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The square-cube law was first mentioned in Two New Sciences (1638).

The square-cube law (or cube-square law) is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. It was first demonstrated in 1638 in Galileo's Two New Sciences. It states:

When an object undergoes a proportional increase in size, its new volume is proportional to the cube of the multiplier and its new surface area is proportional to the square of the multiplier.

Represented mathematically:

$v_2=v_1\left(\frac{\ell_2}{\ell_1}\right)^3$

where $v 1$ is the original volume, $v 2$ is the new volume, $\ell_1$ is the original length and $\ell_2$ is the new length. Which length is used does not matter.

$A_2=A_1\left(\frac{\ell_2}{\ell_1}\right)^2$

where $A 1$ is the original surface area and $A 2$ is the new surface area.

For example, a cube with a side length of 1 meter has a surface area of 6 m² and a volume of 1 m³. If its side length were doubled, its surface area would be increased to 24 m² and its volume would be increased to 8 m³. This principle applies to all solids.

[edit] Applications

[edit] Engineering

When a physical object maintains the same density and is scaled up, its mass is increased by the cube of the multiplier while its surface area only increases by the square of said multiplier. This would mean that when the larger version of the object is accelerated at the same rate as the original, more pressure would be exerted on the surface of the larger object.

Let us consider a simple example of a body of Mass=M, having an acceleration=a and surface area=A of the surface upon which the accelerating force is acting.

The force due to acceleration, $F = M a$ and the thrust pressure, $T = \frac{F}{A} = M*\frac{a}{A}$

Now, let us consider the object be exaggerated by a multiplier factor = x so that it has a new mass, $M' = x 3 M$ , and the surface upon which the force is acting has a new surface area, $A' = x 2 A$ .

The New force due to acceleration $F' = x 3 * M * a$ and the resulting thrust pressure,

$T' = \frac{F'}{A'}$

$\ \ = \frac{x^3 M * a}{x^2 * A}$

$\ \ = x * (M \tfrac{a}{A})$

$\ \ = x * T$

Thus, just scaling up the size of an object, keeping the same material of construction (density), and same acceleration, would increase the thrust by the same scaling factor. This would indicate that the object would have less ability to resist stress and would be more prone to collapse while accelerating.

This is why large vehicles perform poorly in crash tests and why there are limits to how high buildings could be built. Similarly, the larger an object is, the less other objects would resist its motion, causing its deceleration.

[edit] Biomechanics

If an animal were scaled up by a considerable amount, its muscular strength would be severely reduced since the cross section of its muscles would increase by the square of the scaling factor while their mass would increase by the cube of the scaling factor. As a result of this, cardiovascular functions would be severely limited. In the case of flying animals, their wing loading would be increased if they were scaled up, and they would therefore have to fly faster to gain the same amount of lift. This would be difficult considering that muscular strength was reduced. This also helps explain how a bumblebee can have a large body relative to its wings, which would not be possible for a larger flying animal. Air resistance per unit mass is also higher for smaller animals, which is why a small animal like an ant cannot die by falling from any height.

Because of this, the giant insects, spiders, and other animals seen in horror movies are unrealistic, as their sheer size would force them to collapse. However, it's no coincidence that the largest animals in existence today are giant aquatic animals, because the buoyancy of water negates to some extent the effects of gravity. Therefore, sea creatures can grow to very large sizes without the same musculoskeletal structures that would be required of similarly sized land creatures.

Dinosaurs are the largest known land animals in history, but their reptilian circulation system, coupled with higher global temperatures at the time, likely compensated for the limitations of the square-cubed law that would apply to modern land-dwelling megafauna.

[edit] Economics

Economies of Scale: As the capacity of production increases, the cost of producing at capacity decreases. This is due to the ability to produce a larger capacity at a lower average cost. Production plants incur a set amount of fixed costs for their plants regardless of the amount of production. As production increases (volume), the fixed costs are spread out over a larger capacity of production driving the average cost down in portion to the fixed costs (surface area).^[1]

[edit] See also

Wikiversity has learning materials about Proportions

Biomechanics
Allometric law
On Being the Right Size, an essay by J. B. S. Haldane that considers the changes in shape of animals that would be required by a large change in size

[edit] References

^ Besanko, Dranove, Shanley, Schaefer (2007). Economics of Strategy, p.85. John Wiley & Sons, Inc., New Jersey ISBN 9780471679455

[0] Besanko, Dranove, Shanley, Schaefer (2007). Economics of Strategy, p.85. John Wiley & Sons, Inc., New Jersey ISBN 9780471679455

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