Scaling: Why Giants Don’t Exist

Scaling: Why Giants Don’t Exist

Michael Fowler, UVa  10/12/06

 

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.

 

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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.

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