Constructal law

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Design in nature is flow. The two sides of the constructal law are these: we predict the occurrence of design in nature, and we design devices for human use.

Constructal law is a theory that the generation of design (configuration, pattern, geometry) in nature is a physics phenomenon that unites all animate and inanimate systems.[1] The constructal law was stated by Adrian Bejan in 1996 as follows: "For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it."[2][3]

The tree is the natural flow design for achieving flow access between one point and a volume. Alternating trees achieve flow access between two planes. Natural porous media exhibit multi-scale flow structures consistent with the multiple scales and performance of alternating trees.

"Constructal" is a word coined by Adrian Bejan, in order to describe the natural tendency of flow systems (rivers, trees and branches,[4] lungs, tectonic plates [5] as well as engineered forms) [6] to generate and evolve structures that increase flow.[2][7]

The constructal law was proposed in 1996 as a summary of all design generation and evolution phenomena in nature, bio and non-bio. The constructal law represents three steps toward making “design in nature” a concept and law-based domain in science:[3]

  1. Life is flow: all flow systems are live systems, the animate and the inanimate.[1]
  2. Design generation and evolution is a phenomenon of physics.[8]
  3. Designs have the universal tendency to evolve in a certain direction in time.[9]

The constructal law is proposed as a first principle of physics accounting for all design and evolution in nature. It holds that shape and structure arise to facilitate flow. The designs that happen spontaneously in nature reflect this tendency: they allow entities to flow more easily – to measurably move more current farther and faster for less unit of useful energy consumed.[5][10][11][12][13] Rain drops, for example, coalesce and move together, generating rivulets, streams and the mighty river basins of the world because this design allows them to move more easily.[14]

The constructal law asks the question: Why does this design arise at all? Why can't the water just seep through the ground? The constructal law provides this answer: Because the water flows better with design. The constructal law covers the tendency of nature to generate designs to facilitate flow.[15]

The constructal law proclaims that this is why we find a similar tree-like structure in all designs that move a current from a point to an area or an area to a point. The lightning bolts that flash across the sky generate a tree-like structure because this is a good design for moving a current (electricity) from an area (the cloud) to a point (a church steeple or another cloud). The circulatory and nervous systems of biological creatures generate a similar tree-like design because they too are moving currents from a point to an area and from an area to a point.[16]

Although treelike structures are a very common design in nature, they are only one manifestation of the constructal law. In a simple example, logs floating on a lake or icebergs at sea orient themselves perpendicular to the wind which increases the transfer of motion from the moving air body to the water body. A more complex example is the design of animals that have evolved to move mass more efficiently (to cover more distance per unit of useful energy) across the landscape.[17][18][19][20]

This includes the seemingly “characteristic” sizes of organs, the shape of bones, the rhythm of breathing lungs and beating hearts, of undulating tails, running legs, and flapping wings. The constructal law proclaims that all these designs have arisen—and work together—to allow animals, like raindrops in a river basin, to move more easily across a landscape.[21][22] Because human beings are not separate from but a part of nature, their designs are also governed by the constructal law.[18][23][24]

The constructal law defines the time direction of all evolutionary design phenomena. It states that designs should evolve, acquiring better and better configurations to provide more access for the currents that flow through them. It defines in physics terms what it means to be “fittest”, to “survive”, and to be efficient. Not all changes are improvements, but those that stick measurably enhance flow.[7][25] The constructal law states that design generation and evolution are macroscopic physics phenomena that arise naturally to provide better and better flow access to the currents that run through them.

Constructal design occurs at every scale. Each component of an evolving flow system—each rivulet, each tree, and each road—acquires evolving designs to facilitate flow access. As these elements coalesce into larger and larger structures (into evolving river basins, forests and transport networks), a hierarchy emerges such that the varying sized components work together so that everything flows more easily.[14] This is seen in the shape and structure of the neural networks in the brain, of the alveoli in the lung, the size and distribution of vegetation in the forest and of human settlements on the map.[18][24][26]

In the big picture, all the mating and morphing flows on the largest system that surrounds us, the Earth itself, evolve to enhance global flow. For example, trees and other forms of vegetation that move moisture from the ground to the air are components of the larger global system, including forests, river basins and weather patterns, that have the tendency to equilibrate all the moisture on Earth.[15][27] The constructal law states that every flow system is destined to remain imperfect. The direction of design evolution is toward distributing the imperfections of the system, such that the “whole” flows easier (e.g., river basin, animal body, human vehicle).[28]

Evolution never ends. Optimality statements (minimum, maximum, optimum, static, end design, destiny) have only local, limited applicability. The constructal law covers them because it is about the time direction of the evolutionary design phenomenon.

The constructal law is a law of physics - the law of design generation and evolution in nature. The natural phenomenon is not the elimination but the distribution (better and better over time) of imperfection. The distribution of imperfection generates the geometry (shape, structure) of the system.[29][30]

For example, in point-area and point-volume flows, the constructal law predicts tree architectures, such flows displaying at least two regimes: one highly resistive and one with lower resistivity. The constructal-law tendency manifests itself at every scale.[31]

Some domains of application
Application What flows Tree channels: Low Resistivity Interstitial spaces: High Resistivity
Packages of electronics Heat High-conductivity inserts (blades, needles) Low conductivity substrate
Urban traffic People Low-resistance street car traffic Street walking in urban structure
River basins Water Low-resistance rivulet and rivers Darcy flow through porous media
Lungs Air Low-resistance airways, bronchial passages diffusion in alveoli tissues
Circulatory system Blood Low-resistance blood vessels, capillaries, arteries, veins diffusion in capillaries tissues

The constructal law provides a unifying theory of evolution. It holds that inanimate and animate phenomena generate evolving configurations to move more easily. The constructal law also provides the physics definition of life, of what it means to be alive. It states that life means flow and the free generation of design. If the flows stop, the system is dead (in thermodynamic equilibrium). The constructal law is the physics law of life, design and evolution.[1][8][16]

Constructal thermodynamics[edit]

Thermodynamics rests on two laws. Both are first principles: The first law commands the conservation of energy, and the second law summarizes the tendency of all currents to flow from high (temperature, pressure) to low. These two laws are about systems in the most general sense, viewed as black boxes, without shape and structure.

The two laws of thermodynamics do not account for nature completely. Nature is not made of black boxes. Nature’s boxes are filled with evolving, morphing configurations—even the fact that they have names (rivers, blood vessels) is due to their appearance, pattern, or design. Where the second law commands that things should flow from high to low, the constructal law commands that they should flow in configurations that flow more and more easily over time.[29]

Classical thermodynamics versus constructal thermodynamics
Thermodynamics Constructal theory
State Flow architecture (flow structure)
Process Change of structure (design change)
Properties Global objective and global constraints
Equilibrium state Equilibrium flow architecture
Fundamental relation Fundamental relation
Constrained equilibrium states Nonequilibrium architectures
Removal of constraints Increased freedom to morph
Energy minimum principle Evolution toward greater flow access

In contrast to fractal models of observed objects in nature, the constructal law is predictive and thus can be tested experimentally.[32][33][34][35] Many natural designs, animate and inanimate, have been explained and unified by the constructal law,[5][13][14][36][37] for example:

  • Global circulation and climate
  • River basin design: Horton's rules of stream numbers (~4) and lengths (~2), and all the other scaling rules (e.g., Melton, Hack) of river basins all over the world.
  • The distribution of city sizes and numbers, i.e. Zipf's law relating log (size) versus log (rank).
  • The distribution of tree sizes and numbers on the forest floor, which is also a Zipf line of log (size) versus log (rank).
  • The flow of education as a morphing vasculature on the globe, and the rigidity of university rankings.
  • Vision, cognition, and the "golden ratio" phenomenon.
  • The entire architecture of vegetation: roots, trunks, canopies, branches, leaves, and the forest, including the prediction of Leonardo da Vinci's rule, Huber's rule, and the Fibonacci sequence.
  • Pedestrian movement, speeds, and patterns
  • The emergence of urban traffic design
  • The entire morphogenesis of dendritic crystals (e.g., snowflakes), as a flow structure that facilitates the flow of the heat of solidification
  • The scaling law of all animal locomotion (running, flying, swimming): speeds, frequencies, forces and the work spent per unit of mass moved and distance traveled.
  • The evolution of speed in sports.
  • Kleiber's law, the relationship between metabolic rate and body size
  • the relationship between breathing and heart beating times and body size
  • the relationship between the mass transfer contact area and body mass
  • The human bronchial tree with 23 levels of bifurcation.
  • the dimensions of the alveolar sac,
  • the total length of the airways,
  • the total alveolar surface area,
  • the total resistance to oxygen transport in the respiratory tree.

Conflict with Metabolic Theory of Ecology[edit]

Adrian Bejan first published his ideas about the constructal law in 1997.[2] Also in 1997, West, Brown & Enquist published a paper in the journal Science where they come to rather similar conclusions.[38] Both of these theories predict the branching structure of lungs, blood streams, and trees. Both of these theories are based on thermodynamic principles. Yet, regardless of which publications one looks at, these two "camps" with their followers refuse to reference each other. Instead of working together to achieve some sort of mutual ground, they are stubbornly ignoring each other, it seems. The conflict is evident for anyone who has had the opportunity to speak directly to either of the people involved and ask them about these issues. It is not an open conflict, so the only obvious sign that the conflict is there is that they all write about the same things but refuse to reference each other's works, as is usually common in science.

This refusal to recognize the work of others isn't necessarily limited to this conflict though. Both parties (constructal and metabolic) have failed to recognize that they are borrowing ideas from their predecessors. A. Pütter was the first one to include thermodynamic concepts into biology. After him, the mathematician Nicolas Rashevsky put down a lot of work to establish "biophysics" as a science, mostly based on thermodynamics. In the 1940s up until 1960s, Ludwig von Bertalanffy did much work in this area, e.g. defining how individual cells function from a thermodynamic perspective. In a review from 1982,[39] Angelos Economos states that two books published by Rashevsky (1960) and Rosen (1967) contain very similar ideas to the ones expressed both in Constructal Theory and Metabolic Theory of Ecology. These are just a few examples of all the work that has been done in this area, but none of these references appear in any of the work presented on constructal theory, and not in any work presented on Metabolic Theory of Ecology either.

While conflicts between scientific groups may not be uncommon, the conflict exists. One might also proceed to put this forth as a criticism; neglecting to reference previous work can be considered to be a form of plagiarism, or "cheating", which is not something that a scientist should be associated with. Both constructal law and Metabolic Theory of Ecology claim to be original ideas, whereas they actually stand on the shoulders of giants (not only the giants of their predecessors, but also by working on the same things and reading each other's works without referencing).

The comment above is not entirely correct: there is no conflict. Bejan’s Constructal Law was published one full year before West et al.’s model. Furthermore, Bejan cites West et al.’s work, while the reverse is not happening. Finally, Bejan has commented in 2005 about the “conflict” noted by the commentator. Here are the details:

  1. Bejan published his ideas about the Constructal Law in 1996, not 1997. In fact, he published three journal papers [40][41][42] before West et al. published their first paper. In the summer of 1997, Bejan’s thermodynamics book [2] added to his first papers, and defined the field. West et al.’s model also appeared in the summer of 1997. In particular, Bejan’s second paper [41] appeared on 1 November 1996 (see the journal cover of issue number 4), contrary to the March 1997 date given to the issue in which this paper appeared. At the time, the International Journal of Heat and Mass Transfer was experiencing an overflow of accepted papers, and was assigning 1997 dates to 1996 papers.
  2. In a 2005 letter to Physics Today, Bejan [43] drew attention to the fact that in their review published in Physics Today [44] West and Brown did not mention the Constructal Law, the theory, and the precedent. Bejan also drew the distinction between model (description, empiricism, as in the metabolic model) and law (prediction, theory, as in the Constructal Law). West and Brown’s work is not theory. It is modeling, which means empiricism, description, a simple facsimile of the observed. The three or more ad-hoc assumptions on which their model is based are consequences of the Constructal Law, and have been predicted based on constructal theory before West et al.’s 1997 model was published.

The “conflict” noted by the commentator is asymmetric. Bejan referenced West & Brown’s work in his 2000 book (in four places),[45] while West & Brown have been silent about the Constructal Law. The asymmetry may be a good topic for future reviews of the constructal-law field, which includes all the modeling results of West et al. and much more. Bejan called for such a review in his 2005 letter to Physics Today:[43]

“Because West’s three assumptions are consequences of constructal theory, every successful derivation of an allometric law that West and coworkers make is an affirmation of the validity of that theory. In return, every successful invocation of the constructal law in domains far removed from the living flow systems of West and coauthors is an indication not only that their model is correct but that it is an integral part of a theoretical framework that unites biology with physics and engineering. In addition to allometric laws, constructal theory covers many phenomena that fall well outside the biological scaling reviewed by West and Brown. Some examples are turbulent flow structure in various flow configurations, cracks in shrinking solids, the structure of animal hair, refraction in geographical economics, flight, atmospheric and oceanic circulation, and the structure of power plants and refrigeration plants—that is, the architecture of the “human-plus-machine” species.”

The Constructal Law is universally valid, while the metabolic model is not. The overlap that happened by accident in 1996-1997 covers a small territory. In addition to the inanimate, social and non-“metabolic” phenomena noted above in Bejan’s quote,[43] there is the 1/6-power scaling of all animal locomotion (flying, running, swimming)[46] and the 1/4-scaling of life span and life travel.[47] These broad phenomena are predicted based on the Constructal Law, while they cannot be described by using the metabolic model.

Criticisms of constructal theory[edit]

The main criticism of constructal theory is that its formulation is vague.[48][49] The constructal law states that “For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it”, but there is neither a mention of what these “currents” are nor an explicit definition of what “providing easier access” means. As a result, constructal theory is very versatile, but often unconvincing: depending on the choices made for the currents and the “access” to them, it can lead to extremely different results.

The second criticism of constructal theory is that there has been no attempt to prove it from first principles. Contrarily to alternative theories of non-equilibrium thermodynamics,[50][51] there is no proof of constructal theory based on simplified systems of statistical physics. The claim that constructal theory is a fundamental principle of thermodynamics itself has also been disputed.[52]

Alternative theories include:

Responses to criticisms of the constructal law[edit]

Bejan has responded to this criticism [53] by noting that the constructal law is not about what flows, but about the physics phenomenon of how any flow system acquires its evolving -configuration (design) over time. The constructal law is not about optimality (max, min, opt)—it is the definition of “life” in physics terms, and of the time direction of the changes in flow configuration.[54]

Bejan [53] also noted that the phenomenon governed by the constructal law (design in nature) is macroscopic (finite size, not infinitesimal). It is the birth of design and evolution of design in all the parts together. It is dynamic, not static. The evolution never ends. There is no end design, no destiny (max, min, opt).

Bejan and Lorente expanded on this [55] by explaining the difference between a law of physics (e.g., the constructal law) and the many invocations of the law, which underpin many “theories” based on the law. In the section titled “The constructal law versus the second law of thermodynamics” they noted that the constructal law and the second law are first principles. The constructal law is a useful reminder not only of what was missing in physics and thermodynamics (the law of design and evolution) but also of what is present. For example, contrary to the critic’s view, the first and second laws of thermodynamics did not require any “proof based on simplified systems of statistical physics.”

The constructal law is a statement of a natural tendency—the time direction of the phenomenon. It is as non-mathematical as the original statements of the second law:

Clausius: No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.

Kelvin: Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on the system.

A new law does not have to be stated in mathematical terms. The mathematization of the second law statement (and of thermodynamics) came later. The constructal law underwent the same evolution. The 1996 statement was followed in 2004 by a complete mathematical formulation of constructal-law thermodynamics.[56]

Like any other law of physics, the constructal law is a concise summary of observed facts: the natural tendency of flow systems to evolve toward configurations that provide easier access over time. The word “access” means the ability to move through a confined space such as a crowded room. This is why “finite size” appears in the constructal law statement. This mental viewing covers all the flow design and evolution phenomena, animate and inanimate, because they all morph to enter and to flow better, more easily.[55]


  1. ^ a b c T. Basak, The law of life: the bridge between Physics and Biology. Phys Life Rev 8, 249-252 (2011)
  2. ^ a b c d Bejan, Adrian (1997). “Advanced Engineering Thermodynamics,” (2nd ed.). New York: Wiley.
  3. ^ a b A. Kremer-Marietti and J. Dhombres, L’Épistemologie (Ellipses, Paris, 2006)
  4. ^ D. Tondeur, Y. Fan and L. Luo, Constructal optimization of arborescent structures with flow singularities. Chem Eng Sci 64, 3968−3982 (2009).
  5. ^ a b c S. Quéré, Constructal theory of plate tectonics. Int J Design & Nature Ecodyn 5, 242-253 (2010).
  6. ^ D. Queiros-Conde and M. Feidt, Eds., Constructal Theory and Multi-scale Geometries: Theory and Applications in Energetics, Chemical Engineering and Materials (Les Presses de L’ENSTA, Paris, 2009).
  7. ^ a b A. H. Reis, Constructal theory: from engineering to physics, and how flow systems develop shape and structure. Appl Mech Rev 59, 269-282 (2006).
  8. ^ a b L. Wang, Universality of design and its evolution. Phys Life Rev 8, 257-258 (2011)
  9. ^ N. Acuña, Mindshare. Igniting Creativity and Innovation Through Design Intelligence (Motion, Henderson, Nevada 2012).
  10. ^ A. F. Miguel, The physics principle of the generation of flow configuration, Phys Life Rev 8, 243-244 (2011)
  11. ^ A. F. Miguel. Constructal pattern formation in stony corals, bacterial colonies and plant roots under different hydrodynamics conditions. Journal of Theoretical Biology 242, 954-961 (2006)
  12. ^ A. F. Miguel, A. Bejan. The principle that generates dissimilar patterns inside aggregates of organisms. Journal Physica A 388, 727-731 (2009)
  13. ^ a b A. H. Reis and C. Gama, Sand size versus beachface slope – an explanation based on the Constructal Law, Geomorphology 114, 276-283 (2010).
  14. ^ a b c A. H. Reis, Constructal view of scaling laws of river basins. Geomorphology 78, 201-206 (2006).
  15. ^ a b A. F. Miguel, Natural flow systems: acquiring their constructal morphology. Int J Design & Nature Ecodyn 5, 230-241 (2010).
  16. ^ a b A. H. Reis, Design in nature, and the laws of physics. Phys Life Rev 8, 255-256 (2011)
  17. ^ Bejan, A; Marden, James H. (2006). Constructing Animal Locomotion from New Thermodynamics Theory. American Scientist, July–August, Volume 94, Number 4
  18. ^ a b c G. Resconi, Morphotronics and Constructal theory, LINDI 2011, 3rd IEEE Int. Symp. Logistics and Industrial Informatics, Budapest, Hungary, August 25–27 (2011).
  19. ^ L. C. Kelley and K. Behan, Empathy & evolution: how dogs convert stress into flow. Psychology Today 6 August 2012.
  20. ^ L. C. Kelley and K. Behan, The canine mind bows to the Constructal Law. Psychology Today 16 October (2012).
  21. ^ Bejan, A (2010). The Constructal Law Origin of the Wheel, Size, and Skeleton in Animal Design. American Journal of Physics, Vol. 78, No. 7: 692-699
  22. ^ A. F. Miguel. Constructal theory of pedestrian dynamics. Physics Letters A 373, 1734-1738 (2009)
  23. ^ Bejan, A; Merkx, Gilbert A, eds. (2007). "Constructal Theory of Social Dynamics." New York: Springer.
  24. ^ a b P. Kalason, Épistémologie Constructale du Lien Cultuel (L’Harmattan, Paris, 2007).
  25. ^ Bejan, A (2005). The Constructal Law of Organization in Nature: Tree-shaped Flows and Body Size. Journal of Experimental Biology. Vol. 208, No. 9: 1677-1686.
  26. ^ Lorente, S; Bejan, A (2010). Few Large and Many Small: Hierarchy in Movement on Earth. International Journal of Design & Nature and Ecodynamics. Vol. 5, No. 3: 1–14.
  27. ^ Bejan, A; Lorente, S; Lee, J (2008). Unifying Constructal Theory of Tree Roots, Canopies and Forests. Journal of Theoretical Biology 254: 529–540
  28. ^ Y. Ventikos, The importance of the constructal framework in understanding and eventually replicating structure in tissue. Phys Life Rev 8, 241-242 (2011).
  29. ^ a b Bejan, A; Lorente, S (2008). “Design with Constructal Theory,” Hoboken: Wiley.
  30. ^ M. Eslami and K. Jafarpur, Optimal distribution of imperfection in conductive constructal designs of arbitrary configurations. J Appl Phys 112, 104905 (2012).
  31. ^ M. R. Errera and C. A. Marin, A Comparison Between Random and Deterministic Dynamics of River Drainage Basins Formation, Engenharia Térmica (Thermal Engineering), Vol. 8, n. 01, 65-71 (2009).
  32. ^ R. A. Hart and A. K. da Silva, Experimental thermal-hydraulic evaluation of constructal microfluidic structures under fully constrained conditions. Int J Heat Mass Transfer 54, 3661-3671 (2011).
  33. ^ R. A. Hart, M. J. V. Ponkala and A. K. da Silva, Development and testing of a constructal microchannel flow system with dynamically controlled complexity. Int J Heat Mass Transfer 54, 5470-5480 (2011).
  34. ^ Z. Fan, X. Zhou, L. Luo and W. Yuan, Experimental investigation of the flow distribution of a 2-dimensional constructal distributor. Exp Therm Fluid Sci 33, 77−83 (2008).
  35. ^ K.-H. Cho, W.-P. Chang and M.-H. Kim, A numerical and experimental study to evaluate performance of vascularized cooling plates. Int J Heat Mass Transfer 32, 1186-1198 (2011).
  36. ^ D. Haller, P. Woias and N. Kockmann, Simulation and experimental investigation of pressure loss and heat transfer in microchannel networks containing bends and T-junctions. Int J Heat Mass Transfer 52, 2678−2689 (2009).
  37. ^ L. Chen, Progress in study on constructal theory and its applications. Science China, Technological Sciences 55 (3), 802-820 (2012).
  38. ^ West, G B; Brown, J H; Enquist, B (1997). "A general model for the origin of allometric scaling laws in biology." Science 276, 122-126.
  39. ^ Economos, A C (1982). "On the origin of biological similarity." Journal of Theoretical Biology, Volume 94, Issue 1, 7 January 1982, Pages 25–60. ISSN 0022-5193. (
  40. ^ A. Bejan, "Street Network Theory of Organization in Nature," Journal of Advanced Transportation, Vol. 30, No. 2, 1996, pp. 85-107
  41. ^ a b A. Bejan, "Constructal-Theory Network of Conducting Paths for Cooling a Heat Generating Volume," International Journal of Heat and Mass Transfer, Vol. 40, 1997, pp. 799-816 (published on 1 November 1996)
  42. ^ A. Bejan, "Theory of Organization in Nature: Pulsating Physiological Processes," International Journal of Heat and Mass Transfer, Vol. 40, 1997, pp. 2097-2104 (published on 4 February 1997).
  43. ^ a b c A. Bejan, Constructing a Theory for Scaling and More, Physics Today, 2005, Vol. 58(7), p. 20
  44. ^ G. B. West and J. H. Brown, Life’s universal scaling laws, Physics Today, September 2004, pp. 36-42
  45. ^ A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000
  46. ^ A. Bejan and J. H. Marden, Unifying constructal theory for scale effects in running, swimming and flying, Journal of Experimental Biology, Vol. 209, 2006, pp. 238-248.
  47. ^ A. Bejan, Why the bigger live longer and travel farther: animals, vehicles, rivers and the winds. Nature Scientific Reports, Vol. 2, no. 594; DOI: 10.1038/srep00594 (2012)
  48. ^ Kleidon, A (2009). “Nonequilibrium thermodynamics and maximum entropy production in the Earth system: Applications and implications.” Naturwissenschaften, 96, 653-677.
  49. ^ Kleidon, A; Malhi, Y; Cox PM (2010) “Maximum entropy production in environmental and ecological systems.” Philos Trans R Soc Lond B Biol Sci., 365, 1297–1302.
  50. ^ Dewar, R (2003). “Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states.” J. Phys. A: Math. Gen., 36, 631-641.
  51. ^ Dewar, RC (2005). “Maximum entropy production and the fluctuation theorem.” J. Phys. A: Math. Gen., 38, L371-L381.
  52. ^ Bejan, A (2006). “Constructal theory of pattern formation.” Hydrol. Earth Syst. Sci. Discuss., 3, 1773-1807.
  53. ^ a b Bejan, A. (2010). Design in nature, thermodynamics, and the constructal law. Comment on “Life, hierarchy, and the thermodynamic machinery of planet Earth” by A. Kelidon, Physics of Life Reviews Vol. 7, No. 4: 467-470.
  54. ^ L. A. O. Rocha, Constructal law: from the law of physics to applications and conferences. Phys Life Rev 8, 245-246 (2011).
  55. ^ a b Bejan, A & Lorente, S. (2011). “The constructal law and the evolution of design in nature,” Physics of Life Reviews, Vol. 8, No. 3: 209-240
  56. ^ Bejan, A; Lorente, S (2004). The Constructal Law and the Thermodynamics of Flow Systems with Configuration. International Journal of Heat and Mass Transfer 47: 3203–3214

External links[edit]

Popular literature[edit]

  • Maggie Wittlin, "A Finger on the Pulse of the World", SEED Magazine, January 13, 2006.
  • H. Poirier, Une théorie explique l’intelligence de la nature, Science & Vie, 1034, 2003, pp. 44–63.
  • Natural Design with Constructal Theory, Mechanical Engineering magazine, [6]
  • The many and the few, Mechanical Engineering magazine, [7]
  • Laufen = Fliegen = Schwimmen, [8]
  • Jeremy Berlin, Gaudi’s Materspiece, National Geographic, December 2010, pp. 24–27.
  • Tara Bruno, Higher Navel, faster feet, Science World, October 18, 2010, Vol. 67, No. 3, p. 5.
  • A. Boyle, Why March Madness isn’t that mad, MSNBC, 2 March 2011.
  • Jonathan Mitchell, Constructal Law: A Theory of Everything, Studio 360, 2 March 2012.
  • S. Pappas, Fastest swimmers make webbed hands out of water, Live Science, 20 June 2012.