J.-B Conrardy

J Olivier

J Vaxelaire

ELECTRO-DEWATERING OF WASTEWATER SLUDGE: FACTORS AFFECTING KINETICS AND ENERGY CONSUMPTION

Keywords: Dewatering, Electroosmosis, Energy Consumption, Activated Sludge, Sewage Sludge

The treatment of wastewater produces huge quantities of residue (sludge) which contains a large amount of water. A dewatering step is usually carried out prior to further treatment in order to diminish sludge volume and consequently reduce handling and disposal cost. It is commonly conducted thanks to mechanical processes (plate or belt filter presses, centrifuges) which offer a relatively poor efficiency. The water content at the end of the dewatering usually ranges between 65% and 85%. A way to improve the dewatering efficiency is the implementation of an electric field during the mechanical dewatering (electro-dewatering). It has been shown that applying an electric current through the filter cake could reduce significantly the water content of sludge, down to 40%.

A lab-scale device of electro-dewatering has been used to study the phenomena involved in this hybrid process. The experimental set-up consists in a cylindrical chamber where a piston presses the sludge cake on a filter cloth. Two electrodes, one anode and one cathode, are placed at the piston head and under the filter cloth, respectively. Dewatering kinetics and electrical behavior of the sludge cake can be studied with this set-up.

The results of experiments carried out with a constant voltage (or a constant current) show significant correlations between the filtrate flow-rate, the electric current, the electrical resistance of the filter-cake and the water content of sludge. The electric current and the water content of the sludge are found to be key factors that govern electro-dewatering kinetics. Moreover, the electrical resistance through a given cake of sludge is highly correlated to its water content (which decreases during the process). The relationship seems to be valid for a wide range of water content. These results enable a better understanding of how dewatering efficiency and energy consumption are related in the electro-dewatering process.

INTRODUCTION

Wastewater treatment plants generate a very large amount of sewage sludge. The cost of sludge disposal depends directly on volumes to be treated and thus on their water content. To reduce the volume, the sludge is usually mechanically dewatered (centrifuges, plate filter presses, belt filter presses...) but the performances of these processes are generally poor [START_REF] Vaxelaire | Moisture distribution in activated sludges: a review[END_REF]. It has been shown that the implementation of a direct electric current coupled to a mechanical pressing (electro-dewatering) can lead to a considerable reduction of sludge moisture content (Mahmoud et al, 2010[START_REF] Mahmoud | Advances in mechanical dewatering of wastewater sludge treatment[END_REF][START_REF] Tuan | Sewage Sludge Electro-Dewatering Treatment-A Review[END_REF]. Different experimental procedures were proposed in the literature to study the electro-dewatering of various materials. Two classical operating modes, constant voltage or constant current, were generally used, but only few studies proposed a direct comparison between these two operating conditions. Authors usually show that the dewatering kinetics is enhanced when increasing the current or voltage [START_REF] Lockhart | Electroosmotic dewatering of clays. I. Influence of voltage[END_REF][START_REF] Iwata | Analysis of constant-current electro-osmotic dewatering of various solid-liquid systems by considering the creep deformation[END_REF][START_REF] Mahmoud | Electro-dewatering of wastewater sludge: influence of the operating conditions and their interactions effects[END_REF] and that the maximal final dryness of the filter-cake is higher with higher voltage. Nothing has been clearly stated concerning this behavior with tests carried out at constant current (they are often stopped before the maximal final dryness was reached). This is generally due to an excessive rise in temperature in the cake (ohmic dissipation). The electrical quantity which naturally makes a link between the voltage and current is the apparent electrical resistance of the system (electrodes + cake). The present work shows the influence of electrical parameters (current, voltage, moment of electric coupling) on dewatering kinetics. It also studies links between the tests carried out with the two operating modes (constant current or voltage). Moreover, the energy consumption is discussed as part of the overall performance analysis of the dewatering process.

MATERIALS AND METHODS

Experimental unit

The experimental set-up used for the study was previously described by [START_REF] Mahmoud | Electro-dewatering of wastewater sludge: influence of the operating conditions and their interactions effects[END_REF]. It consists in a filtration/compression cell working with a pressure controlled piston moving in a cylindrical vessel of 70 mm inner diameter. The device presses the material to be dewatered against a filter cloth (SEFAR TETEX MONO SK025, CHOQUENET S.A.S, France). The piston and the vessel were made of nonconducting Teflon™ in order to ensure electrical insulation. Two perforated disk electrodes were placed at both ends of the compression chamber. They were designed by De Nora (Italia) and supplied by ECS (Electro Chemical Services, Saint-Genis-Pouilly, France). The anode was inserted against the piston head. It was made of titanium coated with mixed metal oxide (MMO) to prevent its oxidation. The cathode lay under the filter cloth. It was made of titanium. Both cathode and filter cloth were held by a perforated grid made of Teflon™ which was maintained by stainless steel external jacket to ensure the mechanical resistance of the unit.

A DC power supply EV202 CONSORT (maximum 300 V and 2 A), operating under constant voltage or constant current, was connected to the anode and cathode electrodes. The pressure applied to the piston was controlled by a pressurized air system. The mass of filtrate was collected in a beaker and measured against time with a weighing scale connected to a computer. Two digital multimeters (ISO-TECH IDM 73) were used to control voltage and current fluctuations in the electro-dewatering cell.

Sludge

The electro-dewatering tests were carried out with two different types of sludge whose characteristics are listed in Table 1. Activated and primary sludge were collected in two wastewater treatment plants, from aeration tank and primary settling tank, respectively. After transportation to the laboratory, the initial dryness of the sludge was determined by drying at 105°C during 24h . The sludge was stored at 4 °C and used over a period of less than one week in order to limit the variations which may be caused by biological mechanisms. Before each dewatering test, the sludge was conditioned with a cationic polymer and thickened. The polymer was supplied by SNF Floerger. The use of two different types of sludge enables to widen the interest of the results presented in this paper.

Operating procedure

Before each test, the sludge sample was left about two hours at room temperature to reach this temperature. The cationic polymer was added to the sludge, under stirring at 300 rpm for 20 s with a JAR TEST device. Then, the flocculated sludge was thickened by gravity drainage on a lab-scale device [START_REF] Ginisty | Influence of flocculation on sewage sludge thickening and dewatering, 3rd ECSM[END_REF]. The dryness reached after thickening was approximately 6.9% for primary sludge and 5.2% for activated sludge.

For standard dewatering test, the thickened sludge sample was introduced in the filtration/compression cell. A mechanical pressure of 5 bar was applied to the piston without electric current (single mechanical compression) for 2 hours. Then the electric current was applied, either by setting the desired voltage or the desired current intensity for a maximum time of two hours. The pressure was maintained to 5 bar during this step. At the end of the first step the cake dryness reached approximately 20% for the primary sludge and 12% for the activated sludge.

During tests carried out under constant current intensity, significant heating was generally observed as process progressed. Experiments were stopped when the temperature at the head of the piston (anode side) reached 80 °C. The dryness of the sludge at the end of the test was measured after a drying at 105 °C for 24 h. The dryness evolution was calculated from the evolution over time of the mass of filtrate.

With m DS the mass of dry solids and m L the mass of water in the sludge cake (which is a function of time).

RESULTS AND DISCUSSIONS

Two electro-dewatering phases

A first series of experiments was carried out with the primary sludge for different voltages (30 V, 40 V and 50 V) and different current intensities (100 mA, 200 mA and 300 mA), respectively. The evolution of the cake dryness over time and the electrical behavior of the system (variations of electric current and voltage respectively) are presented on Figure 1. According to other results presented in the literature [START_REF] Gingerich | Electroosmotically enhanced sludge pressure filtration[END_REF][START_REF] Mahmoud | Electro-dewatering of wastewater sludge: influence of the operating conditions and their interactions effects[END_REF][START_REF] Citeau | Pressurised electro-osmotic dewatering of activated and anaerobically digested sludges: electrical variables analysis[END_REF] both, the current and voltage play a major role on the dewatering kinetics. Tests carried out at constant voltage shows a rapid dewatering kinetics at the beginning of the process but this kinetics weakens over time. Tests conducted with a constant current intensity show more steady dewatering rates. When looking at the electric current curves (voltage respectively) it is possible to analyze the electro-dewatering process with two distinct phases. During the first phase (phase 1) the current intensity (respectively the voltage) varies with no obvious trend. Then a second phase (phase 2) can be observed where the temporal variations become monotonous. During phase 2, for tests performed at constant voltage, the current intensity decreases over time (Figure 1c) while the voltage increases continuously for tests conducted at constant current (Figure 1d). The transition between the two phases is shown by arrows on Figures 1c and1d. The analysis of the process with different phases was already proposed by [START_REF] Mahmoud | Electro-dewatering of wastewater sludge: influence of the operating conditions and their interactions effects[END_REF] but only from tests carried out at constant voltage. Very similar results were also observed with activated sludge as shown on Figure 2. The dewatering kinetics followed the same trend with respect to the intensity of the electric current: higher currents (or voltages) led to a faster dewatering. Transitions between phase 1 and phase 2 were even more marked (arrows on Figures 2c and2d). From Figure 1 and 2, it seems that the duration of phase 1 depends on the current intensity (or voltage respectively). This phase was shorter when the applied current intensity (or voltage) was higher. When the electric current is switched on, many phenomena may take place such as the starting of electrode reactions and migration of charged species (ions and other charged particles) from one electrode to the other through the sludge cake. The first phase appears as a transient stage of the process before a more steady state (phase 2). Finding the same behavior with the two types of sludge tends to show that the presented results are typical characteristics of the electro-dewatering process itself.

Electric current and filtrate flow-rate

As it was mentioned previously, it seems that a higher current intensity leads to a faster dewatering, even for tests conducted with a constant voltage (in this case the dewatering rate decreases with the current). To complete this result the filtrate flow rate was plotted as a function of the electric current for tests carried out at different constant voltages (Figure 3). The transition between the first and the second phase is represented by an arrow. As it can be seen on Figure 3a no obvious relationship between filtrate flow rate and electric current appeared during the first phase of the electro-dewatering process. At the opposite, during the second phase, a rather strong correlation between the decrease of the electric current and the decrease of the filtrate flow rate was observed. For a given current intensity, the filtrate flow changed in a rather limited range of values. The correlation was found to be valid for the two types of sludge studied in the present work. This result tends to show that this behavior is a process characteristic which does not depend on the sludge origin. Consequently, during the second phase the control of the dewatering kinetics seems to be achieved by the electric current. This behavior can be explained by the fact that the electro-osmotic phenomena are related to the displacement of ions (mainly cations in the case of sewage sludge), which move as the electric charges are released from the electrodes. As the electric current controls the amount of charges released, it is therefore not surprising that electric current play such an important role in dewatering kinetics. However, this result admits some limits. Indeed, the tests carried out at a constant current intensity do not lead to a constant dewatering rate. The filtrate flow rate decreases during the process and cannot depend only on the current intensity.

The influence of other parameters such as the cake dryness and the history of the dewatering process needs to be investigated, as it is proposed in the next section.

Filtrate flux and dryness

The electric current intensity has been shown to be a key factor which impacts significantly the dewatering. However, other parameters such as cake dryness may influence the electro-dewatering. In that way, a series of experiments was carried out with a constant current intensity (300 mA) to study the influence of the dryness of the sludge cake on the filtrate flow rate. The electric was applied to the mechanical process at different times (i.e. at different cake dryness).

Figure 4 shows the variation of the filtrate flow-rate against the filter-cake dryness. Electric current was switched on at different moments, after 2 hours (standard), 30 minutes, 10 minutes of mechanical pressing and also from the beginning of the dewatering process. This series of experiments showed that the filtrate flow rate increased when an electric field was applied. The electric current enables to generate higher flow rates in comparison to a single mechanical pressing, and also higher final cake dryness. Finally, two main curves can be analyzed. The one without electric current (single mechanical pressing) and the one corresponding to the electro-dewatering process (pressure + electric current). It is striking to observe that there were only these two curves and that the flow rate of filtrate "jumped" from one to the other for each experiment. For a given current intensity the flow rate depended only on the filtercake dryness, regardless of the time when the electric current was applied. This result shows that the filter-cake dryness is a key factor affecting the dewatering kinetics as well as the current intensity.

Electrical resistance and dryness

To compare tests carried out respectively at constant current intensity and constant voltage, it was convenient to find a parameter which could be common to these two operating modes. The apparent electrical resistance of the cell (voltage across the sludge cake divided by the current intensity) seemed to be an interesting parameter for this purpose. Consequently, the evolution of this resistance was studied with experiments carried out with various operating conditions: different voltage or current, two types of sludge, different times for the applying of the electric current (during the full dewatering process or after 2 hours of single mechanical compression).

Influence of electrical current or voltage

Figure 5 shows the change in electrical resistance against cake dryness for tests carried out with activated sludge, when the electric field was applied after 2 hours of single mechanical pressing. Figure 5a enables to compare experiments conducted with different currents, namely 200 mA, 250 mA and 300 mA. Similarity between the three experimental curves appears excepted at a low dryness where a discontinuity in the resistance curve can be observed. This "jump" of resistance corresponds to the transition between the two phases mentioned in paragraph 3.1. For the three experiments the dewatering kinetics was different but the relationship between electrical resistance and cake dryness remained the same. Figure 5b shows experiments carried out with different constant voltages, namely 30 V, 40 V and 50 V. The data obtained for both, 40 V and 50 V, were very similar showing a good correlation between the electrical resistance and cake dryness. For experiments conducted at constant current, a single correlation between electrical resistance and sludge cake dryness was also observed although the dewatering kinetics were different in the two experiments. The curve resulting from the experiment conducted at 30 V also fitted the two other ones for a cake dryness ranging between 25% and 45%. Then the electrical resistance increased faster than it did with the two other voltages. It can be assumed that this excessive increase in electrical resistance was due to a poor electrical contact at the cathode. This latter was not in direct contact with the sludge cake and might suffer from a gradual drying when the flow rate was too low (due to low current intensity or voltage). While the cathode dried, the electrical contact became poorer. Otherwise, it is also possible to observe on Figure 5b the transition between phase 1 and phase 2, but the transition is less noticeable than on Figure 5a. The comparison of the whole data showed that the variations of the electrical resistance with respect to cake dryness were very similar from one test to another. The overall resistance decreased at the beginning of electro-dewatering and then increased until the process stopped. The good agreement between the data seems to confirm that the electrical resistance depends slightly on the selected operating mode (constant current or voltage). Furthermore the history of the process (single mechanical pressing or combination with an electric current) does not influence the variation of the electrical resistance. Finding correlations between, respectively, kinetics parameters such as sludge cake dryness or flow rate of filtrate, and electrical parameters (electrical resistance and current intensity), is crucial to understand the link between the dewatering kinetics and the energy consumption of the process.

Overall performance of the process

The electro-dewatering process can be more or less time and energy-consuming. It is possible to determine the overall performance by taking into account these two aspects. Finding a good overall performance is a way to determine what the best operating strategy is to dewater the sludge cake efficiently. By varying the time when the electric current is switched on, the overall time and energy consumption of the process change. The earlier the electrical current is switched on in the compression stage, the faster the kinetics is. However, from an energy point of view, there is no specific need to consume electrical energy while a single mechanical compression allows itself to extract significantly the water. But, if it is assumed that energy expense can reduce operating time, it is thus interesting to define a cost function witch is proportional to the process time and to the spent energy (electrical energy mainly).

,

This cost function can be plotted against the cake dryness for activated sludge (Figure 7). Tests were carried out at constant voltage (50 V) and constant current (300 mA), respectively. For both cases the electric current was switched on at different times of the compression stage. The data obtained show that reaching higher cake dryness leads to higher costs. For tests conducted at constant current intensity the lowest cost (i.e. the best time / energy ratio) was observed in the case where the electrical current was applied after 10 minutes of single mechanical compression (Figure 7a). This optimum value was found according to the chosen cost function. For tests conducted at constant voltage, the lowest cost was obtained when the electric current was applied during the full process (Figure 7 b). Obviously, this was verified only for dryness higher than 13% (which is the maximal dryness reached by mechanical compression). These results may be explained by the fact that the electrical resistance throughout the system was high at the beginning of compression stage. Consequently, the electric current flow through the filter-cake was energy-consuming, especially when the effect of an electric current addition was slight on the dewatering rate. For tests carried out at constant voltage the process was self-balanced. The electric current intensity was low when the electrical resistance was high. As the sludge cake thickness decreased the current increased due to a drop in electrical resistance. At this time the benefit of the electric current was more significant and the process less energy-consuming.

CONCLUSION

The presented work shows with two types of sludge (primary sludge and activated sludge) how electro-dewatering process operates. The tests carried out were used to study the electrical behavior of the sludge over process time. The presence of two distinct phases was pointed out. The large impact of the current intensity through the sludge cake on the dewatering kinetics was demonstrated and a correlation with the filtrate flow rate was highlighted. Furthermore, a comparison between the tests conducted at constant voltage and at constant current was possible thanks to the analysis of the electrical resistance of the system. A strong correlation between the electrical resistance and the filter-cake dryness was established. These data on electrical resistances and filtrate flow rate allowed to point out links between dewatering kinetics and energy consumption of the process. Besides, the overall performance analysis of the process has enabled to find a balance between a fast dewatering and a too high energy consumption.

Figure 1 .

 1 Figure 1. Dryness versus time for tests at constant voltage (a) and constant current (b). Current variations for tests at constant voltage (c) and voltage variations for tests at constant current. Primary sludge.

Figure 2 .

 2 Figure 2. Dryness versus time for tests at constant voltage (a) and constant current (b). Current variations for tests at constant voltage (c) and voltage variations for tests at constant current. Activated sludge.

Figure 3 .

 3 Figure 3. Mass flow of filtrate plotted against electric current for tests at constant voltage with: (a) primary sludge, (b) activated sludge (phase 1 is partially cut for a better reading of the graph)

Figure 4 .

 4 Figure 4. Mass flow rate of filtrate versus sludge cake dryness. With and without electric current.Electric current is added at different times. Activated sludge.

Figure 5 .

 5 Figure 5. Electrical resistance versus dryness for experiments at constant current (a) and constant voltage (b). Activated sludge.

  3.4.2 Influence of the electro-dewatering mode and of the coupling timeThe variation of electric al resistance against cake dryness was plotted on Figure6, for experiments carried out with different operating conditions (constant voltage, constant current intensity, and different times to apply the electric field).

Figure 6 .

 6 Figure 6. Electrical current versus sludge cake dryness. Comparison between constant current and constant voltage modes. Different times where electric current is switched on. Activated Sludge.

Figure 7 .

 7 Figure 7. Cost function plotted against sludge cake dryness for tests at constant current (a) and constant voltage (b) with different time when the electric current is switched on. Activated sludge.

Table 1 .

 1 Sludge characteristics

		Primary sludge	Activated sludge
	Origin	Saint-Thibault-des-Vignes, France (77)	Lescar-Pau, France (64)
	Initial dryness (%)	1.9	0.5
	Mass per sample (g)	500	2000
	Type of flocculant	EM 540 L	EM 640 TBD
	Dosage of flocculant (g/kg of dry matter)	8	8


Alain L'hostis
email: alain.lhostis@ifsttar.fr

Detour and break optimising distance, a new perspective on transport and urbanism

Introduction

Distance, a central concept in geography, can also be seen as a key concept in urbanism and planning if we consider Offner's statement that the task of the planner or the urbanist consists in adjusting distances between functions [START_REF] Offner | L'urbaniste ? Un ajusteur de distances[END_REF]. In the spirit of the definition of urbanism as establishing distances between urban functions, the distances of urbanism cover interventions on transport systems, actions on the spatial distribution of functions in urban and regional spaces, and joint action on both domains such as policies focusing on the transport-land use interaction [START_REF] Webster | Urban land use and transport interaction; policies and models[END_REF]. As Hall stated (1969), distance is at the heart of human, individual and social behaviour. But, if we consider the relatively short list of sources having focused on distances with a theoretical ambition [START_REF] Brunet | Les sens de la distance[END_REF][START_REF] Deutsch | A note on a generalized concept of effective distance[END_REF][START_REF] Gatrell | Distance and space: a geographical perspective[END_REF][START_REF] Hall | The Hidden Dimension Anchor Books[END_REF][START_REF] Huriot | Minimum-Cost Distances in Spatial Analysis[END_REF], it has received less attention than we could expect for such a central concept, despite potential insightful implications for transport, urbanism and spatial planning. We propose here to establish a new perspective on transport and urbanism based on a renewed analysis of distance and of its properties.

This article develop two contributions on geographical distances. Geographical distance is a measurement of spacing between places and refers to the possibility to reach places. This relation to movement links geographical distances to routes in transport systems. The hypothesis that all geographical distances are optimal has been stated (L'Hostis, 2014[START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. Distance influences the decision to realise or to avoid a trip, influences trip choices, influences all kinds of social interactions. For planners and urbanists, who have to solve issues with limited resources, this means that distance, as it conveys an idea of a search for optimality, can be used to take decisions. Furthermore, the analysis of triangle inequality, which is one of the four mathematical properties of distances or metrics, along with positivity, separability and symmetry, and whose role is to ensure the optimal character of distances, reveals key aspects of distances and of geographical spaces [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. Assuming that distances differ from Euclidean straight line is considered as a necessary approach for urbanism and spatial planning [START_REF] Cao | Urban planning in generalized non-Euclidean space[END_REF][START_REF] Friedmann | Toward a non-Euclidian mode of planning[END_REF][START_REF] Hillier | Plan(e) Speaking: a Multiplanar Theory of Spatial Planning[END_REF].

We will now develop the idea that the optimum of distances is, rather counter-intuitively, supported by the existence of detours and breaks in trajectories and movements, and relies on the analysis of the three errors of interpretation of triangle inequality [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. We will illustrate these three principles with concrete situations in the domains of urbanism, spatial planning and transport.

First statement: distances carry an idea of optimality

The first error of interpretation of the violations of triangle inequality consists in considering that distances can be represented by sub-optimal measurements, as in the example given by Haggett (2001: 248). This error resides in considering a sub-optimal measurement of space between locations as equivalent of a measurement of distance, despite its non-metric nature. Some authors see distance as a neutral concept, a geometrical abstraction not directly related to human behaviour. Conversely, we follow L'Hostis's demonstration (L'Hostis, 2014[START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF] that geographical distances are always optimal. In order to address this error of interpretation, we propose to state that distances contain the idea of optimum and that any distance is associated to a search for optimisation.

This first statement entails optimising routes of all kinds, and viewing them as supports of movement and hence as subject to an optimisation process.

An illustration of this principle can be read in the arbitrations realised in urban space design which has become a strategic issue for cities [START_REF] Madanipour | Why are the design and development of public spaces significant for cities?[END_REF]. We consider the case of the surroundings of the Quatre Cantons Grand Stade metro station in Villeneuve d'Ascq (FR) showing that not only planners create spaces but users also participate. A renewal of public space in 2012 adopted the design of this large open space in order to connect the metro station to the university campus, a new stadium and a technology park. Figure 1: Four cities with nonoptimal separation measurements, violating triangle inequality (Haggett 2001, 248) The design opted for an open space to facilitate pedestrian mobility in access to the metro station. The design creates, favours, incites, curbs or lengthens distances; but the user also contributes to the design of space. University residences visible through the row of trees along Avenue Poincaré house numerous metro users. A door in the fence that allows for direct pedestrian access existed before 2012. Perpendicularly to this access, users maintain a desire line visible in the bare ground of the central reservation of Avenue Poincaré. Sociologists have introduced the term of desire line to describe the visible path generated by walkers in landscapes (Simmel, 1997: 171). These desire lines reveal the tensions in the formation of distances, most often resulting from a compromise between least effort and the risks associated with road safety [START_REF] Moody | Shared space: Research, policy and problems[END_REF].

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A rather extreme choice was made in the design of the urban space of boulevard Montebello in Lille (FR): following several road traffic related casualties of pupils in Wazemmes high school on this very busy artery of Lille, thorn bushes in the middle and fences on the side were introduced to prevent straight line and least effort trajectories to the entrance of the high school. the expense of a detour, a safer crossing of the road. The principle of optimality of distances expresses a recommendation to follow, or at least not to depart too much from, desire lines, figures of optimisation, and of solidification of social relationships (Simmel, 1997: 171). Optimality in the pedestrian routes design extends beyond the mere need to pass and to ensure safety. In Quatre Cantons, the designed pedestrian crossing favouring a safe intersection with automobile flow, at the expense of detour for pedestrians, represents a situation where the principle of optimality has not been fully applied. The choice made in favour of safety, and detrimental to optimality, has entailed the continuation of an unplanned practice of public space: the re-emergence of the desire line on the central reservation of Avenue Poincaré, away from the designed, safe pedestrian crossing, shows that the search for optimisation can prove stronger than the invitation to safety proposed by public space design. The objective of safety by design is challenged by practices, and the overall performance, if assessed, would probably be poor. A corrective intervention of the kind of those implemented on motorway-type axis in urban environment, or the one shown in Boulevard Montebello, would be a renunciation to the principles of legibility and openness of space design.

These two examples express clearly the tensions in the design of public space with implications on pedestrian urban mobility, between an open, free and legible space and the principle of separation of flows [START_REF] Buchanan | Traffic in towns[END_REF] inherited from functionalistic urbanism [START_REF] Dupuy | L'Urbanisme des réseaux[END_REF]. The principle of optimality of distance is expressed in the multi-criteria arbitration that opposes straight line to road safety concern.

The second illustration of distance optimality is provided by the analysis of public transport user maps. The layout of a transport network carries an intention of optimality when the user plans his trip. The promoters of the public transport network promise an efficient connection between places, and the user establishes his itinerary by mentally combining coloured lines and nodes that have become conventional for this kind of cartography. The famous London tube map proposed by Harry Beck in 1933 has been praised for its simplicity -its exceptional longevity proving its success-but has also been criticised for being misleading [START_REF] Roberts | Objective versus subjective measures of Paris Metro map usability: Investigating traditional octolinear versus all-curves schematics[END_REF]Scanlan, 2004: 394). The graphical harmonisation of inter-station distances introduces deformations of the topographic map by dilating the centre and compressing the periphery. This deformation gives the false impression that remote locations in the periphery are closer to the centre, and that central stations are further away than they are in topographic space. It is generally accepted that the misleading aspect of the Tube map has contributed to the success of the public transport system, especially among the users residing in the remote periphery [START_REF] Sinnett | Making the case for investment in the walking environment: A review of the evidence[END_REF]. The process of harmonising inter-station graphical distances visually forces the optimal character of centre-periphery itineraries. But, at the same time, it induces the representation of suboptimal routes in the central area. It has become necessary to correct a part of the practices induced by the plan [START_REF] Sinnett | Making the case for investment in the walking environment: A review of the evidence[END_REF] by convincing passengers that between two metro stations located in close by areas, alternative transport modes, especially walking, can be more efficient. car and also by public transport [START_REF] Sinnett | Making the case for investment in the walking environment: A review of the evidence[END_REF]. All public transport networks tend to be saturated in the densest parts of cities. Walking can replace short trips by public transport in the densest areas, providing travellers can consider using alternatives to public transport (Lavadinho, 20094). In Paris and London, most public transport users tend to believe that walking will never be as fast as public transport [START_REF] Lavadinho | The multimodal walker. Potentials for combining walking and public transport at the agglomeration scale[END_REF]. Nevertheless, when a metro delay reaches 15 minutes, the duration of pedestrian routes becomes competitive in the case of London; in Paris this time lapse allows walking the distance of three metro stations (Lavadinho, 2009: 5).

As the map induces sub-optimal routes, it also generates non-optimal practices (L' [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]; by restricting possible choices it shows routes that are not distances because more efficient alternative itineraries exist. Correcting these sub-optimal behaviours supposes an action on street network design and on communication towards public transport users in order to indicate more efficient routes that avoid public transport.

The third illustration is found in works on transit oriented development [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. Developing an urban form favouring the use of railway means, among other actions, increasing urban density around access points of the railway system. In this aim, the Bahn.ville 2 project in the Saint-Étienne (FR) urban region identified the need for a joint observation tool on transport and urban development [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. A central instrument of this observation tool consists in monitoring land and real estate transactions happening inside a ten-minute walkable radius around railway stations [START_REF] Bertolini | Cities on rails: the redevelopment of railway station areas (E & FN Spon, London) Boumoud A, 2012 La grande gare contemporaine et le labyrinthe du transport multimodal: vers une nouvelle approche de la lisibilité[END_REF].

In establishing the ten minutes walkable radii, we distinguish a theoretical measurement as the crow flies and an empirical measurement through the street network. The gap between these measurements indicates areas that could obtain a better access to the railway station providing the pathways were improved. The association of the issues of pathway improvement and of land monitoring for urban densification led the Saint-Étienne urbanism agency (EPURES) to extend the monitoring process to pieces of land that are too small to foster new buildings, but that could serve to improve public space and bike and pedestrian routes. As we see, the action on land is orientated towards an optimisation of pedestrian and bicycle distances which represents a strategic issue for railway station areas. This example shows that the issue of distance optimisation represents one possible levy in the implementation of an urban project.

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Second statement: detours are most often due to an optimisation process

The second error of interpretation of violation of triangle inequality, considering non-rectilinear routes as violation cases (Ahmed and Miller, 2007, page 4;Cauvin, 1984, page 62;Golledge, 1999, page 8;Müller, 1982, page 191;Perreur, 1989, page 133). It consists in a confusion between straight line and optimal route (L [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. This error is associated to the issue of detour, to its status in transport and mobility systems. Is detour an anomaly that should be corrected each time it occurs? Some detours are indeed undesirable, but most of them correspond to a search for optimisation of distances of routes through transport networks. To deal with this error we propose to state that detour in distances is most often a means to optimise movement.

In the general frame of optimality, the idea of detours in distances raises a set of issues that urban planning and urban design has to take into account.

In the scientific literature on transport, detour is, in most cases, seen as a difficulty in moving, either a difficulty imposed by the network or an unnecessary difficulty. In a work on the urban street network and the fragmentation effect of infrastructure, Héran shows that a part of detour is unavoidable, due to the mere presence of buildings and, more generally, to the existence of spaces dedicated to functions other than mobility [START_REF] Héran | Des distances à vol d'oiseau aux distances réelles ou de l'origine des détours[END_REF]. For Héran an incompressible detour lengthens routes by an average 15 to 25 %. But another component of detour, due to the fragmentation effect of infrastructure and traffic, could be reduced, especially concerning active transport modes. In his analysis of urban distances, Héran distinguishes an average normal detour and a supplementary detour that could be reduced. When the street network is sufficiently meshed, detour remains limited. In this context detour is seen as a waste of energy, a loss of efficiency and a handicap for transport modes based on muscular strength. A similar analysis can be found in the literature in ethology [START_REF] Schmidt | How desert ants cope with enforced detours on their way home[END_REF]. Detour gains a positive sense only in the domain of economics, with the famous roundaboutness that describes an investment devoted to a future increase in production. This figure of detour also possesses a positive sense in learning or narration processes; all these contexts are distinct from routing in geographical space. In our domain, detour is generally seen as a problem, an obstacle or a burden in the first place. If detour becomes positive, it turns into a paradoxical idea.

The first illustration of the importance of detour in the optimality of distances, is the study of potential partners of the high-speed rail line project of the Îlede-France south bypass. This infrastructure project is totally included in the Île-de-France administrative region boundary; but five regions in western France have expressed interest in the project (Conseil Économique, Social et Environnemental des Pays de la Loire, 2010), and are potential funders, despite their location very far away from the infrastructure project. The fact that this equipment favours access to the high-speed rail lines of the north, east and south-east, from cities in western France explains the interest expressed by western administrative regions. This infrastructure will develop detours in distances by generating shorter time-distance optimum routes.

More generally, the extreme form of detour is space inversion identified by Tobler (1961: 106) and Bunge (1962: 172). As shown in figure 8, space inversion describes the fact of starting a trip in a direction opposite to the final destination in order to access to a faster or more efficient transport mode. This phenomenon means that an intermodality equipment -such as an airport-located far away will contribute to optimising distances from a given location. Usually, spatial planners and decision makers focusing on the accessibility of a given territory will tend to focus on local equipment, like airports, motorway entrances or railway stations, to improve the distances from and to external spaces. Conversely, space inversion shows a situation where equipment contributing to the external access can be located outside the territory it serves, and hence outside the space of intervention of the decision makers of this territory. In order to illustrate this case we consider three imaginary cities A, B and C during three historical periods before 1950, up to 1990, and beyond. From a situation where road and classic railway networks coexist, the development of high-speed transport leads to the introduction of the aerial mode and the motorway in the 1950s. The second phase of high-speed consists in the introduction of high-speed rail from the 1990s. During the first two phases, the shortest route from city B to city C by road, classic rail or motorway goes directly from B to C. But in the last period, the fastest route is by high-speed rail to A followed by a flight to C. The initial part of the route goes in the opposite direction, according to the space inversion principle. What are the facilities that contribute to the accessibility of the intermediate city? In the first two periods, only local connectivity to fast networks of road, classic rail and motorway, allows the production of direct fastest routes to C. This supposes connections with motorway entrances and railway stations all located in the area where city B decision makers are legitimate to intervene. In the last period, accessibility of the intermediate city is provided by connection to high-speed rail network, intermodal articulation of rail and air networks, and air transport supply in city A. In this case a remote facility located in city A, far from city B has become a strategic asset for the intermediate city despite being located out of its legitimate area of intervention. A relevant policy for developing accessibility of city B cannot rely on a development program of facilities located exclusively in its intervention perimeter.

7/20

As a second illustration of the role of detour in the organisation of distances, we consider a contribution on European accessibility [START_REF] Lennert | Future Orientations for Cities final report[END_REF] based on contact potential indicators [START_REF] Törnqvist | Contact Systems and Regional Development[END_REF][START_REF] Törnqvist | Contact requirements and travel facilities: contact models of Sweden and regional development alternatives in the future[END_REF]. The possibility for daily commuting trips between cities is represented through all the available modal and intermodal chains. In the European context, intermodality plays an important role in long distance transport by combining air and high-speed rail. From the point of view of cities this intermodality sets up what can be called air-rail metropolitan systems [START_REF] Bozzani-Franc | Grandes Vitesses, Métropolisation et Organisation des territoires : L'apport de l'intermodalité aéro-ferroviaire à grande vitesse au rayonnement métropolitain[END_REF]. These systems are based on a high-speed rail access to a major airport, according to a configuration met by 6 cases in Europe, with two main cases around Frankfurt and the axis Lille-Paris-Charles-de-Gaulle [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. Air-rail intermodality generates many detours and space inversion processes. Figure 10 reveals the space inversion by confronting the straight line linking origin and destination, to railway axes leading to airports used in the actual route. These spatial inversions are all the more remarkable since they rely on a terrestrial part of routes.

8/20

Space inversion as the extreme form of detour, has to be considered in spatial planning. Paris-Charles-de-Gaulle airport is an entry point to the world for Lille and the Nord-Pas-de-Calais region [START_REF] Bozzani | L'Intermodalité air-fer à grande vitesse au service du rayonnement métropolitain: étude de l'articulation modale à l'aéroport de Roissy-Ch. de Gaulle au départ de Lille[END_REF][START_REF] Menerault | La face cachée de l'intermodalité[END_REF]; space inversion occurs when the traveller takes a train towards the south in order to reach European destinations located north of Lille. How can local decision makers assume this space inversion that involves getting rid of a rationale of local airport facility development policy and privileging an approach based on transport service? This is the issue raised by space inversion, the extreme form of detour, in the case of Lille.

The third illustration of the importance of detour in the formation of distances is provided by public transport network cartography designed for users. These maps can be categorised according to two modes of representation: most of them adopt a deformed geometrical 1 representation principle, while others use a topographic mode, based on conventional cartographic location of places. In the first case, which corresponds to Harry Beck's classic London Tube map [START_REF] Roberts | Objective versus subjective measures of Paris Metro map usability: Investigating traditional octolinear versus all-curves schematics[END_REF], rectilinear segments remove the local detours of transport lines. In these representations, detours of lines are erased, but another category of detours associated to connections between lines is highlighted.

Any user of public transport systems is highly likely to experience a change of vehicle or a change of modes during her/his trip. In the Paris agglomeration, we estimate that about half public transport trips use several lines or transport systems [START_REF] Massot | Les pratiques de multimodalité dans les grandes agglomérations fran\ccaises[END_REF]. This proportion can reach 70 % in agglomerations like Lyons [START_REF] Richer | Quelle(s) intermodalité(s) dans les mobilités quotidiennes ? L'apport des Enquêtes-Ménages Déplacements (EMD) dans l'analyse des pratiques intermodales[END_REF].

1 In this context, many authors in geography use the adjective topological to refer to the network layout.

Nevertheless, topology is used in the field of anamorphic cartography to deal with transformations of space such as contraction, dilatation and tear. In an objective of clarification, I consider that topology should be confined to this set of issues and not used when simply referring to network layout. 9/20 The Lyons network layout is marked by the will to avoid the creation of a single focal point of heavy lines. Consequently, the share of routes involving several lines is structurally higher than in other equivalent agglomerations.

For instance, the route between the two main railway stations, Perrache and Part-Dieu is not possible by metro without changing lines2 . The design of the simplified user map of which the central section is reproduced in figure 11 follows the principles introduced by Harry Beck for London: lines are turned into rectilinear segments aligned on the eight possible directions of the octolinearity principle. The representation graphically highlights rectilinear itineraries and connection stations. As routes in public transport network depend on the initial design choices of the system, the map expresses a pedagogy of detour, in order to show how to establish distances that are optimal and that assume changing between public transport lines.

Any transport system user has, deep down inside, an intimate reference to the straight line. The representations that are provided to him take this idea into account by erasing detours on the lines as in Harry Beck's London tube map, or, conversely, by elaborating a pedagogy of detour that underlines exchange poles that create spatial and temporal detours.

Detour turns into a positive value, as a means to optimise a route. Fast infrastructures tend to increase the length of detours and contribute to congestion, as Héran shows ( 2009), but this results from an optimisation process by individuals. The etymology of the word detour carries this ambiguity by meaning the tortuousness of routes, and also the fact of getting away from a direct route and reaching a given place. Optimality of distances highlights the search for an optimal route, as sinuous as it may be; in this sense we can state that detour is most often a search for optimisation.

Third statement: breaks participate in optimising distances

Considering a break in an itinerary as a violation of triangle inequality and a cause of sub-optimality of distances (Huriot et al., 1989: 313) is a third error of interpretation of this metric property [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]. Following the views of Brunet, a break represents a reloading of the energy necessary for movement (Brunet, 2009: 16). Consequently, break contributes to the optimality of routes, and hence to the optimality of distances. To address this error of interpretation we state that breaks in movement contribute to optimising distances.

A break is often seen as a factor of sub-optimality in movement. The waiting time before boarding a public transport vehicle is considered as a waste of time for travellers. Nevertheless, any movement in space needs energy loading and reloading, breaks, that are not sources of sub-optimality, but rather contribute to the realisation of optimal routes for persons or objects on the move. Break is necessary in order to provide the momentum for movement. Displacement is not a continuous movement. It contains breaks.The first set of illustrations of this principle is the reflection on urban design for pedestrians, i.e. public space design, that highlights the idea of breaks (Lavadinho, 2011: 145). The objective is to favour an appropriation of space by pedestrians through what Lavadinho calls a lateralisation (2011, page 201), i.e. the possibility to exit one's route for a break, a breathing space, which will provide the momentum necessary to complete one's pedestrian route.

The bridge on river Mur in Graz (AU) materialises the break in a pedestrian itinerary in the form of a two-section footbridge linked through a central island that floats in the middle of the river (The Island in the Mur, 2014). Taking advantage of a detour, the pedestrian has the possibility to physically stop in the café located on the island, but the break can also be simply mental: the route possesses this breathing space, this passage, this instant, that provides rhythm to the crossing and the itinerary considered in a wider scope. The island halfway through the bridge is a break, providing it can be appropriated as such by the walker. The appropriation process relies on a symbolic dimension that allows for an interaction between the subject and his environment, according to a mechanism that Gibson has theorised with the concept of affordance [START_REF] Gibson | The concept of affordances[END_REF][START_REF] Turner | Encoding natural movement as an agent-based system: an investigation into human pedestrian behaviour in the built environment[END_REF]. We can remark that in the case of the island on river Mur, detour and break are superimposed: the straight line of the footbridge is interrupted by the crossing of the island, and the island fosters the facilities allowing break. With this bridge, a break is proposed, but a detour is imposed.

11/20 For Lavadinho, the facilities of break allowing lateralisation during the trip, providing they are placed carefully, can contribute to the implementation of agglomeration-wide pedestrian networks. The rationale of the spatial configuration of these facilities can be compared to the distribution of caravanserais, service-stations, hotels and other places dedicated to loading the energy necessary for movement. It is a matter of optimisation. These facilities of the rest of pedestrians cover a wide variety of types, with, for instance, a free and open space with benches, around the viewpoint provided by a belvedere, an overhang; they can also involve playgrounds in a gamification of urban space (Lavadinho, 2011: 189). As shown in figure 13, the urban space design of the surroundings of the Vincennes regional railway station in Paris (FR) follows these ideas with seats located so that they provide a view of urban perspectives for passers-by, and urban playground with artificial fog. In figure 14, the gamification principle is exemplified in Saint-Gall's Stadtlounge (CH). Passers-by are invited to sit and rest on a couch, as they would do at home or in a café. As Bailly states, the representation of his route by a pedestrian is organised around a series of points, of landmarks, which are essentially symbolic (Bailly, 1990: 268).

12/20 (Lavadinho 2011, 280) Figure 14: Facilities of break in pedestrian itineraries leading to the Vincennes regional railway station [START_REF] Lavadinho | Le renouveau de la marche urbaine: Terrains, acteurs et politiques[END_REF] Break can be symbolic, but can also be spiritual. The crosses are omnipresent in landscapes of rural communes in Brittany (FR) [START_REF] Castel | Atlas des croix et calvaires du Finistère[END_REF]. Always facing west, they bear the name of a place or a saint, and are most often located at a crossroads, and almost always on a roadside. The crosses, as landmarks, provide a sense, a meaning to the way and to the journey. Travellers establish their optimal route by using these landmarks. A villager will use the marking of crosses to indicate a route to a traveller. In addition, the routes of pilgrims are punctuated with these singular points that provide direction, and reassure the traveller afraid of getting lost. Nevertheless, beyond its utilitarian function, the cross also has a spiritual dimension. The pilgrim's route, full of risks, "requires divine protection" (Le Scouezec and Masson, 1982) ̈. The cross provide "support from divine force" through a spiritual break. The pilgrim needs this equipment that allow him to follow the route without getting lost, in all the senses of the expression (Lefebure, 2004, page 60) ́.

If we extend the reflection, we see in this urban function of break the possibility for an aesthetic appropriation of space, in the sense of an artistic experience. Crosses as art-work, but also the location of seating allowing for enjoyment of the urban landscape around the railway, constitute invitations to an aesthetic experience in urban space. This analysis suggests a possible function for pieces of art inserted in public space. Pedestrians in Mamfe in Cameroon can use the numerous statues installed in the public space for this break function. Art can contribute to providing break in pedestrian mobility networks.

The facilities of break represent an urban paradox: they organise the possibility of a break, of a waste of time in a trip, and they also contribute to optimising pedestrian distances in a wider network. In that sense break should be considered as a relevant principle for the design of urban space in order to support a pedestrian oriented urban form.

The second illustration of the role of break in establishing distances is provided by the analysis of railway stations as nodes and places [START_REF] Bertolini | Cities on rails: the redevelopment of railway station areas (E & FN Spon, London) Boumoud A, 2012 La grande gare contemporaine et le labyrinthe du transport multimodal: vers une nouvelle approche de la lisibilité[END_REF]. Railway stations have become key objects in the organisation of transport system alternative to cars. We have 13/20 investigated the perception and representation of railway stations by users. As an illustration, Charlotte, who we asked to draw the railway station space, proposed a representation with the major elements of entrance, ticket offices and platforms. Between the functional entrances and exits of the railway station space, the representation shows the facilities of break available to travellers.

These railway stations are also, as urban places, spaces where service and retail functions develops. The viability of their economic model depends on it. Services to individuals, and particularly food retail, occupy a prominent place among these functions. Recent research show the evolution of food and drinks packages in adaptation to urban lifestyles, made of mobility and immobility, both desired or constrained (Monreal et al.). These evolutions provide an even stronger importance to the urban facilities of break.

And reflections on the design of the La Défense (FR) shopping mall near Paris highlighted the idea of break with a principle of pacing routes, providing the various groups of users with necessary places and facilities for break: rest points, kid's corners, snack breaks (Boumoud, 2012: 183;[START_REF] Roty | Conception et management de projets de pôles d'échanges : la question des parcours des usagers : La Défense, Massy, Val d'Europe[END_REF]. Ambiguity is not absent from this last avatar of urban break: commoditisation of urban space [START_REF] Cronin | Advertising and the metabolism of the city: urban space, commodity rhythms[END_REF] and favouring food regimes with adverse health effects [START_REF] Marmonier | Le grignotage : sa définition physiologique. Mise en évidence de ses conséquences métaboliques et comportementales à court terme chez l'homme[END_REF]) can be criticised. Assessing these facilities of pedestrian break from the point of view of public health involves considering a complex set of issues.

Reflections on shopping mall design can join this problematic through the concept of pedestrian station [START_REF] Amar | Le Piéton considérable, la marche au coeur des mobilité, 1ére phase du séminaire: le partage de connaissances[END_REF]. More broadly, defining the basket of services and retail of the railway stations is a true urban issue that should be considered together with the surrounding urban area supply [START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF].

Railway stations are urban places, but also transport nodes. This is the place where we come to take -and wait for-trains. We get to the railway station to establish a distance, i.e. an optimised route, and yet we come down and wait. The paradox of breaks in routes is here manifest. The dwelling time in the railway stations develops inside a long distance journey, but provides, through lateralisation, the possibility to carry out an activity in the station or in the surroundings. In the time-space paradigm (Chardonnel, 2001;[START_REF] Hägerstrand | What about people in regional science?[END_REF], a tiny time-space prism opens up for the traveller waiting for his train. Once he is aware of the time at the railway station and the actual departure time of his train 3 , the potential for a new trip contained in a larger optimised trip opens up.

In the case of railway stations, as we see, the reflection on the break that optimises the route, allows us to consider railway stations as spaces for lateralisation, of opening of time-space for an activity that develops inside a larger itinerary without altering it. As Amar states in his analysis of the La Défense exchange pole, "the path step, even forced, is enriched with opportunities" [START_REF] Amar | Complexes d'échanges urbains, du concept au projet, le cas de la Défense[END_REF].

3 The importance of time in cities and in railway stations is highlighted by Lallier in his documentary film on railway and metropolises [START_REF] Lallier | Film: La ville sur des rails -L'utopie de la métropole -Predit[END_REF].

14/20 This reflection on break echoes the issues of in-place economy [START_REF] Davezies | L'économie locale «résidentielle[END_REF][START_REF] Terrier | L'Économie présentielle. Un outil de gestion du territoire[END_REF] that, rather than analysing household income at home-place, consists in studying the location of income spending, the spatialisation of which can take a very distinct pattern. The free time of passengers waiting for their train is an opportunity to capture locally an income earned elsewhere.

Beyond the issue of railway stations, in a general context of increasing porosity between professional travel and tourism [START_REF] Kellerman | Daily Spatial Mobilities: Physical and Virtual[END_REF], this reflection entails considering the facilities of tourist break -hotels and restaurants-as elements of attractiveness for professional mobility and not only for tourism. The fact that reloading and break obey organisational constraints of transport and mobility implies that a territorial development aiming at taking advantage of the reloading function should rely on a deep understanding of the functioning of transport systems. Such is the case of the tourist development with relays and restaurants on the N6 road in Burgundy (FR) as path step for Parisian drivers to the Mediterranean coast [START_REF] Bavoux | Structuration du territoire bourguignon : de l'île à l'isthme[END_REF]. More broadly, the idea of break is widely observed in tourist issues if we consider that, most often, the spatial pattern of tourist mobility that consists in a series of path steps to enjoy a view-point, a piece of art or a particular atmosphere; from that perspective, the break, as much as the path, reveals the engagement of the tourist.

Etymology echoes our analysis of mobility: the break refers to the fact of interrupting an activity, and refers also to the time spent on this interruption. The break interrupts and provides rest. Its negative side, the interruption of an effort requiring activity, is justified by the benefits it provides: the rest gives back the strength to return to the route.

Conclusion

From a discussion about the mathematical properties of metrics, we identify three fundamental characteristics of distance, which are optimality, detour and break. We then explore the implications of these properties for transport planning, urbanism and spatial planning.

We state that distances contain the idea of optimum and that any distance is associated to a search for optimisation. We illustrate this idea by analysing pedestrian movement in cities and route choices by means of public transport maps. Spontaneous pedestrian distances shown by desire lines express the strength of straight line and of the search for optimum in the process of formation of distances. This discussion reveals the tensions in the design of public space with implications on pedestrian urban mobility, between an open, free and legible space and the principle of separation of flows inherited from functionalistic urbanism, and supported by concern for road safety. The principle of optimality of distance is expressed in the multi-criteria arbitration that opposes straight line to road safety concern.

Optimality is introduced by design in public transport maps. But the design principles of a simplified graphic can infer sub-optimal routes that, as in the case of London metro, have to be corrected in order to relieve the overloaded central parts of the network. Because distance carries the idea of optimum, even if not unequivocal, it is relevant as an operational concept that has to deal with limited resources.

The second principle we state is that detour in distances is most often a means to optimise movement. The presence and network layout of fast transport systems generates most of the detour observed in geographical spaces at regional scale. This raises the issue of local policies dedicated to accessibility improvement, because the extreme form of detour, space inversion, shows a situation where equipment contributing to the external access can be located outside the territory it serves, and hence outside the space of intervention of the decision makers of this territory. Detour has to be taken into account in regional transport policies.

Detour also exists in urban spaces as in the case of public transport networks. As public transport operations favour an organisation in lines, a given trip is most likely to have to change lines. This is why public transport maps have to develop a pedagogy of detour my means of graphic design highlighting intermodality and exchange poles. Optimality of distances highlights the search for an optimal route, as sinuous as it may be; in this sense we can state that detour is most often a search for optimisation.

The third statement is that breaks in movement contribute to optimising distances. In order to develop agglomeration-wide pedestrian networks it is necessary to introduce the possibility of break in urban space that allows for long distance pedestrian routes. Break for pedestrian mostly provides energy for reloading movement, but can also be symbolic, spiritual or aesthetic. Railway stations represent another facility of urban break. Users get to the railway station to establish a distance, i.e. an optimised route, and yet they come down and wait. The facilities of break represent an urban paradox: they organise the possibility of a break, of a waste of time in a trip, and they also contribute to optimising distances in a wider network. In that sense break should be considered as a relevant principle for the design of urban space in order to support a pedestrian oriented urban form.

This reflection on geographical distances has proven fruitful in the domains of transport, urbanism and spatial planning. Starting from mathematical properties of distances we come to discuss the orientations of urban space design between openness and separation of flows, the distribution of services in the urban form in the aim of establishing efficient pedestrian networks, or the design of urban or regional transport systems that assumes detours.

Figure 2 :

 2 Figure 2: Spontaneous distances of the desire line, the strength of the search for optimum in the Quatre Cantons Grand Stade metro station surroundings (photo L'Hostis 2013)

  Spontaneous pedestrian distances shown by desire lines express the strength of straight line and of the search for optimum in the process of formation of distances. The 2012 project of the Quatre Cantons metro station has neglected this desire line that existed before, privileging a safe crossing of the avenue on a large cross-walk close to the roundabout. Whether created by users or by design, both routes are shown in figure 2 and 3. The first route is approaching the straight line, minimising physical effort, while the second promises, at 3/20

Figure 4 :

 4 Figure 4: Thorn bushes in the middle and lateral fences on Montebello boulevard in Lille to prevent straight line pedestrian movements towards the entrance of Wazemmes secondary school (left) and channel them on safe pedestrian crossings, renunciation to the principles of legibility and openness of space design (Photo L'Hostis 2014)

Figure 5 :

 5 Figure5: Developing a pedestrian link in London transport system in order to correct the misleading, sub-optimal distance induced by Beck's cartography[START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]

Figure 6 :

 6 Figure6: Land use monitoring for increasing density and improving pedestrian routes in the vicinity of railway stations[START_REF] L'hostis A ; Alexandre | Misunderstanding geographical distances: about three errors of interpretation of violations of the triangle inequality" Cybergeo : European Journal of Geography[END_REF]

Figure 8 :

 8 Figure 8: Space inversion(Bunge 1962, 172)

Figure 9 :

 9 Figure 9: Introduction of spatial inversion, extreme form of detour, with the development of high speed in the trip from B to C

Figure 10 :

 10 Figure 10: Contribution of air-rail metropolitan systems in Europe in 2009 to contact potential showing detours and spatial inversion[START_REF] Lennert | Future Orientations for Cities final report[END_REF]

Figure 12 :

 12 Figure 12: Footbridge and island on river Mur in Graz (AU), a break for favouring pedestrian mobility ("The Island in the Mur" 2014)

Figure 13 :

 13 Figure 13: Stadtlounge, an urban installation in Saint-Gall (Switzerland) allows for lateralisation of pedestrians(Lavadinho 2011, 280)

Figure 15 :

 15 Figure 15: Le Keff cross in Saint-Thegonnec, Finistère (FR) landmark for travellers and facility of a spiritual break (photo Annie L'Hostis 2014)

Figure 17 :

 17 Figure 17: Charlotte's mental representation of St-Lazare in Paris (FR), break as an opportunity inside an optimised route

This issue has been tackled with the introduction of the first tramway line of the Lyons network in 2001, connecting the two stations directly. 10/20Figure 11: Central sector of Lyons public transport network, a necessary intermodality (source http://www.tcl.fr browsed in 2014)