Presentation of the Colloquium. The ARPAM Project

My intervention has two parts. The first one is devoted to a general presentation of the Colloquium, through an evocation of the works of the artists who are present among us. Thus that presentation does not address mathematicians in a particular way. It reveals some of the reasons which have directed the scientific organisation of the Colloquium. As its architecture shows, it turns over the art of visualisation of mathematics, either for the general public, or for the one of mathematicians. The second part is devoted to a succinct description of the ARPAM project.

I. Presentation of the Colloquium

I.1 On the foundations of the relations between Mathematics and Arts

As a preliminary comment, it is fitting to say a few words on the relations which tie Mathematics and the Arts : they are so tight that sometimes Mathematics is compared with one of the Fine Arts.

One of the reasons, the main one to my eyes, which solders the arts to mathematics is probably the following : the tangible object, the living being, are not only present in space, and are evolving in space, but are moreover highly elaborated constructions, obtained from the unfolding of the properties of the primordial space.

In other respects, the existence of the object, that is its inward properties of stability, are themselves dependent on the stability of its constituents, of their internal arrangement according the various levels of integration. This existence also depends upon the capabilities of the object to resist against shocks of any kind, of internal or external origin, created by all that makes its environment, close or distant, into space and time. Thus, knowledge of this environment, in all its modalities, is the essential means whereby the being can guarantee its spatio-temporal stability. So we are always brought back to the fundamental problem of the knowledge of the space, of all the richness of its manifestations.

Indeed, mathematicians as much as artists are preoccupied with deepening this knowledge. They succeed by using representations, primarily abstract and scriptural for the mathematicians, more physical for the artists. As both of them are sometimes representing the same objects, one cannot but wonder about the common points of these processes of representation, and that each borrows subjects of representation, resources, discoveries from the other, in such a way that art and mathematics progress together by mutual enrichment.

The recent films by Michele Emmer, one of the pioneers in the study of the connection between mathematics and art, put in the limelight the discreet but significant role of the development of mathematics on contemporary art. The weight of this influence will also be revealed by an inspection of some of the themes on which our artist friends have been working.

Regarding the plastic arts, six topics will be introduced during this Colloquium : the first, titled " perspective and geometry", starts with the artistic technique ; the three following, "polyhedra", "curves", "surfaces", concern mathematical objects of stiff appearance, very classical, which hold an important position in the mathematical universe ; the fifth theme, recurrence and dynamical systems, is more recent, and the development of the computer has given it an important boost ; the sixth and last theme, the "sphere eversion", has likewise the interest of novelty, not only from the mathematical point per se, but also from a pedagogical point of view. To this end, a video was made both for mathematicians and also to popularize mathematics, a consequential video by its cost and by the size of the team which has produced it. Finally, to conclude this first part, homage will be paid to those who make us share joys among the purest, those of music.

I.2 First theme : perspective and geometry

The choice of this theme is partly directed by historical reasons. For at least 32 000 years, when they used to ornament the walls of the caves, the artists have painted on plane or curved surfaces, using sometimes the rules of perspective in a spontaneous manner.

We are indebted to the painters for having founded a rational theory of perspective. According to the roman architect Vitruve, the painter Agatharchus, from the creation of sceneries for the Eschyleus theatre, would have been the forerunner of the theory. Anaxagoras and Democritus would have begun to develop it, but all their works are lost. The Renaissance artists, like for instance Brunelleschi around 1415, in their turn, brought into focus the first known rudiments of this theory for the practice of their art. These elements lead the architect from Lyon, Gérard Desargues, around 1639, to base projective geometry : that is a classical example of the phenomenon of symbiosis between art and mathematics.

Projective geometry holds an important position in mathematics because its interest is much more general than the one of classical Euclidean geometry. Indeed, in Euclidean geometry, the bright source which illuminates the objects is located at infinity. In projective geometry, the bright source lies at any point of space, at a finite distance or not : thus, in this respect, projective geometry contains classical Euclidean geometry.

Many painters are working on canvas which, from the point of view of mathematical representation, are understood as pieces of plane surfaces. The point of their canvas towards which the eyes of the observers seem to converge is called the vanishing point. It plays an important role in the construction of the pictures.

From the mathematical point of view, a surface is an ideal, infinitely thin skin. The diversity of the surfaces is infinite. We are going to stick to perfectly smooth surfaces, without any asperity, as for instance the plane surfaces or the spheres. A plane surface is a very singular surface characterized by the fact that its curvature is null at any point. Apparently, a few things distinguish the plane from the sphere : an important difference (which implies others) lies into the value of the curvature, which is constant at any point in both case, but null in the case of the plane, and non null in the case of the sphere.

The curvature is a local data : the one at a point of a more or less elastic thread is tied to the effect of the internal tensions, to the ability to resist to a stretch at that point. If no resisting force is present, the thread seems to be able to strech out indefinitely, there is no natural curvature, the physical and mathematical curvatures are null. Let us take now an elastic and smooth surface, as the canvas is a little bit. It is a kind of fabric whose stitches are infinitely fine and close. At each point, two elastic and perpendicular threads cross, each one having a local curvature at the point.

From these data, one defines two notions of curvature, first the Gaussian curvature at the point under consideration, which is the product of the local curvatures of each of the two threads crossing over the point. This notion of curvature allows us to classify the smooth surfaces within three categories : the spherical or elliptic ones with a positive curvature, the hyperbolic ones with a negative curvature, the parabolic ones with a null curvature. Among these, are the plane surfaces whose local curvature is null in all the directions. Indeed, when the local curvatures of the threads are for instance positive, their product which is the Gaussian curvature is also positive, as it happens in the case of the sphere. When the local curvature of a thread is positive while the local curvature of the other thread is negative, the Gaussian curvature which is their product is negative, as in the case of the surfaces of some water-towers, and which are generated by the rotation of a piece of hyperbola around one of its symmetry axis.

Let us come back to painting. We are accustomed to look at paintings which mainly are painted on plane surfaces. But why to stick at that ? Would not it be possible to paint on a spherical or hyperbolic surface ? But then what could be the reasons which would lead a painter to put forth his genius on such or such type of surface ? There are the natural data indeed : the painter of caves will practise on the spherical shape of a stone, on those, plane, spherical or hyperbolic of the walls of his cave. But there might be other reasons, as those that the painter Dick Termes will show in detail.

His wish was to represent the whole space, not only what meets our eyes, before us, but also what is on our sides, at the right, at the left, above us, underneath, and behind. He has then taken six different canvas, stretched over a cube, with which he can represent all that space. By blowing inside the cube, not too much strongly in order not to burst the canvas, the cube becomes a sphere, that the topologists like to cover with six curved disks, equivalent to the six faces of the cube.

On each canvas, Dick Termes chooses a vanishing point and represents the part of space that faces it. He will explain us how he chooses his vanishing points in such a way that the partial images join together harmoniously. He proceeds quite as the geometers who construct local representations, then, by using analytical techniques, fit them together to obtain coherent wholes. This point of view is the one of the theorist. Dick Termes 's work is interesting, not only because of his remarkable artistic qualities, but also because he solves a concrete problem of reconciling of images. The artist enriches the corpus of problems brought up to the mathematician, suggesting to undertake a fine study of the junctures between local geometries by changes of the vanishing points on the sphere, and more generally on smooth surfaces of any curvature. One can elsewhere set up the question : given a representation on a sphere, of which types of spaces is it the image ?

I.3 Second theme : polyhedra

Here is an other example of interaction between art and mathematics. We have just met, for the needs of the complete representation of the usual space, a first polyhedron, the cube. Let us notice that the cube has this marvellous properties to be able to be easily mass-produced, and that the stacking of infinitely many cubes in all the directions allows us to fill up the space. This property makes the felicity of one of my friends. One should maybe find him as ill-seeing, if not a little bit foolish, but for him, all the objects of Nature have the shape of a cube. He is so very happy, because he is one of the rare people to be able to answer one the fundamental questions, how does Nature fill up the space ?

Mathematicians have found other polyhedra which, joined, permit to fill up the space. One for instance will quote this result by Poincaré used in the film Not Knot according to which one can make a tessellation of an hyperbolic space, i.e. with negative curvature, using hyperbolic dodecahedra, which are polyhedra with 12 curved faces. Many mathematicians and artists are are captivated by polyhedra. Their study is the starting-point of an important part of the works by George Hart and Charles Perry. Starting with known polyhedra, they proceed to learnedly controlled deformations to get objects which are full of power, of dynamism, and of novelty. George Hart among other things deals with partial but regular simplicial subdivisions on the edges of nested polyhedra. He thus implicitly makes new local groups of symmetry, and contributes to enlarge the theory of the 230 classical crystallographic groups by introducing, over that basis, kinds of algebraic fibers. The proceeding can be widely generalised, leading on the limits to fractal structures, first passing through combinatorial structures of a giddy extent.

Let us suppose that the space is regularly tiled so that all the tiles have the same shape, the same dimension. Let us cut our space by a surface, for instance a plane surface. What is the trace of this tiling on the plane : a regular tiling ? That may happen. Antonio Costa will show us the famous tilings that the Arabian artists have produced in the Spanish city of Grenada, on the walls of the Alhambra palace. Five centuries before us, they almost discovered the fact there exists 17 different really distinct types to tile a plane, each different type of tiling being characterized by a particular family of internal symmetries. We are in presence of motives which, on the plane or in the space, are infinitely repeated. By using a learned play of mirrors constituting a pedagogical tool of great interest, Maria Dedo will explain how she makes these motives discover as well as the set of symmetries which characterise the polyhedra. Such a play of mirrors is used in the film Not Knot, the Poincaré's tessellation of the hyperbolic space appears.

Whereas George Hart enriches the motives of the spatial tilings by the use of methods belonging to static mathematics, Michael Field beautifies the motives of the plane in a rich and elegant manner, by calling upon techniques which are used in dynamics. Indeed, the study of dynamical systems shows us this remarkable phenomenon, the birth of new morphologies at singular moments, for singular values of the parameters. These creative bifurcations can put in light hidden internal symmetries, new shapes of trajectories. The artist and mathematician work these phenomena to create new remarkable ornamental pictures.

I.4 Third theme : knot shaped curves

The curves, the trajectories, the threads without thickness which close on themselves are called knots of topological dimension 1. Their diversity, their interweaving, their infinite variations of shapes immerse the mind into reverie, or on the contrary fix it on perfection. They have inspired among the most impressive works of the sculptors Nat Friedman, Charles Perry and John Robinson.

Knots play an important role in physics and in mathematics. Ronald Brown and Nathaniel Friedmann will show all their artistic and pedagogical value. As one will see in the film Not Knot, there exists very tight connections between polyhedra and knots. Let us take a knot : prick it in some points we call vertices, then stretch the portion of curve located between any couple of vertices up to get a rectilinear segment called an edge ; we have set up a sequence of edges which are closed on itself which has the same topological properties as the initial knot. The skeleton of dimension 1 of a polyhedron made with its edges is then an combination of knots having common parts, and that elsewhere we can separate one from each other in many ways.

I.5 Forth theme : the surfaces

The theory of knots belongs to topology, i.e. to the study of the properties of the space, independently from considerations of distance, the physical meaning of the distance being that of the energetic cost of the transfer from a point to an other. Topology becomes geometry when these supplementary metrical considerations are taken into account in the study of space. Three talks will show us very various shapes of geometrical surfaces, conceived from sometimes very different motivations.

On a technical point of view, Konrad Polthiers' talk addresses means of studies and of representation of minimal surfaces. From the mathematical point of view, they are defined from a notion of local curvature that is different from the Gaussian curvature. The Gaussian curvature at a point is the product of the curvatures of the perpendicular threads that cross at that point. The other curvature, called the mean curvature is simply the half sum of the local curvatures of the previous curves. When the mean curvature of a surface is everywhere null, one says that the surface is minimal because then the value of some energy tied with the surface and depending of the local curvature of the threads is minimal. A noteworthy case is that of soap bubbles, which were deeply studied by the Belgium physicist Plateau in the 18th century. Konrad will project films he did on such surfaces, which sometimes have inspired architects to make important roofings.

The two last talks of this Colloquium will be devoted to other geometrical surfaces. Bruce Hunt will guide us along a rich and beautiful gallery devoted to algebraic surfaces which are not necessarily smooth. The video by the sculptor Helaman Ferguson will show some of them, modelled in stone or in metal.

The ruled sculptures by Philippe Charbonneau, generated by moves of lines, are inspired from some of these surfaces of order 3. Some of theses sculptures have the property to be movable around privileged axis, and introduce symmetries with respect cylindrical or conic elements, which make us think of those one can sometimes see in the gears. François Apéry proposes a generalisation of that constructing process of ruled surfaces by replacing lines by conics. Through the creation of these elements of sometimes branched and with various symmetries algebraic varieties, these sculptures are maybe the indication of an enrichment of the mathematical museum.

Richard Palais is trying to set forth such a museum. He will show us some of the precious objects it contains. In fact Dick Palais' project goes far beyond of a simple virtual museum. In fact, it will be used as an intelligent interactive library of a new style, that will be a very useful tool to inform and to discover, in as much for the mathematician as for the neophyte. I hope you will be charmed by discovering this project, of a high elevation in its conception, to the realisation of which we shall be doubtless numerous to contribute.

I.7 Sixth theme : recurrence and dynamical systems

The study of recurrence and of discrete dynamics has more especially led to the discovery of the fractal world. This one has inspired the realisation of a fascinating video, The Mandelbloom, a marvellous ballet of roses, dancing on a choreography suggested by the events proper to this universe. The fractal phenomenon is tied to the presence of iterative procedures together with scales reductions : one finds it in the methods for solving polynomial equations used by Scott Crass, and in the study of a great number of dynamical systems, such as those studied by Mike Field and John Hubbard. The first development of the fractal theory concerns recurrence equations defined on the space C of Chuquet'numbers, the so-called complex numbers, representing similarities on the plane. Such equations have equally been studied using the Hamilton numbers, the so-called quaternions representing similarities in the usual space. Is it not surprising to find among some sculptures by Charles Perry an anticipation of the representation of some quaternionic Julia sets, discovered more lately, and that we shall find in the Jean-François Colonna virtual exhibition ?

I.8 Last theme : the sphere eversion

Let us imagine Dick Termes located at the centre of a sphere, painting it. Obviously Dick wishes to show his work to the largest possible public. But this public stands outside the sphere. What to do ? Does not the solution simply consist in eversing the sphere ?

In a mail, Stewart Dickson informed me of his present preoccupation, to materialize objects in process of transformation, as for instance the sphere during the steps of its eversion. The precise problem set up by this eversion is to deform the sphere so that its first interior, red and hidden, become in fine the exterior face of the sphere, without by no means injuring the sphere, without causing any traumatism to this infinitely thin skin.

The president of the Association Idem + Arts, Francis Trincaretto, is a surgeon. You will understand that he asked me the origin of this problem, since the work of surgeons would be greatly made easier if they could turn us over like gloves. There is nothing less sure that the mathematicians have much take care of helping the surgeons. Usually, it is rather the contrary that happens.

In a pictorial way, the mathematical origin of the problem is the following. From the "practical" point of view, mathematicians are interested in problems of removal, of transportation. For instance take one of the painted Dick Termes sphere. As a sphere, it is initially a totally smooth surface, without any asperity to the contrary for instance of a mountain, and whose realisation has required a great care. The artist does not look upon it that his work be injured during its transportation from Searfish to Maubeuge. Mathematicians call a carriage a mapping, and to qualify this transport without any accident, use the aquatic term of immersion.

In 1957, the mathematician Steven Smale questioned whether, two transports without accident being given, it would not be possible to pass from the first to the second along a sequence of such transports. In the case of the carriage of the sphere, his answer has been positive, which implies that one can mathematically turn over the sphere without subjecting it to any traumatism. This transport can have anyhow a particularity which sometimes partakes of the science fiction. In the same way that the famous hero of a French story, by the humorist Marcel Aymé, could cross over the walls, it is admitted that the surface may go across itself.

From these data, mathematicians have searched to create procedures to everse the sphere. Bernard Morin, François Apéry and John Sullivan have been important actors of that saga. In order to get for themselves a better understanding of the techniques they had to work out, and also to give a better understanding to their colleagues, the mathematicians have first been led to prepare drawings and iron-wired or papered models. In their workshop, Bernard Morin and Richard Denner will show some of these models inspired by the first methods of eversion developed by Bernard Morin.

John Sullivan has worked on another procedure of eversion using a minimal energetic cost constraint. Stewart Dickson will show some main steps with the help of physical realisations, while John's film, the Optiverse, will allow us to follow the continuous unfolding of the eversion.

This unfolding has inspired the creation by François Apéry of a flexible, luminous steel-wired sculpture, that can be taken into pieces : then one looks at the unfolding of a double covering of the sphere. The object, the demonstration are fascinating.

I.9 De la musique avant toute chose...

"De la musique avant toute chose", in the "old times", pupils in high schools used to learn this verse from the French poet Verlaine. It is possible that, among all the arts, the music be the closest to mathematics by the sensations aesthetic it brings. One can for instance find them again in the ethereal practice of arithmetic. And maybe the Pythagoreans would have not refuted this proportion : the geometry is with respect to plastic art what the number is with respect to musical art. From the past mathematicians who worked on the foundation of the musical art, we can remember the names of Pythagoreas, those of Archytas and Euler. It is on the study of the same foundations or on their expressions that today the musician Michel Deneuve, the mathematicians Carlota Simoes and Yves Hellegouarch, equally a professional cellist, are working. Hellegouarch's mathematical theory of scales precisely describes the most diverse musical situations. We should not thank enough these musicians for having accepted to make us share these sometimes divine suggestive sensations that their art affords to us.

I.10 The objectives of the Colloquium

I would like to conclude this beginning upon the subject by a return to the objectives of the Colloquium. It is not merely a matter of showing the wonderful objects that the artists and the mathematicians, each one on his side, are able to produce. It is not only the question of promoting the fructuous exchanges between these two communities. The Colloquium has not only the simple purpose of serving as a prefiguration to the realisation of the park of geometries, whose main outlines I shall describe in a moment, and to better validate this project in the opinion of the scientific community. The central objective of the Colloquium is of pedagogical nature. In some subtle way, it would like to concern the formation of the mind. In a more apparent manner, it turns around the transmission and the acquisition of knowledge through visual representations, be they static like painting, sculpture or architecture, or more dynamic like the modern audio-visual animations, the qualities of these representations being supported by the resources of the musical art and of the literary art. And so, the important objectives of the Colloquium are to encourage the formation of contacts, the reflection on these animations and their creation in the midst of community structures.

II The ARPAM Project

II.1 The pedagogical data

Two essential reasons, of more or less biological and psychological nature, justify the interest of the project :

1) the first takes as a starting point the saying that "the animated precedes the inanimate one ", and the aphorism of Charles Darwin, äny true construction is genealogical ". From there results that the acquisition of the knowledge and the formation of the spirit, which have a phylogenesis, deserve to be conceived according to a process of ontogenesis which respects this phylogenesis. One will remember that the first developments of science were accomplished within the framework of societies at a pace very slow, appearing even almost fixed to the local temporal glance. The first discoveries were established by the patient observation of structures or objects which appeared immutable. A fixed object has the advantage of being able to be contemplated for as a long time as it is wished so that our senses have the possibility of taking a detailed image of it which can, with much more facility that the fleeting image, print itself deeply in the memory. Therefore is it natural to support any process of training by repeatdly presenting of objects illustrative of the subject matter, in particular, if possible, objects motionless or subjected to slow and repeated movements.

Project ARPAM is a pedagogical project which, as one of its bases, rests on the paramount presentation of fixed objects.

2) Any object which can be seen, touched with the eye, has a semantic field much larger than that of its idealisation, of the abstract object. It has one signifier, one meaning, more or less hidden, and through it, has an emotional potential. As any process of attractive training can only be based on the use of attractive objects, it is advisable to choose the latter with understanding, by taking account at particular emotional effects that they can induce. The sight of certain objects can contract the spirit, to cause in it violent tensions. Others, on the contrary, can produce a relaxation, release the thought of tensions and concern locked up it in a kind of reducing vice, until it is prevented from being receptive on other subjects, in particular the innovations. Many artistic objects have this physical and psychological virtue. Inserted in the process of training, they bring a supplement of heart to science.

Project ARPAM aims at including, in a way more conscious than usual, an artistic dimension in pedagogy, in particular within disciplines considered difficult, like mathematics. The examples, the illustrative objects, through the complex work of our senses, must strike the spirit, and allure it. This is a characteristic of works of art. They can, in thousand ways, present a character of strangeness by their dimension, their unexpected forms, the successful layout of their decoration and their colours, and by their unusual sound effects.

It is on these bases which the project that I will briefly present was drawn up. Although their cost does not allow their multiplication in great number, it goes without saying that other projects of the same nature can be and undoubtedly will be conceived, for, we hope for it, the benefit of the human community. Allow me to recall here that the practice of mathematics tends to abolish the borders of any nature which can separate societies. This practice reveals the universality of the steps of thought. It is a factor of exchange and unity between men. UNESCO carries from there testimony, which made year 2000 not only one year of peace between the people, a wish alas still pious, but also that of mathematics, a wish finally fulfilled.

II.2 General characteristic of the project

The fixed objects chosen are of modest size, on the one hand to res-pect a certain intimacy, on the other hand not to burden the cost with the realization. They are however of sufficient size to impress the senses with a measured strength.

These objects are either gardens, or small buildings called "folies", or sculptures. Although their basic structure is fixed, decorations which they carry are likely to be modified with time. Their principal function is to illustrate concepts and facts belonging to the greatest possible extent of the mathematical field, while evoking the history of their discovery.

In order to reinforce the effect of surprise, the majority of the objects should a priori be hidden. One will discover them at the turning of a path, behind a curtain of trees, by reaching a saddle, or the top of an hillock.

In addition, the objects will not be crowded together as one finds in museums. Relaxing paths will separate them ; this relaxation can also be created by pleasures of the eye, as by physical effort which constrains the work of the thought. The time spent in walking between two objects should be sufficient to make it possible to fix in memory and to assimilate the previous object, and to make the spirit sufficiently available to discover the following curiosity with interest.

These constraints, to which it is naturally necessary to add that of the accessibility, weigh on the choice of a site for the project. There is undoubtedly no ideal site. That which the Town of Maubeuge, offers, in the fortifications created at the 16th century by the geometer Sebastien Vauban, has, by the weight of its famous past, the advantage of the originality.

An introductory panel will be placed at the entry of each garden and each folie. Folders of various types established according to the mathematical level of knowledge of the reader, will give additional information. To the interior of each folie will be placed one or more data-processing consoles for which interactive tools for visualization will be designed. Young researchers present in the folies to answer the possible questions of the visitors will in addition be charged partly to develop these tools, which will have to meet teaching standards.

II.3 A word on the gardens

The gardens, like the folies, bear names. In the preliminary draft, five gardens were envisaged :

A. the garden of symmetries, close to the Seventh Temple

B. the projective park, close to the Cap of Apollonius

C. the phyllotaxic clearing, close to the Horn of Plenty

D. the knotted forest, close to the Knotted Stained Glass

E. the Eulerian gardens, close to Euler' Bridges

The garden of symmetries, the projective park, and the Eulerian garden are typically French gardens. Their creation should make happy a renewed family of gardeners. Free course will be given to their imagination within the geometrical data which we will be able to propose to them. Topiary art, of the size of suitable shrubs, will be renewed. It will not be a question any more of simply cutting parallelepipeds, spheres and cones, but also of reproducing the various shapes of the objects which one meets in the geometry of surfaces.

Here is a very simple example of a flower-bed, illustrating the configuration of Pappus of Alexandria (IVth century after J.C.), the last of the large Greek geometers. This configuration is composed of a polygon with 6 sides or hexagon, whose alternate nodes are aligned on two lines represented on the ground by alignments of red flowers. The sides of the hexagon are represented by alternate alignments of yellow and blue flowers. It is remarkable that the points of intersection of the opposite sides of this hexagon are also aligned. The poets will observe in this configuration the presence of two M intertwined. The modern gardeners could try to execute this flower-bed with a new flower, recently discovered if one believes the scientist newspaper " Nova Biologica ". This flower is very interesting : one could show indeed that it could carry only four colours at most, and that it had a symmetry of a new nature, like the number of nodes of the configuration. This is purely anecdoctal. The scientific name of this flower is : Septembris Malbodiensis (or Maubeugensis Calunaris).

We will see now why the garden of symmetries, whose flower-bed will be able to comprise friezes and plane tilings made up of quite selected flowers, deserves to be placed not far from this folie called " the Seventh Temple ".

II.4 Succinct presentation of some folies

II.4.1 Introduction

Each one is free to conceive as many folies as he wishes. In reaction to the sheep-like and anaesthetizing shape of uniformly cubic constructions which populate our cities, a common point between these projects would be the concern of building folies whose uncommon form stimulates, on the contrary, the spirit and imagination. I have not yet completely defined all those of which I have thought, in particular the three following ones :

" The Knotted Stained glass ", entirely made with transparent material, illuminated from inside, intended to illustrate differential topology.

" The Poincaré' Surprises ", folie whose roof comprises two parts between which circulates a fluorescent fluid. The lower part of the roof is fixed, whereas the other part is mobile and transparent, which makes it possible to follow the evolutions of the fluid.

" The Luminous Torus", a building in the shape of a torus, made partly out of glass. It allows one, from the interior of the torus, to observe the effects, and in particular the caustics, produced by lightbeams projected on a suitable transparent object of variable index of refraction.

Here is the description of six computed folies. The drawings which here accompany them reproduce the original drafts made ten years ago on an old computer.

II.4.2 The Cap of Apollonius

The steps of thought, in particular that of analytical thought, are progressive. After the study of linear properties of objects, of degree 1, comes that of objects of degree 2, the first of the nonlinear ones. These objects are also known as quadratic. They played and continue to play an essential role in geometry, the theory of numbers, and in the applications of mathematics in the physical world, in particular in mechanics. This part of mathematics is partially illustrated by the Cap of Apollonius (born in Pergamon in 262, died in Alexandria into 190). This folie makes it possible to display some basic results of Euclidean geometry in usual space. The shape of the building resembles the cap which the noble ladies wore in the Middle Ages. It is about a truncated cone of revolution, divided by a vertical plane which contains the way out. At the interior, one shows the circles of Dandelin and the principal traditional properties of the conical sections. The higher part of cone is out of transparent material to benefit from the natural light. The remainder of the truncated cone is out of clear metal. Dick Mee will show on one of his beautiful CD-Roms how one can decorate the outside of this truncated cone, isometric with a portion of the Eucliden plane.

II.4.3 The Seventh Temple

Encouraged by the remarks of epistemologists, the mathematicians of the 18th century revealed some structures in the families of objects that they handled, in particular that of group, initially in connection with the numbers, then in geometry. A set of objects has the structure of group, this structure is known as an algebraic structure, if, in particular, these objects can combine between them, and if any object admits a symmetric one. This theory thus makes it possible in particular to study the manner of filling space using standard tiles, where symmetry plays an essential role. The form of the Seventh Temple resembles a little that of certain small temples of Antiquity, and intends to illustrate this significant algebraic theory, in particular, from the visual point of view, through the presentation of tilings of Euclidean and hyperbolic spaces. One of the interests of plane hyperbolic space is to be able to play the role of universal covering of unspecified kinds of smooth surfaces.

Mathematicians established that there are 7 types of possible friezes having internal symmetries. This is why the vertical interior of the folie has 7 faces ; on each face, a removable panel illustrates one of the 7 elements of the types of possible friezes.

17 is a number known as a Fermat number, and Gauss showed that one can build, with the rule and the compass, a regular polygon of which the number on sides is such a Fermat number. But 17 is also the number of elements of the types of tessellations of the Euclidean plane. This is why the vertical outside of the temple has 17 faces. On each face, a removable panel shows one of the 17 tilings. Two adjacent faces are separated by a column in form of braid. The top of these columns carries coloured material polyhedrons, sometimes mobile around an axis of symmetry. Every two years, one will try to launch competitions, at various levels, for the creation of friezes and pavings.

In the interior, the ground carries an example of a paving of the hyperbolic plane. The fact that the interior of the building comprises seven faces invites to pave the plane with hyperbolic Klein triangles of interior angles (pi/2, pi/3, pi/7). This hyperbolic paving is raised on the spherical cupola which is used as the roof of the building. It is a spherical stained glass the colour of which can be either the same as, or complementary to the colours of the paving of the ground. The position of the dome is such that at the zenith of summer, the rays of the sun project exactly the tiling of the stained glass on that of the ground.

Several color fillings of such a tessellation are possible, corresponding to distinct surface coverings. Thanks to a mechanical device, it will be possible to substitute one paving by another. Lastly, from the conceptual point of view, the invariance of properties with respect to a group of transformations makes it possible to introduce the fundamental concept of stability.

II.4 4 The Horn of Plenty

The preceding folie is relates to a paramount state of the universe which fills space with particles and identical forms. The Horn of Plenty corresponds to a second phase of the evolution of the universe, where the descendants of the first generation are identical to their forebears only scaled down by a constant factor which is set at the beginning. The Horn of Plenty thus ad infinitum will illustrate the concepts of recurrence, those of sequence and series, fractals, and thus the premises of analysis.

The building is composed of coupled cells which converge towards a point. The scale factor is naturally the famous golden section [sqrt{5}+1]/2 = 1, 618 ... which comes from the 12th century Fibonacci sequence. One finds this number in the regular pentagon : the length of a diagonal is equal to that of the side multiplied by the golden section. This is why one chose for cross section of the Horn such a pentagon. While varying with constant step the dimension of the pentagon and the rotation angle between two consecutive pentagons, one obtains an unfolding in the space of curves of pursuit giving contours of the Horn. On the walls of the interior rooms, will be drawn or projected the creations made using recurrence algorithms, and thus in particular those of the fractal world. At the interior point of convergence of the rooms will be placed a source of light which will make it possible to irradiate the small final rooms.

I would like to thank Mr. Vitkine for having carried out by infosculpture a first outline of the model of this building.

II.4.5 The Gauss' Observatory

This folie is devoted to differential geometry, notably to that of smooth surfaces in usual space. It is composed of a cylindrical trunk surmounted by a spherical cupola. At a point of this cupola, one makes appear a tangent plane, similar to the cap which English-speaking students sometimes wear, and which will be mobile with the point of contact.

The base of the building is a piece of a Scherk surface, one of the minimal surfaces discovered at the 18th century. One enters the building by one of the cells of this surface. A piece of an other minimal surface, the helicoid, is used as staircase to reach the level of the principal room, whose ground is out of transparent material.

Here still, from the conceptual point of view, one will insist on the concept of stability, whose extremality is one of its metamorphosis, or sometimes one of the substitutes.

II.4.6 The Whitney's Umbrella

Located at the crossway of various branches of mathematics, the Whitney's umbrella was selected to represent algebraic surfaces, as much for historical reasons and mathematics as for the richness of the fundamental concepts which it can illustrate : stability, singularity, stratification, bifurcation.

Being a ruled surface, it is in addition easily constructible. The central part which surrounds the handle will be made of transparent material to better ensure the interior luminosity of the building. Note that the frontal parts of this folie are pieces of swallowtails, which are homeomorphic with the umbrella.

II.4.7 The House of Number

The forms, either the partly circular base one, or the logarithmic curve of the roof, are selected to evoke of course the essential numbers p and e. The input for drawing the base is a piece of the right strophoid with polar equation polaire r = a (cos 2b)/(cos b), which one will compare with that of the Scherk surface previously evoked of the standard form z = Log[(cosky)/(coskx)]. Elements of right helicoids are erected above the towers on the sides of the House.

Arithmetic always inspired a double feeling to me, of great brightness on the one hand, great mystery and darkness on the other hand. Therefore I would have wished to translate this double feeling outside by constructing a building coated with metallized and thus luminous glass plates, but quite dark in its interior, except for some luminescent hearths.

II.5 The future of the project

Thanks to the support of all the participants mentioned in the introduction to this talk, project ARPAM might develop in the years which come, within a few meters from here. It will be necessary to refine it on the technical level to hold account inter alia particular constraints which weigh on the site. In collaboration with the mathematicians, the structural engineers will get busy working on it.

The choice of materials, decorations interior and external of the folies are not always defined. For that purpose, a close cooperation between mathematicians, artists, and perhaps industrials, will be necessary. The project makes sense, and will be perennial, only if we make it our objective to create objects that are beautiful in themselves, jewels which, by their aesthetic qualities, profit from a very strong capacity of attraction, and thus allow everyone to perceive the beauty inherent in mathematics, thus helping to break down some psychological barriers which prevent to enter the mathematical world.

Just like this work of completing the project, the design and realization of animations, meant either for the general public, or for the community of mathematicians themselves, will also require the cooperation and goodwill of professionals coming from all the walks of life, academic or not. Community structures of preparation will have to be set up. Let us all take part in the realization of this project in the service of the works of the spirit, the Beautiful, and the Good.


C.-P. Bruter (1992) Le Parc Mathématique : Éléments pour l'étude de faisabilité architecturale et muséographique, Arpam.