Today, a matter of size. The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
I've done a lot of work on
the similarity between large and small systems.
Years ago I measured the behavior of large objects
at low gravity for NASA, by spinning small objects
in a centrifuge. That elevated gravity has less
effect on small objects because the force of
gravity increases as the cube of size. That's why a
fly with slightly sticky feet can walk on the
ceiling. Scientific modeling is so much fun because
it generates surprises like that.
Now an article in the latest New York Times
calls my NASA work back to mind. It begins by
observing that a mouse eats twenty-two times as
much food per ounce of its own weight as an
elephant does. Is the elephant that much more
efficient? Stop and think about your cats' grocery
bill. Ours eat what seems to be a lot of food for
their size. It turns out that the amount any animal
eats increases only as the three-quarters power of
its mass. Diverse as we animals are, that rule is
A group of scientists at New Mexico's Santa Fe
Institute have been studying modeling laws for
living things, and they've begun to see through
some of the mystery of them. That food/weight rule
is only part of the story. Other things scale as
The larger the animal, the slower its pulse rate.
Blood has to move more slowly in a very large aorta
to keep from choking in tiny capillaries. Conduits
that carry blood or sap keep subdividing to carry
nutrients into all parts of any living thing. The
systems that carry sap to plant extremities not
only look like our arterial tree; their smallest
parts are the same size as ours, because
both have to divide down to near-molecular
There's more: While any animal's surface area
varies as the square of its length, its volume
varies as the cube. That means large animals have
less surface area for cooling blood, and their
metabolic rates have to be slower. And, as we
generate more and more such rules, we find they're
Strength is another scalable quantity. An ant can
lift a hundred times its weight. We struggle
even to lift just our own weight. That's because,
if one animal weighs eight times what another does,
it has only four times the cross-sectional area in
its limbs. And strength increases with
cross-sectional area, not with weight.
Then there's a rule that says the number of species
decreases with the mass of their members. Some ten
million insect species occupy this Earth; 40,000
birds, fish, and mammals; and that one species of
blue whale swims at the peak of the pyramid.
Modeling rules among species aren't new, but this
one reminds us how our lives are interlaced with
those of sharks, ants, and robins.
Change the scale of life and you change its
texture. I saw that dramatically at a Women's NBA
Championship game last year. With the players
scaled down by ten percent in size, the game
changed utterly (and, for my money, it changed for
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
Johnson, G., Of Mice and Elephants: A Matter of Size.
The New York Times, Science Times,
Tuesday, January 12, 1999, pp. D1-D2.
J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook. 3rd
ed., Cambridge, MA: Phlogiston Press, 2004, Click here for a free copy.
For more on modeling and scaling laws, see Episode 68.
My Mechanical Engineering colleague, Roger
Eichhorn, struck by the need for an encapsulation
of these scaling laws in arithmetic form, created
the following summary. (Note that the symbol ^
means "raised to the power of." Thus (mass)^1/4
means the fourth root of mass.)
Heart beats per lifetime is constant (around one
Life span is proportional to (mass)^1/4
Pulse rate is proportional to (mass)^-1/4
Metabolic rate is proportional to (mass)^3/4
Strength is proportional to (mass)^2/3
Population density is proportional to
Average number of offspring is proportional to
Time required to reproduce is proportional to
These show all properties except pulse rate
increasing with mass. Obviously, these rules are
only approximate. For example, little dogs
sometimes outlive big ones, and the human species
has managed to extend its own life beyond that of a
horse. But in the larger overall picture, the rule
is pretty accurate.
Image by Maria Szigmond Baca,
permission of Peter Gordon
The Engines of Our Ingenuity is
Copyright © 1988-1999 by John H.