This lecture reviews a relatively recent body of heat transfer work that bases on a deterministic (constructal) principle the occurrence of geometric form in systems with internal flows. The same principle of global optimization subject to constraints allow us to anticipate the natural (animate and inanimate) flow architectures that surround us. The lecture starts with the example of the optimal spatial distribution of material (e.g., heat exchanger equipment) in power plants. Similarly, void space can be allocated optimally to construct flow channels in the volume occupied by a heat generating system. The lecture continues with the optimization of the path for heat flow between a volume and one point. It shows that when the heat flow can choose between at least two paths, low conductivity versus high conductivity, the optimal flow structure for minimal global resistance in steady flow is a tree. Nearly the same tree is deduced by minimizing the time of discharge in the flow from a volume to one point. Analogous tree-shaped flows are constructed in pure fluid flows, and in flow through a heterogeneous porous medium. The optimization of trees that combine heat transfer and fluid flow is illustrated by means of two-dimensional trees of plate fins. The method is extended to the superposition of two fluid trees in counterflow, as in vascularized tissues under the skin. The two trees in counterflow are one tree of convective heat currents that effect the loss of body heat. It is shown that the optimized geometry of the tree is responsible for the proportionalities between body heat loss and body size raised to the power 3/4, and between breathing time and body size raised to the power 1/4. The optimized structures are robust with respect to changes in some of the externally specified parameters. When more degrees-of-freedom are allowed, the optimized structure looks more natural. The lecture outlines a unique opportunity for engineers to venture beyond their discipline, and to construct an engineering theory on the origin and workings of naturally organized systems. *[S0022-1481(00)02403-8]*

*Shape and Structure, From Engineering to Nature*, Cambridge University Press, Cambridge, UK.

*Advanced Engineering Thermodynamics*, Second Ed., John Wiley and Sons, New York.

*Thermal Design and Optimization*, John Wiley and Sons, New York.

*Availability Analysis: A Guide to Efficient Energy Use*, Prentice-Hall, Englewood Cliffs, NJ.

*Entropy Generation Through Heat and Fluid Flow*, John Wiley and Sons, New York.

*Advanced Engineering Thermodynamics*, John Wiley and Sons, New York.

*Entropy Generation Minimization*, CRC Press, Boca Raton, FL.

*Thermodynamique et Optimisation E´nergetique des Syste`mes et Procede´s*, Technique et Documentation, Lavoisier, Paris.

*Finite-Time Thermodynamics and Thermoeconomics*, Taylor and Francis, New York.

*Endoreversible Thermodynamics of Solar Energy Conversion*, Oxford University Press, Oxford, UK.

*Generalized Thermodynamics*, Editura Tehnica, Bucharest.

*Thermodynamic Optimization of Complex Energy Systems*, Kluwer Academic Publishers, Dordrecht, The Netherlands.

*Energy and the Environment*, Kluwer Academic Publishers, Dordrecht, The Netherlands.

*Desert Animals: Physiological Problems of Heat and Water*, Oxford University Press, Oxford, UK.

*How Animals Work*, Cambridge University Press, Cambridge, UK.

*Scaling: Why Is Animal Size So Important?*, Cambridge University Press, Cambridge, UK.

*The Ecological Implications of Body Size*, Cambridge University Press, Cambridge, UK.

*Size, Function, and Life History*, Harvard University Press, Cambridge, MA.

*Life’s Devices*, Princeton University Press, Princeton, NJ.

*Design and Analysis of Heat Sinks*, John Wiley and Sons, New York.

*Convection Heat Transfer*, 2nd Ed., John Wiley and Sons, New York.

*Air Cooling Technology for Electronic Equipment*, CRC Press, Boca Raton, FL.

*Convection Heat Transfer*, John Wiley and Sons, New York, Chapter 4, Problem 11, p. 157.

*Fractal River Basins*, Cambridge University Press, Cambridge, UK.

*Fractals, Scaling and Growth Far From Equilibrium*, Cambridge University Press, Cambridge, UK.

*On Growth and Form*, Cambridge University Press, Cambridge, UK.

*The Fractal Geometry of Nature*, Freeman, New York.

*Thermodynamic Theory of Structure, Stability and Fluctuations*, John Wiley and Sons, London.

*Fractal Space-Time and Microphysics*, World Scientific, Singapore.

*Morphometry of the Human Lung*, Academic Press, New York.

*Fractals in Biology and Medicine*, Birkha¨user Verlag, Basel, Switzerland.

*Trees and Networks in Biological Models*, Wiley, Chichester, UK.

*Geomorphology*, Prentice-Hall, Englewood Cliffs, NJ.

*Fluvial Processes in Geomorphology*, W. H. Freeman, San Francisco, CA.

*Theoretical Geomorphology*, 2nd Ed., Springer-Verlag, Berlin.

*Geomorphology*, Methuen, London.

*Rivers: Form and Process*, Longman, London.

**43**, to appear.

*Mathematics and Optimal Form*, Scientific American Books, New York.