Street network theory of organization in nature

Journal of Advanced Transportation, Vol. 30, No. 2, Pp. 85-107

Street Network Theory of Organization in Nature

Adrian Bejan

This paper outlines acompletely deterministic ("constructal") theory of why quasi-similar street patterns exist, how they form, and how they grow in time. The function of the street network is to connect a finite area to a single destination point. The new idea is that the network of streets evolves in time, by starting with the optimization of the shape of the smallest area element that is serviced by the network. Next, the optimized area elements are assembled into a larger area element which is againoptimizedforshape. Thissequenceofoptimization&organization is repeated in finite-size steps, toward larger quasi-similar assemblies. The optimization consists of minimizing the travel time between each point of a finite area and a common point of destination. The network is constructed (optimized, organized) in time. Every single geometric feature of the network is the result of pure, deterministic theory: the shape of each area element, the shape of each new (larger) assembly, the optimal number of parts in each assembly, the relative orientation of successive streets, and the optimal width of each street.

The Fundamental Access Optimization Problem

Why are streets arranged usually in clusters (patterns, grids) that look almost similar from block to block, and from city to city? Why are streets and street patterns a mark of civilization? Indeed, why do streets exist?

In this paper I outline a purely deterministic theory that provides answers to these questions in an astonishingly simple and direct way. Furthermore, the implications of this theory extend well beyond transportation, into physiology and thermodynamics. The theory is the result of addressing the following access optimization problem:

Consider a finite-size geographical area A, and a point M situated inside A or on its boundary, Fig. 1. Each member of the population living on A must travel between his point of residence P(x, y) and a point M. The latter serves as common destination for all the individuals who live on A. The density of this traveling population, i.e. the rate of which individuals must travel to M, is fixed and described by n" [people / m2s]. This also means that the rate at which people are streaming into M is constrained, n = n"A. Determine the optimal "bouquet" of paths that link the points P of area A with the common destination M, such that the time of travel required by the entire population is the shortest.

Adrian Bejan is the J. A. Jones Professor of Mechanical Engineering at Duke University, in Durham, North Carolina, 27708-0300, USA. Rcceived March 1996.

86 A. Bejan

Figure 1 Finite-size area (A) covered by a uniformly distributed population (ti") traveling to a common destination (M).

In short, the problem is how to connect or finite area (A) to a single point (M). This problem was stated in the most general and abstract terms, because its solution and its diverse manifestations benefit from such a formulation. It helps, however, to see a real-life problem in this statement, before attempting a solution. The area A could be a flat piece of farm land populated uniformly, with M as its central market, or harbor. It also helps to think in time by beginning with the most ancient type of community that faced this access optimization problem. The oldest solution to this problem was also the simplest: unite with a straight line each point P and the common destination M, and you will minimize the total time spent by the population en route to M.

The straight-lines solution was, most likely, the preferred pattern as long as man (his load, and his ox) had only one mode of locomotion: walking, with the average speed V,. The farmer and the hunter would walk straight to the point (farm, village, river) where the market was located. This radial pattern of access paths can still be seen today, especially in perfectly flat and uniformly rural areas such as the plain of the lower Danube, in Romania. The once ancient (Roman times) market is now a larger village, and the surrounding farmers have become a constellation of almost equidistant tiny villages. As a matter of fact, six seems to be the usual number of surrounding villages, such that the plain is now covered by hexagonal cells with larger villages in the center. When we look at the map, the

Street Network l'heory of Organization.. . ~ 87

honeycomb and BCnard convection patterns come to mind. The radial length in any such "wheel" was set in antiquity by the distance that a pedestrian (or an ox) could cover in a few hours (such that a round-trip to the market place can be made during daylight): the order of magnitude of that distance is 10 km, and is what we see printed on the map today.

The radial pattern disappeared naturally in areas where settle- ments were becoming too dense to permit straight-line access to anyone. Why the radial pattern disappeared "naturally" is also a part of the present problem. Another important development was the horse driven carriage: with it man had two modes of locomotion, walking (V,), and riding in a carriage with an average velocity V, that was significantly greater than V,. It is as if the area A become a composite material with two conductivities, V, and V,. Clearly, it would be faster for every inhabitant (P, in Fig. 1) to travel in straight lines to M with the speed V, . This is impossible, however, because the area A would end up being covered by beaten tracks, leaving no space for the inhabitants and their land properties.

The real, more modern problem then is one of bringing the carriage and the street "near" a small but finite-size group of inhabitants: this group would first have to walk in order to reach the street. The problem is one of allocating a finite length of street to each finite patch of area (A,), where A,ccA. The problem is also one of connecting these street lengths in an optimal way such that the time of travel of the population is minimum.

The Smallest (Innermost) Area Element: The First Street

The approach chosen for solving this problem has a definite direction: from the smaller subsystem (detail) of area A, to the larger subsystem, and ultimately to area A itself. This direction was first recognized in the buckling theory of eddy formation and self-similar development of turbulence (Bejan 198 1,1982). This direction is also the direction of time, or history. It is also the meaning of evolution and growth.

We begin with the observation that the area subsystem to which a street length may be allocated cannot be smaller than the size fixed by the living conditions (e.g. property) of the individuals who will be using the street. This smallest area scale is labeled A, in Fig. 2. For simplicity we assume that the A, element is rectangular. Although A, is fixed, its shape or aspect ratio H, / L, is not. Indeed, the first objective is to find the area shape that maximizes the access of the A, population to the street segment allocated to A,.

Symmetry suggests that the best position for the street segment

88 A. Bejan

is along the longer of the axes of symmetry of A,. This choice has been made in Fig. 2, where L, > H,, and the street has the length L, and width D,. The traveling population density n" is distributed uniformly on A,. Each individual must travel from a point of residence P(x,y) to the (0 ,O) end of the street - this, in order to "get out" of A,. The individual can travel at two speeds, a low speed V, when off the street, and a higher speed V, when on the street.

Y

Hl 0

t

I n" I

Figure 2 The smallest (innermost) elemental area, A,, and the street segment allocated to it.

We assume that the rectangle H, x L, is sufficiently slender (L, > H,) so that the V, travel is approximated well by a trajectory aligned with the y axis. This makes sense from the point of view of the individual who must travel from P(x, y) to (0,O) the fastest: since the street is a faster conduit, it is reasonable to head straight toward it (i.e. downward in Fig. 2). It can be shown that when V, is not sensibly greater than V,, the optimal V, travel is not straight downward but at an angle such that the V, travel is somewhat shorter than the x segment shown in Fig. 2.

The time of travel between P(x, y) and (0,O) is (x / V,) + (y / Vo). The average travel time of the A, population is given by

Street Network Theory of Orgnnizcltion.. . . 89

which yields

f =-+- L l H l ' 2v, 4v,

The elemental area is fixed (A, = H,L,, constant), therefore f can be expressed as a function of H I , which represents the shape of A,:

The average travel time has a clear (sharp) minimum with respect to HI. Solving di / dH = 0 we obtain

and, subsequently, 1 12

L l , o p t = ( l y g ) I ( 5 )

Equation (6) shows the optimal slenderness of the smallest area element A,. This result validates the initial assumption that H, / L, < 1; indeed, the optimal smallest rectangular area should be slender when the street velocity is sensibly greater than the lowest (walking) velocity.

According to equation (6) the rectangular area A, must become more slender as V, increases relative to Vo, i.e. as time passes and technology advances. This trend is confirmed by a comparison between the streets built in antiquity and those that are being built today. In antiquity the first streets (Fig. 2) were short, typically with two houses on one side. In the housing developments that are being built today, the first streets are sensibly longer, with ten or more houses on one side.

The remaining analysis can be shortened based on the observation that exactly the same optimum [equations (4) - (6)] is found by

90 A. Bejan

minimizing the longest travel time (t,), instead of minimizing the average time of equation (1). The longest time is required by the individual who travels form one of the distant comers (x = L,, y = f H, / 2) to the origin (0, 0), and is given by

+ - t =- L l 1 v, 2v,

(7)

Equations (7) and (1) show that the travel time from the most distant comer is exactly twice the area-averaged time, and that the minimizations of tl and i , are equivalent. It is both interesting and important that the optimization of the A, element is of interest to every inhabitant: what is good for the inhabitant with the least advantageous location is good for the community as a whole. The time obtained by minimizing t,, or by substituting equations (4, 5) into equation (7), is

112

ll.M"=($)

At this minimum, the two terms that make up t l in equation (7) are equal to each other. This equipartition oftime principle means that the total travel time is minimum when it is divided equally between traveling along the street and traveling perpendicularly to the street. The next steps in this paper are based on minimizing the longest travel time associated with each (increasingly larger) area element. We will see that the equipartition of time principle governs the optimization of each new area element scale.

Another observation concerns the width of the first street segment, D,. The total "flow rate" of travelers generated by the A, element, and taken out of A, through the origin (0, 0), is n"A,. The same quantity can be expressed as plD,V,, where p1 is the instantaneous number of individuals found per unit of street area in the vicinity of the exit (0,O). In conclusion, the first street width is given by

where both p, and V, are technological parameters. Equation (9) sheds light on the time evolution of the smallest street, which is also the innermost street in a more complex (more evolved, newer) grid. When the smallest street was first built, D, was dictated by the width of one or

Street Network lheory of Organization.. . . 91

two carriages. But as the traveling density (or population, and affluence) increased in time, the constrained D, stimulated technological developments that led to increases in the product plV, that matched the increases in ti". The technological aspect of V, is clear: the family car is faster than the best carriage, even on the smallest street. Increases in p,, on the other hand, were registered as the number of vehicles present on the road increased.

The Second Area Element

Keeping in mind Fig. 1 and the optimal access problem stated on the first page, in Fig. 2 we see the smallest loop of the rectangular grid that will eventually cover the given area A. The question that remains is how to connect the D, streets such that each innermost loop has access to the common destination M. The answer is obtained by repeating the preceding analysis several times, each time for a larger area element, until the largest scale (A) is reached.

Consider then the rectangular area A, = H,L, shown in Fig. 3. This are element represents the next larger size: it consists of a certain number of the smallest patches A,. The purpose of this assembly of A, elements is to connect the D, streets so that the traveling population (n"A2) can leave A, in the quickest manner. We invoke symmetry as the reason for placing the new ("second") street along the long axis of the A, rectangle. In Fig. 3, the stream of travelers (n"A2) leaves A, through the left end of the D, street.

T 1 H2

L2 A Figure 3 The second elemental area, A,, as an assembly of con-

nected innermost elements A,.

92 A. Bejan

For the sake of consistency in notation, we write V, and p, for the speed and traveler density associated with travel on the second street. These parameters are generally not the same as the corresponding parameters of the first street (p2V2 2 p,V, , most likely), because technology evolves in time. The same time arrow is indicated by the growth process in which several A, neighborhoods are now incorporated into a larger community called A,.

With reference to Fig. 3, the longest time of travel occurs between one of the distant corners of A, and the left end of the D, street,

t = - + t L 2

v2

The first term accounts for the portion traveled along the D, street, while the second term represents the minimum time required to travel across the A, element that occupies the most distant corner. Using equation (8) for t l equation (5) for A,, the geometric "assembly" relation L, = H2/2, and the area constraint A, = H,L,, we can rewrite equation (10) to show explicitly the effect of area shape (H,) on the total travel time:

H2 A, ",HZ v l

+ - t =-

The optimal shape of the A, area element is determined by minimizing t2 with respect to H,, and finding again that the time equipartition principle holds,

v 112

L2"p'=(y? A2)

Street Network 7heory of Organization.. . . 93

These results look similar to equations (4) - (6) and (8), however, this time the ratio V, / V, needs not be significantly smaller than one. Note that in this section we did not have to assume that the area element A, is slender: in Fig. 3 the travel through the A, end-corner is always "downward" because of the D, street, not because of the assumption H, < L,, which we did not make. In any case, equation (14) shows that the optimal A, element is more slender when the speed on the second street (the "avenue") is greater than on the first street. The square is the optimal shape of A, when V, = V2.

The geometric relation H, = 2L, and equations (5) and (12) provide the relation between the sizes of the first two area elements, or the optimal number of elements A, that must be assembled into one element of size A,:

As in equation (9), we find that the width of the second street is

or that the street width enlargement factor is

This ratio can be expected to be greater than 1, because of equation (6) and the fact that in a given technological age p2 is of the same order as pl, while V, is sensibly larger than Vo.

The Third Area Element

Continuing the progress from small area elements toward larger elements, imagine next a rectangular area A, that contains a number of the A, elements optimized in the preceding section. This arrange- ment is presented in Fig. 4, which shows that the D, streets of the A, elements are connected by the D, street that serves as axis of symmetry for A,. The speed of travel along the D, street is V,.

A. Bejan 94

T 1 H3

Figure 4 The third elemental area, A,, as an assembly of a large number of A, elements.

The analogy between Fig. 4 and Fig. 3 is clear. If, in addition, we assume that the number of A2elements assembled into A, is large, then the analysis of the preceding section can be repeated. The longest time of travel across A, (from one of the distant corners, to the left end of the D, street) is

t =-+t L 3 3 v, 2,min

This expression can be rearranged by using, in order, equations (1 3, (1 3) and L, = H3 / 2, to obtain

A3 H3 t -- +- , -V3H3 V,

It can be shown that the minimization of t3 with respect to H, yields the aspect ratio

which is a number of order 1 when V, does not differ significantly from V,. This important conclusion, coupled with the earlier conclusion that the aspect ratio of the A, rectangle is of order 1 [equation (14)], means

Street Network 2 k w y of Orgnnization.. . . 95

that the best A, shape is robust (not slender) and occurs when A, contains only a very small number of A, elements.

t L2

1

Figure 5 The third elemental area as an assembly of only two A, elements.

By using equation (21), A, = H,L,, H3 = 2L2 and equation (13) it can be shown that the number of A2 elements that must be assembled optimally into a single A, element is n3 = A, / A2 = 2,12 V, / V,, which, contrary to Fig. 4, suggests that n3 should be a small even number. The number n3 = 2 was used in the A, construction of Fig. 5. This time the D3 street connects just two D2 streets, which meet in the center of the A, element. For this reason, the length of the D, street is only L, / 2. The total time required to get out of A,,

L 3 t =-+t 3 2V3 2,min

can be expressed as

96 A. Bejan

When the average vehicle speed does not change from one street to the next (V3 = V, = Vl) , the A, area is a square, and A, is a rectangle of aspect ratio 2: 1. These shapes were selected for illustration in Fig. 5. The width of the D, street is D, = n"A,/(p3 V,), where p3 is the number of travelers per unit of street area near the exit end of the street. Since A, = 2A2, we conclude that the street width increases from D, to D,,

Equations (24) and (18) are remarkably similar even though the aspect ratios of A, and A, are different. The construction of Fig. 5 and the discussion that followed can be repeated for the case n3 = 4.

Area Elements and Streets of Higher Order

The rest of the sequence consists of repeating what we learned in the preceding section: the best shape of each new area element (Ai) is such that only a small number of elements of the preceding size (Ai-l) are assembled in Ai. Continuing to discover theoretically that Ai = 2Ai-l, and assuming that Vi = Vi-l = . . . = V,, the sequence is one in which two rectangles (e.g. Fig. 5) are integrated into a larger square, and, next, two squares are integrated into a larger rectangle.

The assemblies of order three (A,, Fig. 5) and of higher orders may not be optimal in a precise mathematical sense, because the numbers of parts in each new assembly must be integers [note the discussion above equation (22)]. These assemblies, however, are the best that f i t together.

This sequence is shown in Fig. 6 only for illustration, because it is unlikely to be repeated beyond the street of third generation shown in Fig. 5. The reason is that as the community and the area inhabited by it grow, other common destinations (e.g. church, hospital, bank, school, train station) emerge in A, in addition to the original M point (Fig. 1). Some of the streets that were meant to provide access to only one end of the area element (e .g . D, in Fig. 5 ) , must be extended all the way across the area to provide access to both ends of the street. As the destinations multiply and/or shift around the city, the dead ends of the streets of the first few generations disappear, and what replaces the "growth pattern" of Figs. 2-6 pattern is agrid with access to both ends of each street. The multiple scales of this grid, and the self-similar structure of certain areas (neighborhoods) of the grid, however, can be viewed as the fingerprints of the deterministic,

(Fig. 5 )

" - A4

A, (Fig. 3) -1 Ah

7

time, growth, evolution, development, purpose, life

Figure 6 Higher order area elements in the sequence constructed in Figs. 2-5.

98 A. Bejan

element-by-element optimization & organization principle uncov- ered in this paper.

The optimal access problem solved in this paper was stated in two dimensions, Fig. 1. The corresponding problem in three dimensions can be stated easily: minimize the time of travel from all the points P of a volume V to one common destination point M, subject to the constraint that the traveling population rate is fixed. One application is the sizing and shaping of the floor plan in a multistory building, and the selection and placement of the optimal number of elevator shafts and staircases. The floor plan optimization proceeds according to the method illustrated based on Fig. 2, where Vo corresponds to the travel through (movement out of) the rooms, and V, is the travel along the corridor. The unknown in this first building block is the shape of the floor area A,. Instead of using Fig. 3, to determine the optimal assembly of the A, elements, we stack a certain number (n,) of A, elements on the vertical, such that the next path (V,) is vertical and accounts for the elevator or the staircase. The optimization sequence may be taken to a third assembly (or even higher order assemblies) if the towers optimized in the three-dimensional equivalent of Fig. 3 must be integrated into a larger building with several wings.

The network optimization and organization theory can be extended generally to areas that are populated unevenly, or specifically to highways, railroads, telecommunications, and air routes (e.g. the organization of such connections into hubs, or centrals). Another clear application is in operations research and manufacturing, where the invention of the first auto assembly line is completely analogous to the appearance of the first street (see the third page of this paper).

Optimal Street Widths Subject to Street Surface Constraint

Another opportunity to optimize the street pattern geometry arises in cases where the vehicular speed (Vi, i 1 1) increases with the street width (Di), and the paved surface of all the streets is constrained. To demonstrate the existence of this optimization sequence, let us assume that this monotonic relation is described in the vicinity of the optimum by the power law

Vi = c i Dn (il 1)

where ci and a are positive empirical constants. We begin with the assembly shown in Fig. 3, where a choice can

be made between the sizes of D, and D, when the paved surface

Street Network lleory of Organization.. . . 99

(Ap,,) of the A, assembly is fixed:

Ap,, = D2L2 + n2D1L1 (constant)

The objective is to minimize one more time the travel time of equation (15), which varies as (V!V,)-1/2 - In conclusion, the problemconsists of maximizing the product DID2 subject to the constraint (26), while using equation (25) and the relations developed earlier for L,, L, and n,. This optimization has only one degree of freedom (D1 or D,, or D, / D1). Assuming that c1 = c2 = c1 2, and omitting the algebra, the result for the first optimal width increaie factor is

The same procedure can be applied to each assembly of higher order. For example, in Fig. 5 the question is how to allocate the total paved surface AP,, = AP,, + n, D3 L,/2 among the D,, D, and D, streets, where the relative allocation for D, and D, has already been determined [equation (27)J. The objective is to minimize t3 of equation (19) by selecting the optimal width D,, or a width increase factor such as D3/D2.

The Crucial Importance of the Direction of Time

In Figs. 2-6 we have constructed -building block on building block - the network that minimizes the travel time between all the points of a finite area and a common point of destination. The totally new aspect of this construction is that every single step was determined bused on theory. Figures 2-6 were drawn without "peeking" at a city map: they are not tainted in absolutely any way by empiricism. From the shape of the first elemental area to the higher order assemblies of Figs. 5 and 6, we relied repeatedly on the minimization of travel time subject to constraints (fixed areas, and fixed flow rates of traveling groups).

This purely deterministic approach gave us the optimal shape of each new (next, younger, larger) assembly, the optimal number of parts in each new assembly, the optimal way in which the new parts fit together, and the optimal width of each street. The construction began with the smallest element (Al), in which the travelers diffused (walked) all over A, (see V, in Fig. 2), and proceeded step-by-step

100 A. Bejan

(eventually through coallescence, or pairing) toward assemblies of larger size.

The deterministic power of the method stems from its direction - from small parts to a larger assembly - which happens to be the correct direction oftime. To see how absolutely essential time is to this theory, let us reconsider the access optimization problem in reverse, i.e. by looking at Fig. 1 from the present. We see an area that is already covered by a large (developed) network that starts out of M and proceeds along smaller and smaller streets toward (we hope!) each point of the area A. What we know is summarized in Fig. 7. There is a hierarchy of street sizes, which is indicated by the subscript j . Each street size is characterized by the travel speed Vj, width Dj and length L.. The number of streets of the jth size is Nj. The sequence staris from M, where j = 0 and No = 1, and continues toward smaller streets up to j = n, where the number of street sizes n is unknown.

k----h--+I

-- area A

Figure 7 A postulated network of streets connecting a finite area (A) with a destination point (M).

The total travel time from the deep end of this "general" network to the other end (M) is

n L. n L. t =,z -J=,z 7 J g J = O V . J z 0 C . D .

J J J

Street Network Theory .f Orgarrizcrriotr.. . . 101

in which we have used the constitutive relation (25) with j in place of i. The total paved surface constraint is

n

A = N . D . L . (constant) (29) P,g j = O J J J

The first observation is that the time tg can be minimized only with respect to the street widths (Dj). It cannot be minimized with respect to the street lengths (Lj): street numbers (Nj), and number of street sizes (n), and certainly not with respect to the relative geometric orientation (layout) of two successive street sizes. The optimal street widths can easily be derived from equations (28,29),

where the Lagrange multiplier (h) is given by

The only geometric implication of the reversed-time optimization approach is that, proceeding away from the common destination, the street widths must shrink by the same factor,

D . c , N , l / ( a + l ) (q) = (cj+; <+]) (32) opt

For example, if we set cj = cj+!, and if we write (empirically, based on observation) that in a certain neighborhood the streets bifurcate (Nj+, = 2Nj) then equation (32) becomes

Furthermore, since equation (33) is not of the same type as equation (27), this street shrinking rule fails at the narrow (large j) end of the sequence.

The reversed-time analysis based on Fig. 7 shows that ( 1 ) the optimal (minimum time) street pattern cannot be predicted, and (2) why it cannot bepredicted. The most important conclusion is that the process of optimization and organization is time itself. The network is predictable in its finest detail only if its correct time direction is recognized.

102 A. Bejan

Origin of the Deterministic Theory of Networks

In closing, let me remind the reader the first instance in which the proper time direction of the process of self-organization was recognized and used in a deterministic manner (Bejan, 1981, 1982). In the buckling theory of turbulent flow the flow field was constructed in time, by starting with the smallest scale and continuing in steps of geometrically similar building blocks toward larger scales that eventually covered the field.

The first such example -the two-dimensional shear layer - is reproduced here in Figs. 8 and 9. To proceed correctly and theoretically in time is to predict the flowfrom left to right in Fig. 8, which is the correct direction of the flow. The theory showed that the shear layer must always begin with a laminar length, the transition to a buckling (meandering) shape occurs when the viscous time becomes longer than the rolling (eddy formation) time, and the flow evolves as a sequence of geometrically similar assemblies (eddies) the sizes of which increase stepwise (Fig. 9). Note that Fig. 8 is a close-up view of the starting section (the tip) of Fig. 9.

The many predictions made for the first time possible by this "construction theory" of turbulent flow have been reviewed on several occasions, most recently in Bejan (1995). What we showed with the help of Fig. 7 for streets also holds for turbulence research: the reversed-time approach does not work, and is in fact responsible for the lack of purely theoretical progress on turbulence. Indeed, the pre-1981 approach was based on the reversed-time view, specifically, on the "cascade" idea that large eddies break down into smaller eddies (Richardson, 1926). More recently, the cascade idea was also refuted by Gibson (1991).

Figure 8 Stable laminar length in the beginning of a shear layer (Bejan, 1982).

Street Network neory of Orgaiiizarion.. . . 103

Half-angle of visual growth

Local wavelength

Figure 9 The constant-angle growth of the turbulent shear layer, as the repeated manifestation of geometrically similar (buckling, hB - 2D) building blocks (Bejan, 1984).

The laminar section (Fig. 8) and the viscous diffusion across it are analogous to the A, element (Fig. 2) and the Vo travel across that element. The convective roll-up and the successive assembling of small rolls into a larger roll are analogous to the higher velocities Vi (i 2 1) and the sequence of quasi-similar geometric figures shown in Fig. 6. The flow too acts as a "composite material" with two different "conductivities" -a low conductivity (viscous diffusion) and a high conductivity (eddy rotation, convection).

The buckling theory of turbulent flow, and now the construction theory of street patterns make us even more aware that the structure of geometrically similar (optimized & organized) systems can be predicted in great detail.

Streets as Live Networks

To appreciate how much is new in the deterministic theory of street patterns, it is important to note that one end of the street network (the one formed by higher-order assemblies, Fig. 6) is not entirely new. It was proposed in physiology as a three-dimensional heuristic model for the vascular system (Cohn, 1954), where it is known empirically that each tube is followed by two smaller tubes,

104 A. Bejan

i.e. each tube undergoes bifurcation. Cohn proceeded from large tubes toward smaller tubes. It is also known that the tube diameter must decrease by a constant factor (2-I i3) during each bifurcation. This diameter reduction factor has been derived based on flow resistance minimization (Thompson, 1942), and is the only theory- based notion present in the algorithms used to reconstruct the circu- latory and pulmonary trees, or other tree-shaped "live networks" that appear in nature (trees, roots, leaves, river basins, deltas, lightning). The description of these geometric constructions has been popular- ized through the advent of fractal geometry (Mandelbrot, 1983; Barnsley, 1988).

In spite of these advances, man's ability to anticipate the architecture of living networks on a purely theoretical basis remained limited to one result: the constant factor for diameter reduction during branching. The construction steps that were left to be determined theoretically are the number of new (smaller) ducts formed during branching (why two branches, and not six?), the relation between the branch length and the length of the mother duct, and the position of the smaller branches relative to larger branches, i.e. the manner in which the network fills the volume. Another extremely important aspect that awaited an explanation is why the theoretical diameter reduction factor 2-Ii3 fails to describe the sizes of the smallest ducts. In other words, why does the heuristic construction of an algorithm-based network break down at a sufficiently small scale? What is that small scale? Or, using the terminology of fractal geometry, what is "the inner cut-off", and why must it exist?

A close examination of the theory presented in this paper will show that theoretical answers have been given to all these questions, of course, by using the street optimization & organization problem, not the fluid flow resistance problem. As in turbulence theory, there is a fundamental difference between the approach followed in the past and the approach proposed in this paper. In all the theoretical studies of living fluid networks, the network was first seen and accepted, and then it was broken down repeatedly (e.g. through "bifurcation"), beginning with the largest duct and proceeding toward smaller scales. It was this "fracturing" point of view that made natural fluid networks ideal examples of "fractal" geometric constructions. The point of view exercised in the present paper was precisely the opposite: the network was constructed from the smallest elements by using optimized building blocks (e.g. through "pairing", Fig. 5), and proceeding toward larger scales.

In the natural sciences, the commonality of the living network architecture (trees, rivers, lightning, vascular and pulmonary systems) was left to the "explanation" that it is the fruit of a process of

Street Network l3heory of Organization.. . . 105

self-optimization and self-organization. The present paper places a completely deterministic method-a theory-behind the word "self".

Even before the present work on street patterns, I applied this theory successfully to other optimal-access networks (fluid flow, heat current): these results are summarized in a forthcoming book (Bejan, 1998).

We need to be clear about what the present theory represents. This line of inquiry is less about networks than it is about method: a completely deterministic theory of optimized & organized systems that evolve in time. It is about predicting natural patterns that, until now, could not be predicted. This method is the purely deterministic step that had been ruled impossible by contemporary physics and mathematics. Writing these closing comments I just discovered that the fundamental problem stated on the first page is known in mathematics as the Steiner problem (Courant and Robbins, 1941). Ac- cording to Bern and Graham's (1989) review, "the solution to this problem has eluded the fastest computers and the sharpest mathematical minds," and its solutions "defy analysis". The present paper provides a purely theoretical and analytical alternative worthy of attention.

Constructal Theory versus Fractal Geometry

The frozen geometric image - the fractal description of living networks - is only partially correct, because it misses the genesis of all such phenomena, which lies in the smallest (finite, now predictable) length scale. What is wrong with the fractal description - and by wrong I mean totally upside down - is the time arrow of the description. Contrary to Richardson's (1926) cascade, which was adopted unchanged by Mandelbrot (1983), in a turbulent flow field eddies do not break down, from the large to the small, all the way down to the viscous length scale. The eddy cascade proceeds in the opposite direction in time. Starting with the smallest eddy, eddies coalesce (as in Figs. 8 and 9) and arrange themselves into larger and larger structures the sizes of which increases stepwise. The oldest geometric feature is the smallest, and the youngest (most recent) feature is the largest -after all, this is the meaning of growth. Never mind that eddies of the smallest (elemental) size are being formed all the time, intermittently yes, but without interruption. So are the smallest branches on a tree. The point is that at the start of its existence, the system consisted of one element of the smallest size: this is also the smallest geometric feature that is found in the system's structure at subsequent points in time. The largest eddy and the largest branch (the tree trunk) are geometric features ofthe present.

106 A. Bejan

If "fractal" is an appropriate Latin-based word* for breaking things (Mandelbrot, 1983), i.e. for the opposite of the direction in which natural systems evolve, then the appropriate word for the geometry and evolution of optimized and organized natural phenomena is constructalt.

Some would argue that fractal geometry has nothing to do with time, and they would be right. The mathematical results (geometrical images) produced by the repetitive algorithms are frozen in time. The assumed (postulated) algorithm can certainly be executed in both directions, from the largest scale to the smallest, and from the smallest to the largest. As a descriptive aid for natural phenomena, however, the fractal description represents a clear choice, namely from the largest scale all the way to size zero. The word fractal has the concept of time built in it: the act of breaking something evolves in time from large pieces to smaller pieces. It is not that the mathematician is wrong to be an artist and paint a lifeless image that resembles the instantaneous image of a natural system. As a way to think about predicting the morphology of natural systems, however, the fractal paradigm is oriented backwards. The theoretician must still predict the algorithm postulated by the mathematician, or cre- ated by the artist.

As noted in the discussion of Figs. 8 and 9, the correct time sequence of self-similar natural phenomena was reported in 198 1 - 1982 as a repeated, geometrically similar construction in turbulence. It is a remarkable coincidence that this idea came at the same time as fractal geometry, when it was certainly unrelated to work that was being done in mathematics. In view of the growth experienced by both fractal geometry and buckling flows (Bejan, 1995), it is even more amazing that the two fields have not intersected until now.

Acknowledgment. This work was supported by the National Science Foundation. Figures 1-7 were drawn by Kathy Vickers. I must thank two friends and fellow engineers who drew my attention to street networks, Constantin Comanita (Director of Engineering, City of Galati, Romania) and Ruben Ledezma (President of EMTRAVENCA, Caracas, Venezuela). 1 also thank the Editor, Prof. Charles M. Harman, for his guiding comments on an earlier version of the manuscript.

* from the Latin verb frungere (to break), which survives unchanged in both Italian and Romanian.

f from the Latin verb constriikre (to build), which survives as construire in both Italian and Romanian.

Street Network lleory of Organization.. . . 107

References

Barnsley, M., 1988, Fractals Everywhere, Academic Press, Boston. Bejan, A., 1981, "On the Buckling Property of Inviscid Jets and the

Origin of Turbulence," Letters in Heat and Mass Transfer, Vol.

Bejan, A., 1982, Entropy Generation through Heat and Fluid Flow, Wiley, New York, chapter 4.

Bejan, A., 1984, Convection Heat Transfer, Wiley, New York, chapter 8.

Bejan, A., 1995, Convection Heat Transfer, second edition, Wiley, New York, chapters 6-9.

Bejan, A., 1998, Advanced Engineering Thermodynamics, second edition, Wiley, New York.

Bern, M. W. and Graham, R.L., 1989, "The Shortest-Network Prob- lem," Scientific American, January, pp. 84-89.

Cohn, D.L., 1954, "Optimal Systems: I. The Vascular System," Bulletin of Mathematical Biophysics, Vol. 16, pp. 59-74.

Courant, R. and Robbins, H., 1941, What is Mathematics? Oxford University Press, London, pp. 354-361.

Gibson, C.H., 1991, "The Direction of the Turbulence Cascade," In Some Unanaswered Questions in Fluid Mechanics, Trefethen, L.M., Panton, R.L. and Humphrey, J. A. C., eds., ASME paper

Mandelbrot, B.B., 1983, The Fractal Geometry of Nature, W. H . Freeman and Company, New York.

Richardson, L. F., 1926, "Atmospheric Diffusion Shown on a Dis- tance-Neighbor Graph," Proceedings of the b y a t Society, Lon- don, Vol. A 110 (756), pp. 709-737.

Thompson, D'A. W., 1942, On Growth and Form, Cambridge Uni- versity Press, Cambridge, United Kingdom.

8, NO. 3, pp. 187-194.

91-WA/FE-1.

Documents

Street network theory of organization in nature