Paolo Simonetto

David Auber

Daniel Archambault

Archambault

Fully Automatic Visualisation of Overlapping Sets

Keywords: Descriptors (according to ACM CCS): I, 3, 3 [Computer Graphics]: Picture/Image Generation-Display algorithms P, Simonetto, D

Visualisation of taxonomies and sets has recently become an active area of research. Many application fields now require more than a strict classification of elements into a hierarchy tree. Euler diagrams, one of the most natural ways of depicting intersecting sets, may provide a solution to these problems. In this paper, we present an approach for the automatic generation of Euler-like diagrams. This algorithm differs from previous approaches in that it has no undrawable instances of input, allowing it to be used in systems where the output is always required. We also improve the readability of Euler diagrams through the use of Bézier curves and transparent coloured textures. Our approach has been implemented using the Tulip platform. Both the source and executable program used to generate the results are freely available.

Introduction

Visualisation of hierarchies, usually described by a set of elements and a cluster tree that groups the elements recursively, has been widely studied over the last decade. Many techniques such as Treemaps [START_REF] Shneiderman | Treemaps: a spacefilling approach to the visualization of hierarchical information structures[END_REF] have been proposed as interactive solutions for the exploration of this type of data. However, the containment metaphor, where a parent of the hierarchy strictly contains all of its children, is limited when analysing modern datasets. With increasing frequency, these classifications are no longer cluster trees, but, rather, they are taxonomies where an element can belong to multiple groups at the same level of the hierarchy. Thus, the classification of the elements is no longer a cluster tree but rather a cluster DAG or directed acyclic graph.

A cluster DAG can always be transformed into a cluster tree through unfolding, where branches of the DAG are duplicated. Unfolding has been used [KMBG07, SMO * 04] for the visualisation of company relationships and metabolic networks respectively. However, these works demonstrated that in practical cases the number of duplications can be very high. These duplications result in a visualisation that is far more complicated than it needs to be. Approaches, which are not based on data duplication, can be developed once techniques for visualising sets and their intersections are available. Approaches based on Euler diagrams seem to Figure 1: Example of an Euler diagram rendered with our approach. The red set contains world monuments, the green set contains things that are typically Italian, and the blue set contains things that are typically French. The red set shares the element "Colosseum" with the green set and the element "Eiffel Tower" with the blue set as demonstrated through overlaps of the regions.

be promising candidates for solutions to these types of problems.

In this paper, we present an algorithm for the automatic generation of Euler diagrams. Our technique is fully automatic, and, unlike previous work, does not have any undrawable instances of input sets and their intersections. This visualisation technique could be used whenever it is necessary to deal with overlapping sets of elements, like in social [START_REF] Palla | Uncovering the overlapping community structure of complex networks in nature and society[END_REF] or biological networks [START_REF] Bader | An automated method for finding molecular complexes in large protein interaction networks[END_REF], complex query display [START_REF] Verroust | Ensuring the drawability of extended Euler diagrams for up to 8 sets[END_REF], engineering diagrams [START_REF] Howse | Generating Euler diagrams[END_REF] and many other application areas.

Related work

Euler diagrams [START_REF] Euler | Lettres à une princesse d'allemagne[END_REF] are graphical representations that depict sets and their intersections. In an Euler diagram, a set is a region of the plane bounded by a curve and intersections between sets are depicted through overlaps between these regions as shown in Figure 1. Unfortunately, there is no widely accepted definition of an Euler diagram. Many definitions exist and differ substantially from each other in the types Euler diagrams they can draw [START_REF] Fish | Defining Euler diagrams: choices and consequences[END_REF][START_REF] Fish | Formal issues in languages based on closed curves[END_REF][START_REF] Stapleton | Properties of Euler diagrams[END_REF]. A variety of algorithms exist using different definitions for an Euler diagram, each with its own set of undrawable instances [START_REF] Chow | Generating and drawing areaproportional Euler and Venn diagrams[END_REF][START_REF] Howse | Generating Euler diagrams[END_REF][START_REF] Fish | Euler diagram generation[END_REF][START_REF] Verroust | Ensuring the drawability of extended Euler diagrams for up to 8 sets[END_REF]. However, all these algorithms suffer from undrawable instances, where the algorithm is unable to produce output for some collections of sets.

The problem of generating Euler-like diagrams for any collection of sets and their intersections is a relatively new topic. An Euler-like diagram generalises the notion of an Euler diagram to include disconnected sets, sets with holes, and other constructions which are usually not permitted. Since we focus on Euler-like diagrams for the rest of the paper, we use will almost always use the term Euler diagram when describing our algorithm to mean Euler-like diagram. Simonetto and Auber [START_REF] Simonetto | Visualise undrawable Euler diagrams[END_REF] analysed the conditions which resulted in instances of undrawable Euler diagrams. They demonstrated that allowing holes and disconnected sets was sufficient to resolve all undrawable input instances. However, this theoretical work did not present techniques for visualising Euler-like diagrams, rather the results were drawn by hand. Subsequently, the authors presented an algorithm to resolve undrawable input instances and compute set adjacency [START_REF] Simonetto | An heuristic for the construction of intersection graphs[END_REF], but the work did not present an algorithm for drawing the resultant diagrams.

To our knowledge, the only existing algorithm that generate Euler-like diagrams for every input instance has been developed by Rodgers, Zhang, and Fish [START_REF] Rodgers | General Euler diagram generation[END_REF][START_REF] Rodgers | Embedding wellformed Euler diagrams[END_REF] as an extension of the algorithm of Flower, Fish, and Howse [START_REF] Fish | Euler diagram generation[END_REF]. However, their approach differ from the one we are presenting for several reasons. boundaries. In our method, textures and smooth bounding curves for each set mostly handle these cases, as shown in Figure 1, making our approach easier to implement.

Both approaches subsequently improve the initial drawing of the diagram by using a force-directed layout algorithm. However, in our approach, a structure called grid graph is used as an approximation for the set boundaries. This approximation aims to be a good starting point for the layout stage, rather than being visually similar to the final boundaries. This simplifies the process and leads to a clearer division of the stages. Also, it is important to note that adding edge-node repulsive forces to the model does not guarantee that no new crossings will be incurred [START_REF] Bertault | A force-directed algorithm that preserves edge crossing properties[END_REF]. In our approach, we use a force-directed algorithm that guarantees nodes and edges will not cross during iterations. Finally, by inserting set elements before executing the force-directed algorithm, we can produce diagrams where the size of a set is proportional to its cardinality, making better use of space in the plane.

Definitions and Algorithm Overview

Before we present our approach, we should review some concepts formally defined in previous work. Given a subset S of the sets in the Euler diagram, a zone corresponds to the region in the plane where all and only the elements of S intersect. Examples of zones are shown in Figure 2(b). A zone is also the subset of elements contained by region. Classes, defined previously in Simonetto and Auber [START_REF] Simonetto | Visualise undrawable Euler diagrams[END_REF], are the sets depicted in the drawing of the Euler diagram. Specific classes are denoted with capital letters, and zones are denoted using letter combinations as shown in Figure 2.

At a high-level, the approaches used by previous algorithms are relatively similar. All of the approaches identify three steps as illustrated in Figure 3: that represents the structure of the Euler diagram, associating nodes of the graph with zones, and selecting edges between zones that should be adjacent. 2. Draw the intersection graph: compute a planar drawing of the intersection graph 3. Construct the set contours: compute the contours of the classes using the drawing of the intersection graph.

Algorithm

The input to our algorithm consists of a listing of the classes in set notation. For example, the input to generate the diagram shown in Figure 1 The heuristic of Simonetto and Auber [START_REF] Simonetto | An heuristic for the construction of intersection graphs[END_REF] is used to generate the intersection graph from the input sets. In this work, the authors describe a greedy approach that ranks all intersection edges with a metric. This metric encodes the contribution the edge would make in creating a well-formed intersection graph. At each iteration, the edge with the highest metric value is inserted into the graph in a way that is similar that Kruskal's algorithm for computing a minimum spanning tree. The algorithm terminates when no more edges can be inserted, either because they will create an intersection graph that is not planar or because they do not significantly contribute to intersection graph structure.

In the heuristic, the metric was developed to encourage the creation of intersection graphs that correspond to a proper Euler diagram whenever possible, i.e. Euler diagrams without disconnected sets. Moreover, the algorithm drives the selection of graph edges toward more readable Euler diagrams. For example, the metric discourages the adjacency of unrelated sets and encourages the adjacency of zones that share a larger number of common sets.

As this paper focuses on the visualisation of overlapping sets, the remainder of this paper concerns itself with drawing Euler diagrams. We accomplish this in four steps:

1. Construct a planar drawing of the intersection graph 2. Build a grid graph around the intersection graph, mapping nodes of the intersection graph to zones.

Insert the elements into their zones and apply iterations

of PrEd [START_REF] Bertault | A force-directed algorithm that preserves edge crossing properties[END_REF] to refine the drawing. 4. Select colours and textures for the classes and use Bézier curves to draw the class boundaries.

In the following sections, these four steps are described in detail.

Drawing the Intersection Graph

Recall that each node of the intersection graph represents a zone, and an edge links two nodes if their zones are adjacent. Therefore, a drawing of the intersection graph decides positions and shapes of the zones and classes in the Euler diagram. Our drawing of the intersection graph must be planar and as regular as possible, avoiding large variations in edge length or angular resolution.

To obtain an initial planar drawing of the intersection graph, we use the algorithm of De Fraysseix et. al [START_REF] De | How to draw a planar graph on a grid[END_REF]. The output of this algorithm is guaranteed to be planar but is quite unsatisfactory with respect to the criteria described above as shown in Figure 4(a). To overcome this problem, we apply the PrEd [Ber99] force-directed algorithm. In PrEd, no edge crossings are incurred or eliminated. As there are no crossings in the initial layout, the resulting layout will also be planar. The drawing will generally conform more closely with the above criteria as shown in Figure 4(b). Applying PrEd at this stage of the algorithm is not strictly necessary, but it allows for faster convergence and helps the approach avoid local minima. plicitly by tracing the contours around an initial layout of the intersection graph. In our approach, the mapping between a node of the intersection graph and a region of the Euler diagram is explicitly done through a second graph called the grid graph. The result of a grid graph computation is shown in Figure 4(c).

Building the Grid Graph

The grid graph encloses each node and edge of the intersection graph in its own non-overlapping region of the plane. Thus, by building the grid graph, we will identify regions of empty space around nodes and edges of the intersection graph. We call these regions node-regions and edge-regions respectively. As long as these regions are of limited size, zone overlaps that do not correspond to set intersections are avoided through a planar drawing of the intersection graph.

To construct the grid graph, we start by computing a set of non-intersecting circles around the nodes of the intersection graph. Each circle centred on a corresponding node in the intersection graph layout and will be used to define the noderegions and the edge-regions of the grid graph directly.

Computing the Set of Circles

The position of each circle is given by the position of each node in the layout of the intersection graph. Thus, we can choose a single radius R for all circles that is just small enough to avoid all overlaps. If we inscribe our node regions on the interior of these circles, we are guaranteed that the node and edge regions of the grid graph do not overlap.

For these reasons, we choose an R which is less than half the minimum node-to-node and edge-to-node distance. In our system, we use:

Constructing the Node-Regions

We will now inscribe a polygonal region on the interior of each circle, defining the node-regions. To guarantee that the centre of the circle is inside the polygonal region, adjacent polygon vertices in a anticlockwise traversal form a maximal central angle of 2π/3 denoted α.

Let us consider a circle centred on each node of the intersection graph. Each edge of the intersection graph divides the circle into several sectors. More formally, consider the set G ⊆ E of edges incident to a node v ∈ V of an intersection graph I = (V, E). As the layout of the intersection graph is fixed, we can consider each edge e j = (v, w j ) of G as a vector e j = -→ vw j . If we sort the |G| vectors by increasing angle from the x-axis into the ordered sequence e 1 , e 2 , . . . e k , e k+1 , e 1 = e k+1 , adjacent elements define sectors around v. When v has no incident edges, we will consider the entire circle as a single sector.

For each sector (e i , e i+1 ), we select a set of ⌈γ/α⌉ polygon vertices where γ is the central angle of the sector. The polygon vertices are evenly spaced δ radians apart on the circumference of the circle. The process is shown in Figure 5.

Constructing the Edge-Regions

When constructing the node-regions, the circle around each node of the intersection graph is divided into ordered sectors. Thus, each edge of the intersection graph is a boundary of an ordered sector. Given a node of the intersection graph n ∈ V and edge of the intersection graph e ∈ E, with e incident to n, we will denote c(n, e) and a(n, e) as the rays emanating from n that bound e on clockwise and anticlockwise traversals of the polygon vertices respectively. Let p c(n,e) and p a(n,e) be the polygon vertices supporting these rays. The edge-region for the edge e = (v, w) of the intersection graph is therefore bounded by the following four edges of the grid graph: (p a(v,e) , p c(v,e) ), (p a(w,e) , p c(w,e) ), (p a(v,e) , p c(w,e) ), and (p a(w,e) , p c(v,e) ). An example of an edge region is shown in fig. 6.

Element Insertion and Boundary Refinement

Once an initial planar layout of the grid graph is defined, we will run PrEd to refine the drawing. But before we run it a second time, we insert the elements inside the classes and compute the edges that will be used in the drawing of the class boundaries. Figure 4(d) shows the result after this step.

Inserting elements into their respective zones is relatively straightforward. Each zone in the grid graph is bounded by a polygon inscribed on the interior of a circle of radius R and the maximum permissible central angle is 2π/3. Thus, the closest a polygon edge can ever be to the centre of the circle is R/ √ 3. Thus, the set elements are randomly placed inside a circle of this radius.

To determine which edges of the grid graph define a class boundary, we have to consider subgraphs of the intersection graph or class schema as defined precisely in Simonetto and Auber [START_REF] Simonetto | Visualise undrawable Euler diagrams[END_REF]. A class schema is an induced subgraph on the intersection graph such that it includes all the nodes belonging to the same class. The grid graph edges that define a class are the edges that enclose the schema without crossing it. If we consider the edge (v, w) in Figure 6 as its own class, then the class schema will be formed just by v, w and e. We can therefore remove the grid graph edges (a 4 , a 1 ) and (b 2 , c 1 ), because they intersect e. Of course these grid graph edges may be used by other classes, so we must consider all classes before deleting these edges.

Euler Diagram Rendering

Many Euler diagram generation algorithms stop after the class boundaries have been identified by closed curves, and elements are placed inside their appropriate zones. However, as our drawings are intended for visualisation tasks, we should pay close attention the rendering of the diagram. In this section, we discuss how we select colours, textures, and bounding curves for classes.

Colour Selection

Humans have trouble distinguishing more than about six to eight colours [START_REF] Ware | Information Visualization: Perception for Design[END_REF] in a complex diagram. If we use transparency and allow colours mix at each intersection, we can see why minimising the number of colours used becomes a priority. On the other hand, different colours should be used for different classes in the diagram.

To circumvent this problem, we generate another graph where each node is a class of the diagram and an edge indicates that two classes overlap. We run the node colouring heuristic of Welsh and Powell [START_REF] Powell | An upper bound for the chromatic number of a graph and its application to timetabling problems[END_REF] on this graph. The output of this algorithm suggests a small number of colours and textures that can be assigned to the classes of the diagram. Assuming that the graph can be coloured with less than c colours and we have c colours and textures at our disposal, it is guaranteed that no two sets which overlap will be assigned the same colour and texture combination. Currently, we use c = 8.

Textures

We implemented the work of Byelas and Telea [START_REF] Byelas | Texture-based visualization of metrics on software architectures[END_REF] to select textures regions of the classes overlap. By using textures in addition to colour, we are better able to represent regions more distinctly than by simply using colours or class boundaries alone. Intersection patterns between the textures also can help indicate the classes to which a zone belongs and helps the approach scale to larger Euler diagrams. if two incident edges e 1 and e 2 of the grid graph share the same set of classes, the orientation of l is given by the line perpendicular to the bisector of the angle between e 1 and e 2 . (b) Case 2: if three or more incident edges, e i , i = 1 . . . n, are associated with different sets of classes, and there exists an edge e k that is the union of all these classes, the orientation of l is the orientation of e k (c) Case 3: if neither of the above-cases apply, we cannot have continuous Bézier curves at this grid graph vertex. Thus, we place the control points along their associated grid graph edges. That way, no new boundary intersections are introduced.

Bounding Curves

We convert the polygonal boundaries of classes into continuous Bézier curves, making them easier to follow. We transform each edge of the grid graph into its own Bézier curve, respecting the following properties:

• do not introduce any new set boundary intersections or change inclusion or exclusion of the set elements • smoothly join Bézier curves at grid graph vertices

The curve drawn for each edge e = (v, w) has a control polygon of four control points: P 0 , . . . P 3 . The points P 0 and P 3 are always coincident with the nodes v and w. The position of v is also used to calculate the position of P 1 and the position of w is also used to calculate the position of P 2 . For this reason, we say P 1 is related to v and P 2 is related to w.

In order to respect all the above conditions, we must choose the positions of P 1 and P 2 in such a way that:

• no control polygon overlaps with another control polygon or set element • when two control polygon edges meet at a common grid graph vertex, they must be collinear along the line l

The line l can have any orientation in the plane. The orientation of l is calculated individually for each grid graph vertex in the following way as shown in Figure 7: 1. if two incident edges e 1 and e 2 of the grid graph share the same set of classes, the orientation of l is given by the line perpendicular to the bisector of the angle between e 1 and e 2 2. if three or more incident edges, e i , i = 1 . . . n, are associated with different sets of classes, and there exists an edge e k that is the union of all these classes, the orientation of l is the orientation of e k 3. if neither of the above-cases apply, we cannot have continuous Bézier curves at this grid graph vertex. Thus, we place the control points along their associated grid graph edges. That way, no new boundary intersections are introduced

We calculate the maximum displacement of each control point along l that does not introduce any new crossings. This displacement is calculated using a function of PrEd [START_REF] Bertault | A force-directed algorithm that preserves edge crossing properties[END_REF].

Results on Real World Data

We tested our approach on a few real world datasets. The Internet Movie Database (IMDB) is an online database of information related to films, actors, and the cinema. It also ranks films according to viewer ratings.

To show how our approach can scale to larger datasets, Figure 8 is the diagram generated for sixty of the top seventy films of the database with full credited casts for each film. All the discarded films were non-overlapping. The diagram categorises over two thousand actors. Although the diagram is complex, it is still fairly readable. Subsequently, we selected a subset of twenty films from the top forty highest ranked films. For each film, we considered the first twenty actors in credits order. Figure 9 shows how the films, the sets in the diagram share cast members. In this diagram, we can make a few interesting discoveries. First of all, there are three set of classes, labelled {A, B, C}, {N, Q} and {H, L}, that have extensive overlaps. These films are actually film trilogies: "The Lord of the Rings", "Star Wars" and "The For our second example, shown in Figure 10, we show the full credited cast of the "The Lord of the Rings" trilogy. First off, all the major protagonists are in a single zone as shown in Figure 10(a) and in the close-up in Figure 10(b). This zone corresponds to the intersection of the three movies. Miranda Otto, the actor who plays Eowyn, appears only in "The Two Towers" and "Return of the King", is an exception in the upper left corner of the figure. We can also see that Billy and Katie Jackson, the children of the director, Peter Jackson, made cameo appearances in all three films.

Future work

Our algorithm for drawing Euler-like diagrams can be improved in many ways. As seen in Figure 11, it is often possible to compute better shapes for classes. The best way to get well formed contours more consistently would be to improve the force-directed algorithm stages of our system. We would also like to improve the execution and convergence speed of PrEd. As PrEd comprises nearly 95% of the computation time, speeding up this algorithm would greatly improve the As seen in this picture, at the moment we cannot guarantee that new regions will not be created.

execution speed of our system and allow us to scale to even larger datasets.

In this work, we focused primarily on algorithmic contributions required to draw collections of sets. There are many other problems associated with the visualisation of sets that have yet to be explored: interactive manipulation of sets, how sets could be extended into multi-level settings, and the visualisation of dynamically changing sets. Moreover, we should investigate how this technique could be applied to applications in the humanities, sciences, social sciences, as well as many other fields. More specifically, we would like to apply this technique to the problem of visualising overlapping graph clusters, which arise in pathway visualisation of metabolic networks or protein complexes in protein-protein interaction networks.

Conclusions

Our work provides a fully automatic method for the generation of Euler-like diagrams. The system produces a drawing for any collection of input sets. It uses the work of Simonetto and Auber [SA08, SA09] as a basis. In order to circumvent undrawable instances, it must disconnect regions or introduce holes into them. Our approach easier to implement than other competitive previous work, because it does not try to prevent collinear boundaries. By using colours and textures more effectively, we are no longer required to solely rely on contours, allowing us to scale to larger dataset sizes.

FirstFigure 2 :

 2 Figure 2: Classes, zones, and labelling conventions. (a) We use capital letters to denote classes. In this example, M denotes the set of monuments, I the set of Italian things, and F the set of French ones. (b) We use sequences of lower case letters to label zones. For example, the intersection between I and M is labelled im.

Figure 3 :

 3 Figure3: High-level procedure for drawing Euler diagrams. First, construct the intersection graph. Second, draw the intersection graph using a planar graph drawing algorithm. Third, draw the set boundaries around all nodes in the same class.

  would be: Monuments = {Eiffel Tower, Colosseum, Taj Mahal, Big Ben, Great Wall of China, Statue of Liberty} France = {Wine, Baguette, Eiffel Tower} Italy = {Colosseum, Pasta, Pizza}

AFigure 4 :

 4 Figure 4: Overview of our drawing algorithm. (a) Initial output of the FPP algorithm. This layout of the intersection graph is unsatisfactory in appearance. (b) The algorithm draws the intersection graph with PrEd [Ber99]. (c) The algorithm constructs a grid graph around the layout of the intersection graph. (d) Elements of the sets are added to the drawing, and the algorithm runs several iterations of PrEd. In the four subfigures, nodes of the intersection graph are red, the nodes of the grid graph are green, and the elements of the sets are blue.

  I = (V, E) is the intersection graph and d is a function which defines the distance between elements.

Figure 5 :

 5 Figure 5: Constructing a node-region. (a) The edges incident to the node of the intersection graph define three sectors, bounded by (e 1 , e 2 ), (e 2 , e 3 ), and (e 3 , e 1 ). (b) Considering only the red sector, we are able to define the vectors f 1 , f 2 δ radians apart. (c) The intersections between the vectors f i and the circumference of the circle are taken as vertices of the node-region.

Figure 6 :

 6 Figure 6: Construction of an edge-region. An edge of the intersection graph divides the circles into the sectors A, B and C. The polygon vertices computed in the previous step are ordered clockwise and anticlockwise. In the diagram, the edges of the grid graph which define the edge-region of e are: (a 4 , a 1 ), (b 2 , c 1 ), (a 4 , c 1 ), and (b 2 , a 1 ).

Figure 7 :

 7 Figure7: Construction of smooth Bézier curves at grid graph vertices. (a) Case 1: if two incident edges e 1 and e 2 of the grid graph share the same set of classes, the orientation of l is given by the line perpendicular to the bisector of the angle between e 1 and e 2 . (b) Case 2: if three or more incident edges, e i , i = 1 . . . n, are associated with different sets of classes, and there exists an edge e k that is the union of all these classes, the orientation of l is the orientation of e k (c) Case 3: if neither of the above-cases apply, we cannot have continuous Bézier curves at this grid graph vertex. Thus, we place the control points along their associated grid graph edges. That way, no new boundary intersections are introduced.

Figure 8 :

 8 Figure 8: A diagram showing sixty highest rated movies in IMDB with full credited cast.

Figure 9 :

 9 Figure 9: An Euler diagram of IMDB movies composed by about twenty movies. (b) An overall look. The sets have been labelled with capital letters. A -The Lord of the Rings -Part II (2002). B -The Lord of the Rings -Part I (2001). C -The Lord of the Rings -Part III (2003). D -American History X (1998). E -The Matrix (1999). F -Taxi Driver (1976). G -Goodfellas (1990). H -The Godfather: Part II (1974). I -Fight Club (1999). J -Raiders of the Lost Ark (1981). K -Apocalypse Now (1979). L -The Godfather (1972). M -Se7en (1995). N -Star Wars: Episode V (1980). O -The Shawshank Redemption (1994). P -The Dark Knight (2008). Q -Star Wars (1977). R -The Usual Suspects (1995). S -American Beauty (1999). T -Léon (1994). Capital letters labelling sets are placed manually, but every other part of the diagram generation was automatic. (b) A close-up of part of the diagram. In these figures, images of the actors are shown when they are available in the directory.

Figure 10 :

 10 Figure 10: Cast of the trilogy "The Lord of the Rings". (a) The diagram generated. "The Fellowship of the Ring" is red, "The Two Towers" is green, and "The Return of the King" is blue. (b) The zone shared by all three films. In these figures, images of the actors are shown when they are available in the directory.

Figure 11 :

 11 Figure11: Direction for future work. The original diagram can be improved simplifying the shape of the curves. This mock-up was created with a version of the algorithm that is under development. As seen in this picture, at the moment we cannot guarantee that new regions will not be created.

submitted to Eurographics/ IEEE-VGTC Symposium on Visualization(2009)

Construct an intersection graph: construct a planar graph submitted to Eurographics/ IEEE-VGTC Symposium on Visualization (2009)

Acknowledgements

This research is partially funded by ANR-BBSRC Systryp project and the INRIA Gravité project.


Fractional Dynamics, Tiling Equilibrium states and Riemann's zeta function A Le Méhauté1 , L. Nivanen, A El Kaabouchi, Q. A. Wang ISMANS,44 Ave. F.A. Bartholdi, 72000 Le Mans, France Summary : Il is argued that the generalisation of the mechanical principles to other variables than localisation, velocity and momentum leads to the laws of generalized dynamics under the condition of continuous and derivable space time. However, when the fractality arises, the mechanics principles may no more be extended especially because the time and space singularity appears on the boundary and creates curvature. There is no more equilibrium state, but only a horizon which might play a same role as equilibrium but does not close the problem -especially the problem of the invariance of the energy -which requires two complementary factors: a first one related to the closure in the dimensional space, and the second to scan dissymmetry stemming from the default of tilling the space time. A new discreet time arises from fractality. It leads irreversible thermodynamic properties. Space and time singularities lead to the relation between the above mentioned problematic and the Riemann zeta functions as well as its zeros.

Historical context

The relationship between the dynamics and thermodynamics was developed in a conflicting historical context in which the Newtonian mechanics, as the Queen of the sciences, has always driven the paradigms and made many eminent and unquestionable scientists, such as Boltzmann Duhem or Prigogine, expiatory victims accused to be deviationist (I. [START_REF] Stengers | Tangente : Les fractales : Art, nature et modélisation[END_REF].

The mechanics was formed gradually over four centuries along successive stages dating back to the smart experience of Galilée (1564Galilée ( -1642) ) which turns out to initiate the theoretical quarrels between Leibnitz (1646-1716: mv ) and Descartes (1596-1650 : 2 mv ) on the one hand around the vis viva (living forces), and posthumously with Newton (1642-1727) on the other hand. Although it was Galilée and Leibnitz who claimed the measurability of acceleration, it was Newton who won the first round with the expression linking the force to the mass (

2 2 1 / r m m F  
) for two body system which is the first foundation for Kepler laws.

The next step of the mechanics is called Lagrangian event by [START_REF] Stengers | Tangente : Les fractales : Art, nature et modélisation[END_REF]. This step begins with the principle of d'Alembert (virtual work) who suggested the independence of external forces with respect to internal bonds (dipoles). These forces, if deriving from a potential, are independent of the kinetic global state of the system (in motion or not). This leads to the second law of Newton  m F  updated by Euler (1707-1783) and then by Lagrange . All systems and motions, reversible or irreversible, are commonly represented by an ensemble of points with neither spatial nor temporal extension. Consequently, due to the hidden permanent internal/external correlations, the simple concept of friction, depending on velocity and, in practice, on spatial extension, has been dropped out. So, there is no irreversible loss. The equilibrium of interior links (having central role) makes the notion of cause, which is completely balanced by the effect, disappear. The universality of mechanics emerges naturally from the absence of dissipation. This reasoning applied to motion results in the principle of virtual work. The conservation of the living force T leads to the state function called Lagrangian (

V T L  
). It is the states associated with Lagrangian that can be related to the work (W) and gives sense to the potential V. At this step, the causality match precisely the effect, property which excludes all loss functions and gives rise to state function. Only the initial and final states count in the formulation of the dynamics. Hence there is an optimal path, or, put it in another way, there is a variational principle stuck on this optimal path. From these principles, the mechanics is the sole theory axiomatically constructed on the paradoxical time reversibility (Albert, Coveney, Prigogine). The momentum mv p  is introduced into it as a thermodynamic intensive variable which is nothing but the velocity itself. The energy of variation, or of motion, is fixed by the differential of the intensity p such that vdp du 

. Finally the energy gets a quadratic form 2 2 / 1 mv u  which gives rise to very useful quadratic properties (Deheuvels) allowing analytical solution of dynamic problems. Inversely, the fine properties of the quadratic form become the intrinsic properties of this mechanical theory.

Lagrange will have two descendants whose diverging visions turns out to be conceptually formulated in very different even conflicting manners: irreversible in both time and phase space (see below) for Carnot and reversible with time symmetry for Hamilton who swear allegiance to the queen of sciences and his paradigms with his two beautiful symmetric

equation x H p      and p H x     where ) , , ( t p x H
is the Hamiltonian given by a sum:

H=T+V. Then the conservation of energy, that is the absence of dissipation, requires 0    t H .

We will come back to Carnot (1796Carnot ( -1832) ) at the end of this paper since his result on irreversibility implicitly guides our reasoning. It is Hamilton (1805-1865) who takes the role of guiding line. As shown above, Hamilton constructed a very concise formulation in establishing the equivalence between the degree of spatial freedom and the degree of dynamical freedom. The velocity is postulated as a variable independent of another one: the position, hence an N body system in 3 dimensional space has 6N degree of freedom forming the phase space and then a principle of conservation arises. This choice of Hamilton leads to the definition of a quantity called Hamiltonian (

V T H  
) whose conservation is in the heart of mechanics2 . Therefore mechanics has to consider non conservative properties through extensions/approximations which preserve the beauty of the quadratic properties. The system can be considered as an object on its own thanks to the principle of Poincaré ) 0 (  rotdiv or equivalently to the absence of monopole which possibly will break, at the boundary, the symmetry imposed by the principle of d'Alembert. The Hamiltonian approach leads to the normal modes as the solution of the dynamic, that is t i e  , a solution of the operator

2 2 2 x t c   
which is symmetrical in t and x, and invariant with respect to the inversion of t and v, meaning the absence of time arrow and space orientation. The harmonic analysis naturally becomes a central tool of the mechanics.

In the perspective of confirmation of the emergence of irreversibility from statistics, Badiali, after Souriau, added to the above analysis some complementary elements, as for example the symplectical step (Libermann, Souriau). He extended (2008-9) the analysis of [START_REF] Stengers | Tangente : Les fractales : Art, nature et modélisation[END_REF] to quantum mechanics by making the different construction steps of the mechanics into hierarchical stages: the Heisenberg step, the Schrodinger step and the Feynman step, each of them giving rise to subtle additions of new concepts and not only new point of view on the existing concepts. These additional concepts (Badiali) lift the degeneracy of physical laws which remained circumstantial despite their generalities [C* algebra (Murphy) associated with Heisenberg and the algebra of von Neumann in statistics, fractal trajectories, the role of automorphism and of time with the Tomita theorem and the KMS conditions applied (or not) to the path integrals of Feynman (Feynmann)]. Following A. Connes (A. Connes) and [START_REF] Roveli | Qu'est ce le temps ? Qu'est ce que l'espace[END_REF] approaches, Badiali's work is an interesting deepening into dynamical foundation of thermodynamics ( ) êxp( H Z

 

). Admitting the multiplicity of paths (Feynman), it offers a more fundamental understanding of the ambiguous status of time in the absence of geodesic but also  the exponential character of partition function in the standard case  the status of the H theorem in statitical mechanics and the meaning of Loschmidt-Zemerlo's controversy ( inversion of all velocities of all particles )

 the status of the Lagrangian action  

1 2  t ML
as relativistic invariant which is distinct from the Euclidean action    Hdt  The absence of Hamiltonian invariant in General Relativity from the factual curvature of space-time

The work of Badiali sheds light for example on the question whether or not exists a time constant beyond which it is possible to introduce the thermodynamics constraints and irreversibility since inside of which the physics loses both the notion of means and its experimental attributes. One more time the key concept of time seems completely associated with the concept of velocity which appears a posteriori as a generic character of the mechanics (Targ). But one can ask the following question, i.e., why the reversibility of time might be a justification of the notion of velocity which imposes de facto, not de jure, the non commutativity of space-time (Connes, Roveli), and how the heuristic hypotheses might have been later changed to commutativity from the space-time homogeneity principle (Noether).

Such an assertion has been questioned by an engineering work on the dynamics of 'energy' transfer in fractal geometry [START_REF] Méhauté | and Fillipi Introduction au transfert d'énergie en milieu fractal[END_REF][START_REF] Méhauté | Transfert Process in Fractal Media[END_REF]. Contesting, from the consideration of fractal geometry, the universality of Hamilton point of view, the authors questioned the status of velocity as a dynamical state variable in fractal media. This position opposes the distribution theory and experiments against the opinions ofsome authors searching for the marginal adaptation of classical mechanics to the fractal limit conditions (Gouyet, Stauffer, Vannimenus). In such a media, there exists an intrinsic anomalous link between space and time, determined by the fractal structure, which is more general than the notion of velocity. This new variable (

1  t L d
) is needed by the consideration of the non differentiability (Tricot) of 'generalized phase' space and hence by the approach of Fourier transformation. This transformation leads to inter-scale strong correlations inside the set of Hamiltonian normal modes (states) or related coarse-graining analysis. The pertinence of this idea was initially supported by the observation that the diffusion constant represents only a particular case of that new class of generalized variables (

1 2  t L ).
The Fourier transform, as the main tool of this distribution approach, implies a singular selfsimilar tiling of space-time (Bedford, Hirsch). By using adequate gauges this tiling can be  either Euclidean (d=1) if the space is continuous, without energy in the dipole (d'Alembert) but with arbitrariness of rules for the choice of self-similar tiling (topology of torus), and with velocity as a state function of the dynamics.  or, if d=2, hyperbolic (Toubiana) by the consideration of discreet space as an ensemble of singular points coupled with self-similar topology of punctuated torus (Bedford). We will all the same emphasize that there are two classes of possible tiling, that will upset the viewpoint which is after all orthodox.

In fact, whatever is the topological difference of self-similar space-time tiling (fundamental group and domain) between the Euclidean (d=1) and hyperbolic ( 1  d

) approaches (Berline), both tilings require to ground the time concept via a dynamics (Badiali, Bedford). The tiling points out the scales carried by two automorphisms: one concerning the space and one concerning the time. The last one

 A A 
is based on a variable  , where must be normalized from space either in the external form of velocity (Euclidean), or in an internal form of a time constant 1   which requires an heuristic hypotheses on space time relationship. Badiali (2009) reported the same kind of problem for the Feynman's path integral of quantum Mechanics on the basis of the theorem of Tomita-Takazati -there exist a one parameter group of automorphism which leaves the algebra globally invariant. In fractal geometry this parameter is complex (Le Méhauté 1997) and involves non commutativity with respect to time (see below). Hence a specific dissipation arises. This is easy to understand. In the Euclidean case, due to the fact that the only singular point is the, somewhat arbitrary, original point of the axis, the paving leads to the torus topology of the dynamics (Hirsch). This introduces the operator

2 2 2 x t c   
into a formulation associated with the existence of normal modes of the space-time manifold and of cycles, or their equivalent in quantum mechanics (Hilbert space, eigenvalues or eigenfunctions). In this approach, the reversibility is obviously the consequence of our arbitrary adjustment of the scales we use for the dynamics (coarse graining without specific correlations due to the only spatial character of the Fourier transform of a velocity). We are here at the centre of the paradox which opposes the mechanics to the thermodynamics and Hamilton to Carnot; a paradox stressed by Prigogine and his students who, after De Donder, turned out to make the irreversibility (viewed from mechanics) a metaphysical property relative to the arbitrariness of imposing ad hoc approximation of measure on a continuous and homogeneous space-time (Noether). This conclusion is a petitio principii since it assumes a priori the absence of correlations and of the pertinence of the basic Hamiltonian concepts such as velocity.

If we keep the debates at the same level of analysis, the situation may partially change if the space-time or the phase space is considered as a distribution of singularity (Schwartz) and more specifically a scaled set of singularities (Le Méhaute, Mandelbrot). As a matter of fact, if space-time become discreet, due to some mathematical constraints imposed by the singularities (Schwartz), the impossibility (unless for the case where the time is infinite) of interpreting the singularities gives rise to non commutating properties at infinity. Therefore irreversible factors emerge from collective properties of these singularities. The analytical and topological reasons are the following. The singularity exists on its own and is independent of whatever analysis scale and approach, especially for the initial or final equilibrium state with

  , 0 t
, that is, for all 0 ,    in Fourier space. Every test function (Schwartz) used to characterize the singularity leads an energy u  as a mathematical asymptote, and a limit distribution r  . This test function must be a coarse graining procedure characterized by renormalisation properties. It is in practice the test function which allows an access to the singularities. Like derivation it implicitly carries the variable of time through the automorphism driven by the procedure. It is just this function which couples in Fourier space the intrinsic properties with the external characteristics of the test signal. In practice, and except in a few number of cases, the convolution, which points out the correlations, breaks the Hamiltonian properties and introduces a new time factor.

The modalities for Poincaré tiling of the standard tree-hyperbolic space-time (Berline, Hirsch, Toubiana) partly mimic the exponential relaxation process in the Fourier space and clearly give rise to geodesics (analogy between arc of circle and transfer exponential function leading standard harmonic ratio, see Figure 1). The coarse graining may lead to some irreversibility factors through the discreet variable

   Z
=R as well as through the relaxation time

RC c     / 1
(Figure 1). Nevertheless, the above analogical viewpoint of irreversibility is still to be strengthened from the consideration of algebraic topology so that it is more robust against possible objection from mechanics. Therefore we should move farther to fractal environment starting from Poincaré tiling ( Figure 1) and introducing fractal correlations.

Scaling correlations

As we have previously indicated (Le Méhauté 2005), the topology coming from the tiling in fractal medium can be founded neither on the periodic fundamental domain of the Poincaré type [stemming from the orthodox hyperbolicity (Bedford, Berline)] because the analogy pointed out above seems an ad doc representation, nor a fortiori on the simple torus, but on the punctuated torus opened on the boundary. We assert that it is just this singular punctuation, carrying the geometry correlations inside it, which leads the dynamic to intrinsic irreversibility. It is imposed by the space time parameterisation of the geometry (manifold of geodesics). Through an extended harmonic ratio

         c v u / /  which controls its value, ) ( Z  i c   / u v
the angle over the punctuation (at infinity) definitely breaks the symmetry associated with the operators accessible to the normal modes and makes the physics on the torus questionable but also the physics on Poincaré punctuated torus (Bedford). In order to show all the subtleties related to the emerging punctuations opened on the boundary, we will make a detour by the analytical foundation of electrodynamics, a son of the differential geometry.

External algebra and their reductions

The orthodox physics is based on the homogeneous space-time structure (Noether). According to Kant, space-time is an a priori frame of human conscience but, according to the disciplines, science needs to classify the operating variables in the right space-time for the description of reality. Both points of view have to be related. Starting from the example of electrodynamic variables, we can interpret the laws of electrodynamics (Le [START_REF] Méhauté | Batteries as fractal object[END_REF][START_REF] Méhauté | Introduction to wave phenomena and uncertainty in fractal space in Quantum mechanics, diffusion and chaotic fractals Edt[END_REF] as shown in Figure 2 in a 3 dimensional schematic view, with the extensive variables on a back plane, the intensive variables on a first plane separated by a certain 'distance' which represents the matter . The spatial gradients are given in horizontal direction and the temporal gradients in the vertical direction. The time is necessary for the expression of flux and moments. However, it cannot be extended without giving a sense to the notion of velocity. The velocity appears as the key factor to give a meaning to space time differentiable manifold and even to give a meaning to external algebra, that is to rise the concept of matter (here seen through L and C)

    t c B rotE ; 0  divB ;  q divE  ; c t c E rotB     
, the quadratic form of energy

2 2 2 1 2 1 B E U   and B cE S  
by analogy to the mechanics concerning the momentum density. The electromagnetic wave emerges from the diagram with the forms

0 2 2 2      E t c E ; 0 2 2 2      B t c B
. The suppression of the potential vector A simplifies the diagram which has only one source of extensity and leads to a diffusive form

0      t U C U
. The quantum mechanics is only a reduced form of the same geometrical approach even if the latter is, for this purpose, transformed into the complex space:

) , ( ) ( 2 ) , ( t r r V m t r t i                
. Figure 2 is a simple representation of the geometric differential properties which support the Maxwell equations (Spivak).

This diagram can be simplified in the way of E. Vieil [START_REF] Vieil | [END_REF](Vieil, -2010) ) according to the principle of d'Alembert, by considering that the notion of scalar and vectorial spatial gradient naturally enters into the definition of state. According to Vieil, the link defines the dipoles. Like Lagrange and d'Alembert proposal and following the expression of mechanical energy ( Fdl U   ), Vieil considers two classes of coupled variables which, independently from their thermodynamic nature (active for q U or reactive for q  ), obey the conservation of energy.

dq E U q q   q q q dp I   
q is the transferred extensive quantity similar to the momentum in mechanics, q E is the related field 1, from which the transfer arises, q I is the flux of q, that is q  and q p is the momentum (field 2 : B) from which the force can be derived. We then get Figure 3. Far from that point it authorized the analysis of the coupling between different fields (Energy conversion).

Although Vieil turns out to be able to find the concept of dipole, and hence the concept of discreet space, the advantage of his approach with respect to the generalized expression of the external algebra in Figure 2, that is to delete some variables in order to keep only the temporal derivation, is legitimate only if the notion of velocity is a priori taken for granted. In view of this implicit hypothesis and in the frame of an analogy with mechanics, it is possible to suggest from Vieil's proposals one further step with a homogenous version of the diagram by extending the formalism into reciprocal space, i.e., by using, to represent the temporal part, the Fourier transformation (with the normal mode of frequency  i ). We get then Figure 4.

This approach turns out to transform the variables of Vieil, especially     q q q p E I , 0 ,  , into singular points of the fundamental topologic domain based on an automorphism of the Poincaré type (Figure 4), so that the variables are associated with a dynamics over q which is invariant from the viewpoint of the dipole energy ) 0 ( q E . of the dynamics founded on the Vieil's variables. Here the fundamental extensity q in space-time gives rise to an ensemble of fundamental variables in frequency space. These variables acquire physical sense only asymptotically in accordance with the exponential relaxation (the hyperbolic geodesics represented by cyclic arc). The reversibility in time and frequency is conserved by the factual symmetry. One notices that in order to work out this diagram, Vieil cancelled the spatial gradient (implicit he admit the principle of d'Alembert).

This figure allows us to introduce the following links:

 between the gauges of measure related to velocity   t L / . The Fourier transformation of the velocity ) (  is associated with a length and ) ( Z the impedance of the dynamics is therefore related to an 'interface' for exchange. This relationship seems very conceptual but it naturally comes from the simple experience with complex medium (storage of energy, viscoelastic dynamics, thermodynamics of crystallization and percolation). In Fourier space, a flux is only a charge (capacity) proportional with the factor Z, to a force (impedance, see Ohm law U=ZI for example). In the absence of the source   , the flux density is equal, with a geometrical constant, to the force.

Hence, in Fourier space, Z / 1 is nothing but the measure of the interface of energy storage or exchange (length, surface, etc). From this which is the basement of the TEISI model (Le [START_REF] Méhauté | and Fillipi Introduction au transfert d'énergie en milieu fractal[END_REF], a generalization of the concept of capacity named Fractance is authorized. Hence it is in practice possible to put in the diagram two different representations, i.e., the representation based on   t L, and the one with

    i Z ),
( which take into account the concept of velocity. Assuming the presence of a temporal horizon, Z is a variable linked to ) (  by a relation of Mobius type

) ( 1 1 ) ( 1      Z
. The term 1 indicates the presence of a source of dissipation   , that is of a resistance, so a dissipative term clearly related to the behaviour at infinity ( on the punctuation). This relation expresses both the pertinent transformation of displacement associated with a source which can be shifted according to our will and a pertinent inversion of the variable associated with the possible construction of reciprocal tiling (for example the diagram of Brillouin in solid state physics). Among t q

) ( q I ) 0 ( q E q p L  i -q R ) ( Z consequences the relation 1 ) (    i
which implicitly expresses the pertinence of the concept of velocity and resolves the problematic by asserting the conservation of measure in two reciprocal phase spaces  

] ,t L and     i Z ),
( . The introduction of the complex variable i is due to the arbitrariness of  with respect to , or to the conservation of certain velocity.

Figure 4 surprisingly keeps the Hamiltonian characteristics . The transformation of Fourier of a velocity is a length ) (  a gauge of the reciprocal space. In this framework, R appears as a limit of

) , 0 (     Z . The two extensities   ) ( ,  q I q have a spatial localization in the sense of ) 0 (    but the two intensities   q q p E ), 0 (
, which are respectively the force constituting the spatial pivot, and the momentum the frequency characteristic, have singular position on the axis. The physical sense of the diagram is obvious. In space-time, there is no other extensity in the thermodynamic sense than the dipole   q q  , accessible at long time limit of the dynamics. The dipole   q q  , is in addition the sole source of the energy ) 0 (

q E
. The resistance is strictly related to the geometrical distance of the dipoles. If the space time is homogeneous, this distance is related to a length and the resistance is an ad doc distance.

The above analysis sheds light on the physical signification of the constants L and C. These constants are of spatial nature: progression and regression in the definition of the dipole through C, and cycle in the definition of the moment through L. The simplest understanding of these constants is supported by the homogeneity of the state functions in the sense of Euler

        q S q S   
where S is certain measure in space-time-matter. In this way, whenever we write q CU q  , the expression dq E U q q   takes place by the time derivation

q t q U RCd U   where   RC
appears as a time constant. Obviously, this constant plays a central role whenever the dynamics is described with a series of stationary states without any correlation, hence the orthodox physic arises from differential integer analysis: All this above statements are relative to an elementary examination of the extended differentiable paradigm of the mechanics. But what happens if the phase space becomes fractal and non differentiable? A non standard heuristic reasoning (Hoskins, Le Méhauté) can lead to conceptual difficulties concerning the notion of measure (Tricot) and the notion of dynamic and geodesics which can only be reached via distribution theory (Schwartz).

operator      1 t or x v t      0 x D t 2 2 0      x v t 2 2 0 2 2 2      Geodesic   t   or   1 0   Lt v Exponential response   1 2 0   t L D Diffusive response   1 0   Lt v Wave response
To summarize the advantage of the orthodox viewpoint -presented above, which nevertheless conjugates the Fourier time-space  

  i Z ), (
with the standard time-space   ] ,t L -, as a seed for future develoments, it naturally suggests, from geodesic geometry, the extension of the analytical form of the function  

) ( Z to the form   ) (  Z
. This analytical extension of first order (half circle, zero angle on the boundary, standard harmonic ratio, Poincaré paving) into fractional order representation (see below) conserves the hyperbolic character of the geodesic (circle arc), expands the harmonic ratio into fractional one, restricts its Fourier space-time expansion and gives rise to the boundary singular behaviour. The related topological extension, widely justified experimentally (see reference in [START_REF] Méhauté | Geometry fractales Hermes Paris[END_REF]1997), reinforces, in addition, a consequence of the heuristic hypothesis formulated in the TEISI model (Transfert d'Energie sur Interface Self-similaire). This model is founded on the projection of fractal phase space onto the Fourier space and on the conservation of measure in that space by taking into account the fractal property in a generalization of the concept of

velocity through   1 ) ( / 1      i .
This detour by the fractality will shed light on the real limits of the mechanistic philosophy if one keeps the point of view of Noether, of d'Alembert and of Hamilton for answering the question of irreversibility as a simple consequence of our cognitive incapability and of approximation (Prigogine). This detour leads to the consequence of the geometry of the complex environments, that is, of the generalized metric of the space-time, on the determination of irreversibility. Let us now enter into the details of the analysis

Equilibrium and periodicity

What does the concept of resistance mean? The answer can be given indirectly by the reasoning around the differential equation of first order which describes the proportionality of the flux to the force pushing the system to equilibrium. This equation will be later expended under the form of fractional differential equation. We used here the word equilibrium which, as shown below, contains certain ambiguity. To understand it, the basic differential equation

)] ( ) ( [ 1 ) ( t U t U RC t U dt d  
 must be used. The horizon represented by the equilibrium

) (  U
imposes to the dynamics a source   conducted by an external force which behaves as teleonomical characteristic. The Fourier transform of this equation is nothing but

  iRC Z   1 1 ) ( , the impedance of a parallel RC circuit with ) ( ) ( ) (      U U Z
. The solution of that equation is an exponential function

 / ) ( t e t U  
where RC 



. We get the notion of time constant from this particular case. The presence of R in  is an explicit dynamical expression of the existence of an asymptotic state called equilibrium (horizon) but also it signals a thermodynamic irreversibility. In other words, the question is: is there an equilibrium thermodynamics state only under the condition of the existence a priori of a dynamics having exponential solution? As a matter of fact, the exponential is the only function which contains, locally, all the information carried globally by the dynamics. It is the only function that is analogous to its derivation. Put it in another way, one does not define intrinsic equilibrium just confined in its local behaviour without reference to irreversible global process characterized by the presence of R. Let us observe that the exponential formulation guarantees also the periodicity of the standard state through the link arising via the standard algebra (standard relation between summation and multiplication (Nivanen)). This periodicity can also be found from tiling the space. Is it associated with the presence of hyperbolic geodesic giving a sense to the least action and to the variational calculus, in both configuration space and Fourier space? Does it authorize the reduction of chaos to certain form of determinism mechanistic (Mandelbrot 2004, Prigogine)? Some (mathematically coherent) answers to these questions can be found in the fractality and fractionarity of the dynamic.

Fractality, fractionarity and associated topology

On the basis of the experience in the fields of storage of energy in fractal media (Le [START_REF] Méhauté | and Fillipi Introduction au transfert d'énergie en milieu fractal[END_REF], one of the authors has concluded long ago that geometry is the source of entropy. This conclusion was latter confirmed by the propagation of electromagnetic waves in heterogeneous media [START_REF] Jonscher | Dielectric relaxation in solid[END_REF][START_REF] Méhauté | Generalized electrodynamic analysis in fractal media[END_REF] as well as in viscoelastic media (unpublished). Conversely, many authors show that fractal born easily from flows of aggregation, percolation, deposition etc (Sapoval). If these observations does not have sufficiently robust conceptual basis to relate geometry and irreversibility they bring to the laboratory sufficient reasons to address theoretical questions based on the dynamics of non standard hyperbolic geometries.

The introduction of irreversibility into mechanics via the emblematic equation of first order is a tool to extend the mechanical analysis into Poincaré hyperbolic environment (Figure 4). Nevertheless, as shown by the TEISI model, the operator of fractional derivation in time (Poldubny, Samko, Spanier):

     i d TF t  ) (
describes the fractal like dynamics. Hence the generic first order equation suggests a simple canonical extension of the dynamics using a fractional order operator. It is this operator which, making a breakthrough in the standard differential analysis, questions with respect to irreversibility, the paradigms of differential physics and then the mechanics . This can be briefly remind in table 1 without reference to the relationship between these operators and the fractal geometry established elsewhere (Le [START_REF] Méhauté | Geometry fractales Hermes Paris[END_REF][START_REF] Méhauté | Flèches du temps et géométrie fractale Hermes paris[END_REF],

x f t      0   x D t      0 2 / 1 2 / 1 x f t r 2 2 0             Lt f 0 Power law response   1 2 0   t L D Diffusive response      t L f r 2 0 ,

Fracton response

Table 2: Fractional differential operators of different orders. Note the difference from the corresponding operators in Table 1.

The interest of the generalization of differential analysis is to get geodesic and differential link between q I q  and q q E p  controlled by a unique transfer function of the form

      i Z   1 1 ) (
. This analytical form is associated with an -hyperbolic geodesic in Fourier space (arc of circle). With a nonexclusive example d / 1   for a -fractionnal differential form in a fractal space of dimension d, the geodesic can be related to an energy transfer through a d-fractal interface according to the -differential equation  L . This difference will play a major role in the understanding of irreversibility (Figure 8)

)] ( ) ( [ 1 ) ( / 1 / 1 / 1 t U t U RC t U dt d d d d          (Le
The figure 5 discloses the restriction of the Poincaré hyperbolic geodesic (standard semi circle) which can produce a symmetry breaking (boundary properties). This fractal consequence on the dynamics implies the disappearance of state functions and the rising of a nonextensive algebra (Nivanen) for the assessment of the thermodynamic extensity and intensity variables addressed for instance by Vieil's analysis. One of the consequences of the formal analysis related to the symmetry breaking is a generalization of the normal exponential and logarithmic functions used in an extended statistical thermodynamics developed for complex media (Nivanen). More over, in the framework of the statistical mechanics, this extension leads to the generalization of the statistical laws taking into account the incompleteness of the estimation field of the random variables (Ou,Wang) and to a more general mathematical measure on non extensive set (El Kaabouchi).

Although power laws does not lead to the convergence of the temporal series, this analysis paradoxically maintains the concept of horizon ) (t U  . This concept is factually extended to the notion of equilibrium as an expansion of internal scaling correlations. This equilibrium in the sense of a quadratic form, is the result of a conjugation between two complementary L t

     i Z   1 1 ) (     i R   L   t  i
power laws behaviour with characteristics  et 1- ,taking into account the fractional order of the singularity on the boundary.

Even on waiting if possible, of the construction of the generalized exterior algebra, the algebraic and geometric formulations show a proximity, noticed long time ago, between the diffusive operators and the operator of order ½. Let us remind that the diffusive operators conjugate conventionally the equation of first order and the equation of continuity. Such a definition may probably be bypassed if we observe that, with 1 )

( 2    i ,
      i Z   1 1 ) ( with 2 / 1  
, we might believe that the introduction of the fractality expands the transformation of the dynamics solely into a double effect (i) scaling (through   ), on the one hand, and (ii) a rotation of axis without temporal consequence on the complex plan (through  i ), on the other. However, as shown by the representation using the double basis (Figure 1 to 5), there is an additional major effect than mentioned above. The geometric treatment of the parameterization of ) ( Z imposes a lifting of the degeneracy of the punctuated singularity on the boundary (short or long time)  

    t L ,
. One may say that the initial Poincaré like singularity at the boundary, a point so that it has neither spatial nor temporal extension (Figure 1), acquires an emerging structure coming from the correlations related to the -distribution. The fractality rises time-space singularity.

The splitting of the singularity on the boundary, due to the collective properties of the hyperbolic geodesic, affects the dynamics through the border properties in the space

    i Z ), (
. They implicitly impose a curvature of all space time   t L, . Besides the scaling properties, the dynamics undergoes the effect of acceleration hidden behind the fractionality. The notion of time can not be reduced to a simple cyclic series (state functions from Fourier transformation or eigenfunctions in Hilbert space) since crucial space time correlations are partially lost in the sole geodesic. There is an irreducible space time residue which closes the energy invariance and therefore the entropy measure as a state function whenever a fractal dynamics takes place.

The additional transformation related to the splitting of singularity on the boundary plays the role of the closure of the fractal system in order to put it in contact with a thermostat. This closure returns the vocation of Noetherian invariant to the energy (Kosmann-Schwarzbach), i.e., to the relative quadratic form 3 . This quadratic closure shows another advantage: it gives a nominal equivalence to the two complementary parts of the arcs of circle, and then reduces, as shown in Figure 5, their inverse transformation with respect to the pole within two parallel straight lines, affected by the similar, but not equal, singularities on the boundary. More precisely, these two singularities denoted by (

   R , ,     1 ,R
) are complementary one to another in the quadratic form. This analysis turns out to modify the diagram to construct the Figure 6. This figure underlines both the role of the splitting singularity in a representation leading to 3 Although the conceptual tools of extension of exterior algebra are not mature at this stage, one can attribute the existence of magnetic monopoles to such property (unpublished).

the tiling extension, and the two singularities

1 ,    R , 2 1 ,     R
sharing R as a common variable linking the fractal with the thermo-state (periodic conditions for whole boundary; quadratic global form) , that is with a certain value of temperature.

Let the arcs of circle be denoted by

) (  Z and ) ( 1    Z
, we can generalize the periodic and non periodic hyperbolic tiling of the complex plan (Berline) using these geodesics. It is in fact possible to use the elementary tile from Figure 5 in order to cover the space. The tiling takes place in an imperfect manner with two distinct components, for the pavement: a covered and an uncovered singular zone. Despite its operational simplicity -it is in fact possible to reproduce simply the techniques of tiling of a plan and of the circle of Poincaré -this tiling is worthy to be mathematically deepened since it contains obviously open questions about the topological, and analytic formulation of the related automorphisms and associated dynamic.

Figure 6 : Extension of the Poincaré fundamental domain required for the singular tiling of the space-time when the dynamics is controlled by a fractional differential equation. We can notice the splitting of the energy related to the appearance of the singularities at the boundary as well as the complementary metric character of these singularities. Such a diagram opens plenty of questions in topology as well as in the physics of fractal media. The appearance of the singularity at the boundary is clearly related to the concept of emergence in physics. The rising of this singularity may even sheds light upon the question of emergence in complex systems.

Let us notice that the to semi circle are related to backward and forward geodesics. What will interest us now is not only the tiling question but the question of degeneracy of the tilling which occurs when 2 / 1   . We noticed previously the specific role of the factor 2 / 1   which returns the problematic to the case of diffusion in a field of Noetherian space-time. Extended at tiling, the singularities represented by the factor ½ return the spatial and temporal problematic to the question of the zeros of the Riemann zeta function. What happens if 2 / 1 1      ? The Figures 5 and6 clearly indicate a mechanism of fusion-confusion of singularities and dynamics. The angle of phase becoming equal to 4 /  , the resonance of the quadratic form becomes the limit of the two fractal forms of

) ( 2 / 1  Z ; R R  2 . 1 ,   , 1 ,    R = 1 1 ,     R , 2 / 2 / 1 , R R L  
. In practice, the imperfect pavement proposed above and the related geodesics degenerates into a dual singularity. (Le Méhauté, 2009). The problem of zeros of the zeta Riemann function arises.

) (  i 0        1 ) ( 0 , 0   q E ) 1 (    i      i Z   1 1 ) (

Zeta function

The zeta function is given by a sum of a series of power functions

    1 1 ) (    s s with   i s  
. One recognizes (Le [START_REF] Méhauté | Contribution of non integer differential operator NIDO, to the geometrical understanding of Riemann's conjecture[END_REF]Méhauté, -2009) ) in the real part of the zeta function the hyperbolic distance measured on a fractal arc of circle given by ) (  Z which allows one to plunge the whole problematic relative to this series and also to its avatars ( ) (s  function and L functions) into the complex plan. We also recognize, in each of the forward (+) and backward (-) arcs of the intersected circles

) (  Z , ) (  Z , ) ( 1    Z and ) ( 1    Z
the real part of the four symmetrical components of the Riemann function, i.e.,  

) 1 ( ), 1 ( ), ( ), ( s s s s       where   i s   .
We know in addition that the zeros of the zeta function are discreet and infinite in number since they are related to the prime numbers (Edwards, Slater). The zeros come in practice from an ultra-degeneracy over the tiling. Because the variable  is free and used for the screening of the complex plan, what happens from the degeneracy of the tiling of chaotic zeta trajectory with  is relative only to the functions

) (  Z ) (  Z ) ( 1    Z ) ( 1    Z
? This degeneracy appears for 2 / 1 



.  , which is convergent with 2 / 1   , will transform the intersection of the arcs into a fusion of the arcs, making the incomplete tiling disappear. There is no more difference of parameterization between backward and forward. The generalized

harmonic relation       2 / 1 / / c v u   
sends to the infinity the quadratic resonance factor underlining the complementary  and 7) and therefore the resonance of the quadratic substructure managing the energy. This is just the content of the Riemann conjecture which occurs as a necessity when the dynamic representation of the singularities from

  1 transfer function ) ( i Z (Figure
) (  Z ) (  Z ) ( 1    Z ) ( 1    Z
vanishes. The time gives to the conjecture its specific representation (Figure 7 and 8) and, through a derivation of order ½, returns the problematic in the continuous field of the type  

2 2 0 2 x t v    to a classical diffusive x t D   2 / 1
and to an almost symmetric gravitational problematic

  2 / 1 / 1 x t    
. The precision 'almost' points out the fact that to exchange L to t we must take into account a shift R which introduces a fundamental irreversibility; the price for the emergence of the matter. The Riemann conjecture tells us that, beyond the equilibrium in the traditional sense as a result of exponential relaxation, and for all fractal dynamics, there is no equilibrium in the strict sense of the word but a conjugated state of two complementary 'dynamics' in  and   1 . This duality of the system, -thanks to the additional independent variable  playing the role of temperature-, can make the system converge punctually to a non dissipative state (zero of zeta function) which is therefore stable state. This happens in some particular situations where the chaos becomes again the Brownian like chaos with the dissociation of the properties of the space from the dynamics (velocity) according to Hamiltonian properties i.e., integrable character. From the work of Stengers, we notice the whole cognitive difficulty in the treatment of irreversibility. It is because the Hamiltonian system is integrable that the balance between cause and effect is reached and that subsequently the notion of irreversibility disappears. On the contrary, the new vision introduced by the fractal dynamics is the paradigm of a geometrically irreversible dynamics.

 

Figure 8: This is the case of =1/2 giving rise to the degeneracy of the tiling. The geometry partly disappears here. The time constant of the sub quadratic form tends to infinity. The space time can no more be tiled as usual. More precisely, the residual discreetness is only reducible within two complementary discreet subspace-times

  2 / 1 2 / 1 , t L   and   2 / 1 1 2 / 1 1 ,   L t  
. The complete analysis requires (i) a complex representation of the time The -dynamics carries on itself a requirement for taking into account the fractal set of singularities and the correlations between scales (dendritic growth, aggregation, extremum phenomena, etc) hence the irreversibility appears naturally (Le [START_REF] Méhauté | Batteries as fractal object[END_REF]. If the dynamic is controlled in the frame of chaotic environment, this environment must impose an ½ fractional dynamic to have any hope to reach a non dissipative Hamiltonian properties. The fact that the zeros of the zeta function are related to the stable quantum Hamiltonian in Hilbert's space, is just a consequence of fractal space time game of hide and seek around time and space of the dimension 1 or 2.

     0 2 / 1 i t  , (ii) a complex representation of the space     0 2 / 1 il L  (

Facing the mechanics: the thermo-dynamics

L t   2 . 1 1 1 ) (   i Z   2 / 1 L    2 / 1  i R  2 / 1 t  2 / 1 1 t  2 / 1 1 L 
The breaking of the symmetry, -evidenced by the presence of the singular zeros of the Riemann zeta function-, is introduced here from the fractional dynamics. This dynamics is very different from the one induced by the baker's transformation (no trajectory and no point object in phase space but only the expanding or contracting lines [START_REF] Prigogine | Physique temps et devenir[END_REF]). However, the fractal chaos and the fractional dynamics related to -differential forms do not exclude the concept of trajectory which can be found through the generalized hyperbolic geodesics underlining ) ( 

Z

. Precisely, if one excludes the particular cases 2 / 1   , the dynamics which interpolate the rare non dissipative signatures are in general irreversible.

2 / 1   is characterized by an intrinsic hidden quantified irreversibility. Despite the history of the Queen of sciences which makes integrable system, optimal trajectories, variational principles, asymptotic equilibrium state, and the paradigms fixing stable and unforgettable references with plenty of symmetries, the fractional dynamics turns out to indicate the contrary, i.e., the generality of the dissipative system. Although the dynamics with chaotic character are still related to optimal trajectories, these trajectories raise dual singularities on the boundary. They

are irreducible in     i Z ), (
space as well as in the space   t L, . These singularities are dual also because the collective correlations of the trajectory bear additional 'elements', -we mean emerging organisation and group transformation-, on the trajectory and its boundary, which makes the system couple with the thermostat or the environment. For 2 / 1   the dual singularity is its own 'thermostat'. That is why the idea of the relation between irreversibility and approximation keep probably a part of truth.

The respectful criticism of the works of Hamilton from fractional analysis, leads the emergence of the geometrical irreversibility. Carnot belongs to the world of the engineering, a world of action, of objective intelligence, of creativity, and of progress; a world grounded by the following question in mind « what happens if… ? » (Stengers). Like many engineers who constructed the innovation progressively without experimental facilities, Carnot, first followed by his heir Clausius, who coined the term 'entropy'4 , completed with conceptual vigour the work of Watt and then Thomson, by taking the irreversibility as an initial data and not as a by-product of mechanics. By inventing a cycle composed of isothermal and adiabatic processes, Carnot mimicked, in the field of caloric technology, the equilibrium of Lagrangian forces with the hypothesis that the state of matter can split the standard space-time (Stengers). The principle of causality does not correspond only to the laws of nature but to a will which drives the action and faces the reaction. Even if the system is opened it acquires a new kind of universality. Contrary to the ideal regular pendulum which denies the difference between the cause and effect in conserving one in another, a clock, introducing the irregularities of mechanical world and the correlations between the irregularities, is a device based on non harmonic relation between space and time     

1 / /  c v u  
. Notice that there is no need of the presence of the observer in order that the space-time geometry gives rise to the irregularities. It suffices that the homogeneity of space time, postulated by Noether to disclose the invariance of energy, be broken in some catastrophic circumstances. Independent from the presence or not of the observer, the internal energy of a clock is not conserved. Many years will be necessary for Clausius and Thomson to shed light on this question and to introduce the concept of entropy TdQ dS  as a state function. In the analysis given above, the group of transformation associated to the dual singularity plays the same role as the Carnot's cycle and therefore the dual singularity contains some quantum of entropy and negentropy. Many decades will be yet necessary for understanding this assertion as well as many others related to the singularity of the topological opened punctuation.

The geometrical irreversibility is a major characteristic at least for the systems affected by scaling laws. This is exactly what shown by the battery, a 'faitiche' of the thermodynamic counter culture (Le [START_REF] Méhauté | Batteries as fractal object[END_REF]. The engineering of the battery, -a controllable source of dissipation associated with the fractal geometry of the electrodes , hence a seed for irreversibility-, offers a right experimental way to understand the importance of the fractional dynamic in the world of energy. The engineering problem of battery optimization for example, makes the geometrical structure of the matter a deep source of adjustable dissipation. Hence it is requiered to consider the fractionality of the dynamics of transport and of exchange. The -dynamics contains all the appropriate singularities for the emergence of dissipation and of the associated scaling laws. Among these laws, the Peukert law (Le Méhauté 1989) which points out a scaled covariance of space-time (Nottale) and the emergence of new type of invariance with respect to the time (see also WLF laws or the relation temps/temperature). The time, as a sensor, is in practice a parameter of exploration of the Riemannian manifolds affected by singularities. These manifolds cannot be reduced to their local Euclidean projection. In this simple state of equilibrium -or of stationary-their surfaces vibrate according to the well defined normal modes even if these modes are of infinite number (Hilbert space) and controlled by the boundary conditions. This first approximation forgets the cuttings, the collages, and other overlapping of surfaces, or, in a word, the singularities which give rise to dissipative structures. The fractal based manifolds authorize the use of -fractional equation to disclose the 'anomalous' behaviour. It was a chance to have to our disposable devices (batterie, tires, cables etc), but also singular matter (supraconductivity and fluidity or monopoles) to strengthen the new point of view proposed herein and to check it experimentally. In spite of the lack of mathematical studies on the topology associated with fractal geometry, these devices and this singular matter offer an opportunity to disclose the relationship between -dynamic, geodesic and tiling and to emphasize its role in the dissipation of energy and even in some circumstances with the decrease of entropy (emergence). A lot of work must still be done with the fractional operator opening new scientific territories which have to be explored. It will be done if we just drop the umbrella of the Hamiltonian mechanics which tells us nothing about our freedom of creation and nothing but a beautiful immobile eternity of a world without life and creation (Stengers).

Figure 1 :

 1 Figure 1 : Hyperbolic tiling of a complex plan using of dynamical functions on

Figure 2 :

 2 Figure 2 : 3D representation of the empirical variables for the Maxwell equations of standard electrodynamics.The time is necessary for the expression of flux and moments. However, it cannot be extended without giving a sense to the notion of velocity. The velocity appears as the key factor to give a meaning to space time differentiable manifold and even to give a meaning to external algebra, that is to rise the concept of matter (here seen through L and C)

Figure 3 :

 3 Figure3: Graphic representation of the variables of all physical process according toVieil (2009). One notices the disappearance of spatial gradient with respect to the Figure2. This diagram authorizes the extension of the mechanic concept to other field of the thermodynamics (electricity, chemistry, surface and thermal properties). Far from that point it authorized the analysis of the coupling between different fields (Energy conversion).

Figure 4 :

 4 Figure 4 : Double representation in   t L, and in       i Z , (

  [START_REF] Méhauté | and Fillipi Introduction au transfert d'énergie en milieu fractal[END_REF]. The diagram 5 shows this particular mode of representation in the space   t L, conjugated to the space 

Figure 5 :

 5 Figure 5 : Elementary form of the fractional transfer function of order  (impedance) on the basis   t L,

Figure 7 :.

 7 Figure 7 : A schema showing the approach leading to the zeros of the zeta function of Riemann from the expressions of different ) (  Z

.

  iii) a quasi-commutative exchange between both 'transfer functions' Z (discreet or not), subject to the introduction of the irreversible terms R which belongs to R takes into account the shift associated with the non commutative permutation of   t L, , that is, the non commutative relation between diffusive constant distribution of prime number. Due to the introduction of a shift into the exchange t L  irreversibility is therefore a factor of the discreet space time structure when =1/2, that is at the degenerate state. There is Time/Temperature sub-relation due to incomplete algebra of space time(Nivanen).

  BIBLIOGRAPHIEAlbert D, Time and chance, Harvard University Press London 2000 Arnold V.L., Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989 Badiali J-P., Relation time thermodynamics. A path integral approach . arXiv 0902 [V] [Cond. Mat.Stat.Mech.] 5 fevrier 2009. and Questions about the time, to be published Bergé P., Pomeau Y., Vidal C., L'ordre dans le chaos Hermann paris 1984 Bedford T., Keane M., Series C. Edt. Ergodic theory, Symbolic dynamics and hyperbolic spaces, Oxford Science Publication

  This figure contribute to the geometrical representation of the operator Laplacian u

																												x
	(	(	. div	grad	( u	)		rot	. rot	(	) ) A	) as well as to d'Alembertian	(	c	2		x 	d	2 t	) u	(	x	,	t	)	and give rise to
	normalizable forces	F			qE		JB	. The limit conditions impose, among other things, the
	continuity equations	dt dq	 divJ		0	upon the dynamics. Geometrically, this results in the laws
	formulated through the Maxwell equations. Hence we find	0

Table 1 :

 1 Differential operators of different integer order for different dynamical processes.

Corresponding author: alm@ismans.fr

This conservation is crucial for most of the Hamiltonian systems, although a hamiltonian system may not conserve its energy (Arnold) in some special cases where there are, for example, nonconservative forces.

'Entropy' was invented from the Greek words en meaning in and tropē meaning transformation or turning towards implying dissipation of mechanical energy into non available energy.