Scaling
Michael Fowler, UVa 10/10/06
Galileo begins “Two
New Sciences” with the striking observation that if two ships, one large
and one small, have identical proportions and are constructed of the same
materials, so that one is purely a scaled up version of the other in every
respect, nevertheless the larger one will require proportionately more
scaffolding and support on launching to prevent its breaking apart under its
own weight. He goes on to point out that
similar considerations apply to animals, the larger ones being more vulnerable
to stress from their own weight (page 4):
Who does not know that a horse falling from a height of
three or four cubits will break his bones, while a dog falling from the same
height or a cat from a height of eight or ten cubits will suffer no
injury? ... and just as smaller animals
are proportionately stronger and more robust than the larger, so also smaller
plants are able to stand up better than the larger. I am certain you both know that an oak two
hundred cubits high would not be able to sustain its own branches if they were
distributed as in a tree of ordinary size; and that nature cannot produce a
horse as large as twenty ordinary horses or a giant ten times taller than an
ordinary man unless by miracle or by greatly altering the proportions of his
limbs and especially his bones, which would have to be considerably enlarged
over the ordinary.
To see what Galileo
is driving at here, consider a chandelier lighting fixture, with bulbs and
shades on a wooden frame suspended from the middle of the ceiling by a thin
rope, just sufficient to take its weight (taking the electrical supply wires to
have negligible strength for this purpose).
Suppose you like the design of this particular fixture, and would like
to make an exactly similar one for a room twice as large in every
dimension. The obvious approach is
simply to double the dimensions of all components. Assuming essentially all the weight is in the
wooden frame, its height, length and breadth will all be doubled, so its volume—and
hence its weight—will increase eightfold.
Now think about the rope between the chandelier and the ceiling. The new rope will be eight times bigger than
the old rope just as the wooden frame was.
But the weight-bearing capacity of a uniform rope does not depend on its length (unless it is
so long that its own weight becomes important, which we take not to be the case
here). How much weight a rope of given
material will bear depends on the cross-sectional
area of the rope, which is just a count of the number of rope fibers
available to carry the weight. The
crucial point is that if the rope has all its dimensions doubled, this
cross-sectional area, and hence its weight-carrying capacity, is only increased
fourfold. Therefore, the doubled rope will not be able
to hold up the doubled chandelier, the weight of which increased
eightfold. For the chandelier to stay
up, it will be necessary to use a new rope which is considerably fatter than
that given by just doubling the dimensions of the original rope.
This same problem
arises when a weight is supported by a pillar of some kind. If enough weight is piled on to a stone
pillar, it begins to crack and crumble.
For a uniform material, the weight it can carry is proportional to the
cross-sectional area. Thinking about
doubling all the dimensions of a stone building
supported on stone pillars, we see that the weights are all increased
eightfold, but the supporting capacities only go up fourfold. Obviously, there is a definite limit to how
many times the dimensions can be doubled and we still have a stable
building.
As Galileo points
out, this all applies to animals and humans too (page 130): “(large) increase
in height can be accomplished only by employing a material which is harder and
stronger than usual, or by enlarging the size of the bones, thus changing their
shape until the form and appearance of the animals suggests a monstrosity.”
He even draws a
picture:
Galileo understood
that you cannot have a creature looking
a lot like an ordinary gorilla except that it’s sixty feet high. What about Harry Potter’s friend Hagrid? Apparently he’s twice normal height
(according to the book) and three times normal width (although he doesn’t look
it on this link). But even that’s not
enough extra width (if the bone width is in proportion).
There is a famous
essay on this point by the biologist J. B. S. Haldane, in which he talks of the
more venerable giants in Pilgrim’s Progress, who were ten times bigger than
humans in every dimension, so their weight would have been a thousand times
larger, say eighty tons or so. As
Haldane says, their thighbones would only have a hundred times the cross
section of a human thighbone, which is known to break if stressed by ten times
the weight it normally carries. So these
giants would break their thighbones on their first step. Or course, big creatures could get around
this if they could evolve a stronger skeletal material, but so far this hasn’t
happened.
Another example of
the importance of size used by Galileo comes from considering a round stone
falling through water at its terminal speed.
What happens if we consider a stone of the same material and shape, but
one-tenth the radius? It falls much more
slowly. Its weight is down by a factor
of one-thousand, but the surface area, which gives rise to the frictional
retardation, is only down by a factor of one hundred. Thus a fine powder in water---mud, in other
words---may take days to settle, even though a stone of the same material will
fall the same distance in a second or two.
The point here is that as we look on smaller scales, gravity becomes
less and less important compared with viscosity, or air resistance—this is why
an insect is not harmed by falling from a tree.
This ratio of
surface area to volume has also played a crucial role in evolution, as pointed
out by Haldane. Almost all life is made
up of cells which have quite similar oxygen requirements. A microscopic creature, such as the tiny worm rotifer, absorbs oxygen over its entire
surface, and the oxygen rapidly diffuses to all the cells. As larger creatures evolved, if the shape
stayed the same more or less, the surface area went down relative to the
volume, so it became more difficult to absorb enough oxygen. Insects, for example, have many tiny blind
tubes over the surface of their bodies which air enters and diffuses into finer
tubes to reach all parts of the body.
The limitations on how well air will diffuse are determined by the
properties of air, and diffusion beyond a quarter-inch or so takes a long time,
so this limits the size of insects.
Giant ants like those in the old movie “Them” wouldn’t
be able to breathe! The evolutionary
breakthrough to larger size animals came with the development of blood
circulation as a means of distributing oxygen (and other nutrients). Even so, for animals of our size, there has
to be a tremendous surface area available for oxygen absorption. This was achieved by the development of
lungs---the lungs of an adult human have a surface area of a hundred square
meters approximately. Going back to the
microscopic worm rotifer, it has a simple straight tube gut to absorb nutrients
from food. Again, if larger creatures
have about the same requirements per cell, and the gut surface absorbs
nutrients at the same rate, problems arise because the surface area of the gut
increases more slowly than the number of cells needing to be fed as the size of
the creature is increased. this problem
is handled by replacing the straight tube gut by one with many convolutions, in
which also the smooth surface is replaced by one with many tiny folds to
increase surface area. Thus many of the
complications of internal human anatomy can be understood as strategies that
have evolved for increasing available surface area per cell for oxygen and
nutrient absorption towards what it is for simpler but much smaller creatures.
On the other hand,
there is some good news about being big—it makes it feasible to maintain a
constant body temperature. This has
several advantages. For example, it is
easier to evolve efficient muscles if they are only required to function in a
narrow range of temperatures than if they must perform well over a wide range
of temperatures. However, this
temperature control comes at a price.
Warm blooded creatures (unlike insects) must devote a substantial part
of their food energy simply to keeping warm.
For an adult human, this is a pound or two of food per day. For a mouse, which has about one-twentieth
the dimensions of a human, and hence twenty times the surface area per unit
volume, the required food for maintaining the same body temperature is twenty
times as much as a fraction of body weight, and a mouse must consume a quarter
of its own body weight daily just to stay warm. This is why, in the arctic
How high can a
giant flea jump? Suppose we know that a
regular flea can jump to a height of three feet, and a giant flea is one
hundred times larger in all dimensions, so its weight is up by a factor of a
million. Its amount of muscle is also up
by a factor of a million, and when it jumps it rapidly transforms chemical
energy stored in the muscle into kinetic energy, which then goes to
gravitational potential energy on the upward flight. But the amount of energy stored in the muscle
and the weight to be lifted are up by the same factor, so we conclude that the giant flea can also
jump three feet! We can also use this
argument in reverse—a shrunken human (as in I shrunk
the kids) could jump the same height as a normal human, again about three
feet, say. So the tiny housewife trapped
in her kitchen sink in the movie could have just jumped out, which she’d better
do fast, because she’s probably very hungry!
