Mathematical Park.
Claude Bruter, one of the authors of this article, had formulated the concept of the Mathematical Park as far back as in September 1989 at a symposium in Leeds, UK, devoted to popularization of mathematics. At this event D. Blane, professor from the University of Monash (Australia), during his presentation told that one day while walking around the city with his students, he noticed that the round platform was paved with small stone tiles, and suggested that students use this fact to determine the approximate value of the number π.
This story suggested to C. Bruter an idea of a park designed for walks and mathematical activity, the concept of which he then sketched in basic terms. Lately C. Bruter talked about his project to other participants of the symposium, in particular, the famous French mathematician J.P. Kahane, who supported his intention to promote this idea.
Shortly thereafter, the Association for the Implementation and Management of the Mathematical Park for Walking and Mathematical Activity, ARPAM (Association pour la Réalisation et la Gestion d’un Parc de Promenade et d’Activités Mathématiques) was established under the chairmanship of C. Bruter [6]. In essence, the ARPAM project refers to a long tradition, according to which scientific, in particular mathematical, activity took place during walks in a beautiful natural environment. It is well known that Plato (428/7‑348/7 BC) talked with his followers in the gardens of the Academy (Ἀκαδημία). Plato’s pupil Aristotle (384‑322 BC) preferred to expound his teaching while walking in the countryside of the Lyceum, and his school was called Peripatetics, from the ancient Greek word περιπατητικός, which means “walking around”.
The first Administrative Council of the Association included French mathematicians as L. Schwartz, laureate of the Fields Award, and mathematicians from other countries known for their activities focused on the popularization of mathematics and mathematical education, as the well-known mathematician Ian Stewart.
In July of 2010, the founding conference of the European Society of Mathematics and Art (ESMA) took place at the Institute Henri Poincaré in Paris and elected C. Bruter as its president. Since that time, one of the main activities of ESMA was the project of the Mathematical Park, which requires the combined efforts of different specialists in the field of science and art [7].
To implement the best successfully the artistic and educational tasks of the Project, a large area would be required. This will allow visitors during their walks in the Park to rest from their previous impressions before immersion into new ones. Therefore, it is very important that the territory of the Park would have expressive natural features, such as the rough terrain, picturesque water formations, small woodlands, meadows, etc.
At the same time, it is obvious that in the real conditions it is not always possible to find a suitable site with such a large area and with such natural qualities. Because of this situation, compact variants of the Park having a reduced set of pavilions may be proposed for a smaller area taking into account the wishes of the local scientific community. Even “mini-parks” could also be organically included in the existing ensembles of city parks and recreational areas of large cities.