ARPAM

THE MATHEMATICAL PARK

The ARPAM Project

General characteristic of the project

The fixed objects chosen are of modest size, on the one hand to respect 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.

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.

A word on the gardens

The gardens, like the folies, bear names. In the preliminary draft, four main gardens were considered:

A. The garden of symmetries (close to the Seventh Temple)
B. The projective park (close to the Cap of Apolonius)
C. The phyllotaxic clearing (close to the Horn of Plenty)
D. The Eulerian garden (close to the Eulerian 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 colors at most, and that it had a symmetry of a new nature, like the number of nodes of the configuration. This is purely anecdotal. The scientific name of this flower is : Septembris Malbodiensis .

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 ".

Brief presentation of some folies

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. Here is the list of a first series of folies :

A. The Cap of Apollonius
B. The Seventh Temple
C. The Horn of Plenty
D. The Euler Bridges
E. The Gauss' Observatory
F. The Whitney's Umbrella
G. The House of Number
H. The Luminous Torus
I. The Poincare's Surprise
J. The Knoted Stainglass

A. 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 Euclidean plane.

B. 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 families 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 familiy 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 colored material polyhedrons, sometimes mobile around an axis of symmetry. Every two years say, 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 color of which can be either the same as, or complementary to the colors 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.

C. 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.

D. The Euler Bridges

This is a resting area devoted to a schematic reconstitution of a part of the city of Koenigsberg, where it was necessary to build seven bridges to cross over the local river and its arms. Euler set up and solved the first problem in graph theory, the first non metric mathematical theory. The notion of connexity which does appear plays a fundamental role in topology.

E. 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.

F.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.

G. 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 pi and e. The input for drawing the base is a piece of the right strophoid with polar equation 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.

H. The Luminous Torus
I. The Poincare's Surprises
J.The Knotted Stained Glass Window*

It is based on what can be called a singular representation of the Boy surface that has a key role and position in the 2-sphere eversion process.

This representation has a vertical axis of symmetry. When the object is cut by a plane perpendicular to this axis, it becomes a kind of flower with three equal foils, the angle between two foils being 120°. This “flower” is the base of a fiber-space ; the fibers are ellipses that all meet in a singular point on the vertical axis. Two such ellipses, being relatively close one to the other, form the boundary of a singular Möbius band with the previous singular point as a singularity. Some well-chosen ellipses, materialized in chrome steel, constitute the main part of the frame of the folly that has two parts : one visible from the outside, and the other, that can only be visible from the inside. The floor lies on a perpendicular plane to the axis, located at approximatively one third of the height of the construction, starting from the singular point. It is transparent so that one can see the lower part of the construction from the inside. The lower part lies inside a cylinder plastered with broken mirrors which reflect the internal lights illuminating the frame. A few Möbius bands are materialized in stained glass windows. Looking up and down, one can see them entirely from the inside of the folly.

The outside part of the folly is made of three pieces, but only two will serve as rooms for visitors. A series of vertical arches made of the same material as the frame will cover the entrance of the two symmetric rooms. Convenient transversal rods to the elliptic frame will define panels and allow to cover all the outside part with glasses and mirrors. Some of these panels will be stained glass windows showing either tilings or knots and braids with symmetries. Beams of search-lights of appropriate colour will illuminate the folly, and close by outdoors sculptures.

The future of the project

It will be necessary to refine it on the technical level to take into account particular constraints which will 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.

Reference

- C. P. Bruter (1992) Le Parc Mathématique : Éléments pour l'étude de faisabilité architecturale et muséographique.
- Proceedings of the Isama Conferences 2002 /(The Luminous Torus),/ 2003/ (The Poincare's Surprises), /2008/ (The Knotted Stained Glass Window/).