Last revised: Monday, 16 January 17 00:09:36 Europe/London

Old fashioned questions in arithemetic often used to take the following form;

"It takes 3 minutes for the drain in a bath to drain 100 litres of water. How long would it take to drain 75 litres?"

Such thinking suggests to us that arithmetic relationships are dependable, and, far too often, this idea that changing the scale of the problem has no impact on the methods we ought to use to address it becomes reified - not even considered.

Consider the same problem, slightly altered;

"It takes 3 minutes for the drain in a bath to drain 100 litres of water. How long would it take to drain 1,000,000 litres?"

The answer is not 30,000 minutes. 1,000,000 litres of water weighs 1000 tons - it would demolish the house.

Of course, this is a facile example. But more serious cases turn out to have even stronger effects. An often used scale example refers to the strength of an ant;

"If an ant were the size of a person, it could carry a house."

In fact, if an ant were the size of a person, it couldn't even carry itself. In admittedly crude terms (the shapes are not circles and spheres, and ant's bodies are not homogenous), the carrying capacity of legs goes up with the square of size (pi x radius squared), but the mass of the ant goes up with the cube of size (pi x radius cubed). So in multiplying the size of the ant by 100, we increase the carrying capacity of its legs by 10,000, but increase its mass by 1,000,000 - forcing the legs to bear 100 times the stress they were built for.

Whenever the scale of application of some method or model is increased by an order of magnitude or more, consider the impact of this from first principles; be prepared to realise that significant modification - or even a completely new approach - is needed.

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