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Animals spinning their wheels: nature anticipated mankind in the development of one of civilization's fundamental machines.


No topic is more "mechanical engineering" than the wheel. When the wheel appeared, he movement of humanity jumped to new dimensions, higher speeds, longer distances, and less effort per unit of mass moved through a distance. If engineering is the kitchen of civilization, then the wheel is the key ingredient.

Today we take the wheel for granted, because it is everywhere. Older generations were more keenly aware of where we came from, and commemorated the wheel in the emblems of cities, business groups, trade unions, and engineering departments in universities. It is good that we maintain these images. The icon for "settings" on my iPhone consists of several wheels, even though there are no wheels inside.

Along with complacency comes arrogance. "Everybody knows" that nature did not invent the wheel. The famous Harvard biologist Stephen Jay Gould wrote a book about natural history and gave it the title Kingdoms Without Wheels.

The common wisdom is that humans invented the wheel and that it does not exist in nature. This idea places humans in a world distinct from and higher than all the other animals. Darwin must be rolling in his grave.

The common wisdom is wrong. But first, here is a brief reminder of why the wheel was such a dramatic change in how humans move. The work, W, spent on sliding a mass, M, through a horizontal distance, L, is equal to the weight, Mg, times L and a coefficient of friction, [mu]. With wheels placed between M and the ground, the work formula remained the same (W = [mu]MgL) but the coefficient [mu] decreased considerably.


The time direction of this change, from high [mu] to low [mu], is in accord with the constructal law of design and evolution in nature, which states that all flow systems (including human movement) persist in time by changing into configurations that flow more and more easily.

Humans and their loads found an easier way to move on the map, just as river basins find better tree-shaped flow designs every year. Seepage in the wet mud is not eliminated by the birth of the river channel, because seepage continues to improve flow by finding new channels. Similarly, when humans got their stuff off the ground and rolled with it, sliding was not eliminated. It persists today, at speeds and scales small enough to be comparable with the movement that existed before the wheel. For example, when we stock the shelves in grocery stores, we slide cans or boxes into place. On top of the old design of movement that slid loads across the ground, a better one with reduced friction was added.

An Evolutionary Design

The natural emergence of the wheel design can be predicted by using the constructal law in two ways. First, consider the evolution of the wheels made by humans. In the beginning, the wheel was a solid disk. The wheel and the ground made contact over a narrow strip on the rim. The stresses were distributed nonuniformly in the disk. The highest stresses were in the vicinity of the contact strip. Most of the wheel body was not stressed.


Less material is needed when the maximum allowable stresses are distributed more uniformly through the loaded structure. When the design requires less material, it becomes lighter. A single column with uniform cross-section requires the least material to support a weight. The stresses in the column are distributed uniformly. The volume of the column is a tiny fraction of the volume of the solid disk.

The column is a much lighter organ than the disk to carry on the vehicle, but one column is not enough to serve as a wheel. Three or more columns, a rigid rim, and a rigid track are required to prevent the body from falling. Fewer columns are lighter, and this constructal-law direction for easier movement in time is confirmed by the evolution of wheel technology from solid disk to spokes supporting a rim.

Second, imagine the horizontal movement of a terrestrial animal as a rolling body. Imagine the human body, or the front or back half of a quadruped.

The leg is a single column. Many bipedal creatures can stand still on one leg. There are many kinds of birds that sleep that way. To walk requires the forward movement--in essence, a falling forward from the one-legged stance.

The order of magnitude of the speed of falling forward is the same as the speed of falling down, namely V ~ [(Rg).sup.1/2], where the distance above the ground (R) is the f body length scale, which is the same as the length of the leg. The body mass scale is equal to the density of the body times the length scale of the leg cubed. Coincidentally, because M ~ [??]R3, where [??] is the body density, the speed of locomotion is also recognized as V ~ [M.sup.1/6][g.sup.1/2][[??].sup.-1/6], in agreement with the known characteristic speeds of all animals (runners, fliers, and swimmers), and with the world-record human speeds in running and swimming during the past 100 years.

To maintain this horizontal speed, the body design requires a second column, which must also have the ability to absorb shocks and to elongate itself to reposition the body weight to its traveling height (R). That's why legs are most efficient when they come in pairs. The second leg is brought forward in time to catch the body before it falls too far or accelerates too much toward the ground. This function is like the cooperation of the spokes of a wheel. But because it can be done by consistently cycling two spokes, which are paired legs in animals, that is the way nature does it.

A third leg would continue the work of the first two, but it would increase by a factor of 3/2 the mass of the organs that the animal must carry along in order to have locomotion. Thus, the third beat of this rhythm is executed by the first leg, which swings outward, from behind the body, to take the position that the third leg would have occupied. This alternative is much lighter and faster, and (in accord with the constructal law) it is the natural design of rolling locomotion.


The legs, as two columns swinging back and forth, perform the function of an entire wheel-rim-track assembly. They do it in record fashion-one wheel with just two spokes and with uniformly stressed material in each spoke. No wheel is stronger and lighter than this. The animal body is both wheel and vehicle for the animal mass that moves on the surface of the Earth.

Changing Speeds

Nature evolved not only the design of wheel-like movement, but also the design for changing speeds.

Larger bodies move faster--that is, as the average of speed over a lifetime. The cheetah may be able to outrun any other land animal in a short sprint, but like all cats, it spends much of its time sleeping and watching. The cheetah can reach running speeds exceeding 100 km per hour for a distance of about 500 meters. Then it must rest or die.

The body mass of a cheetah is generally less than 40 kg. A medium-size man has at least twice that mass, and humans generally move more mass faster and farther than cheetahs over long periods of time. Considering the movement over a lifetime, humans are bigger, faster, and more economical vehicles of animal mass than cheetahs.

Because bigger means faster, greater speed could be found by increasing the height of the body mass above the ground.

Animal bodies have shapes with multiple scales. A simple body shape is the elongated body of a serpent. A quadruped's body is much taller and more massive than a serpent's, but it is also elongated: its length is greater than the height of its torso.

Evolution toward higher speeds points toward designs that are taller. This agrees with the evolutionary design of animal locomotion: quadrupeds occurred after swimmers and crawlers, not the other way around.

The animal becomes faster by orienting its longer dimension vertically, i.e. by making itself taller. The constructal-law direction is from long to tall, and this too agrees with the evolution of animal locomotion: bipedal locomotion evolved after quadrupedal locomotion. The vertical orientation increased the size of R. In the formula for speed, in which velocity correlates to the square root of R times g, a greater value of R contributes to higher speed.

There are many examples of the animal design for changing speeds. A human has two speeds: walk and run. A horse has three speeds: walk, trot, and gallop. The human and the horse increase their speeds by increasing the height from which their centers of gravity fall during each locomotion cycle.

From the walk to the gallop, the horse body movement changes abruptly such that the amplitude of the jump increases stepwise. The animal body with three different designs for movement (rhythm) is like one vehicle with one engine and a gearbox with three speeds.

Engineering and Nature

The evolutionary designs of nature have arrived at wheel-like locomotion and at changes in body movement that result in changing speeds. The designs developed by humans are late comers to this long evolutionary sequence. They come from the same natural tendency to move on Earth more easily, to go with the flow. The human wheel and gearbox were not copied from nature. They are not fruits of biomimetics. These artifacts are part of our own evolutionary design for moving our mass on the landscape.

Engineering makes a contribution to understanding design in nature--a contribution that the other sciences cannot make. Biologists and geophysicists argue that one cannot witness and test "evolution" because of the enormous time scale of the phenomenon. Engineers bring an important idea to the current debate of design and evolution in nature. Yes, we can witness and test evolution during our lifetime, by studying the evolution of our designs and technologies. These evolutionary designs illustrate the time direction of the constructal law, which unites animate and inanimate design phenomena.

To Learn More

This article is based on three recent research articles:

A. Bejan, 2010, "The constructal-law origin of the wheel, size, and skeleton in animal design," American Journal of Physics, Vol. 78[7], July, pp. 692-699.

J. D. Charles and A. Bejan, 2009, "The evolution of speed, size, and shape in modern athletics," Journal of Experimental Biology, Vol. 212, pp. 2419-2425.

A. Bejan and S. Lorente, 2010, "The constructal law of design and evolution in nature," Philosophical Transactions of the Royal Society B, Biological Sciences, Vol. 365, pp. 1335-1347.

See also the constructal-law portal at

This research was sponsored by the National Science Foundation and the Air Force Office of Scientific Research.

Adrian Bejian is the J.A. Jones Distinguished Professor of Mechanical Engineering at Duke University. He can be reached at
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Author:Bejan, Adrian
Publication:Mechanical Engineering-CIME
Geographic Code:1USA
Date:Jun 1, 2011
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