Square-cube law - AbsoluteAstronomy.com

The square-cube law (or cube-square law) is a principle, drawn from the mathematics of proportion

Proportionality (mathematics)

In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

, that is applied in engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

and biomechanics

Biomechanics

Biomechanics is the application of mechanical principles to biological systems, such as humans, animals, plants, organs, and cells. Perhaps one of the best definitions was provided by Herbert Hatze in 1974: "Biomechanics is the study of the structure and function of biological systems by means of...

. It was first demonstrated in 1638 in Galileo's

Galileo Galilei

Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

Two New Sciences
Two New Sciences
The Discourses and Mathematical Demonstrations Relating to Two New Sciences was Galileo's final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned...

. It states:
Represented mathematically:

where

is the original volume,

is the new volume,

is the original length and

is the new length. Which length is used does not matter.

where

is the original surface area and

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 the dimensions of the cube 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.

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,

and the thrust pressure,

.

Now, let us consider the object be exaggerated by a multiplier factor = x so that it has a new mass,

, and the surface upon which the force is acting has a new surface area,

.

The New force due to acceleration

and the resulting thrust pressure,

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 can be built. Similarly, the larger an object is, the less other objects would resist its motion, causing its deceleration.

Engineering Examples

Airbus A380
Airbus A380
The Airbus A380 is a double-deck, wide-body, four-engine jet airliner manufactured by the European corporation Airbus, a subsidiary of EADS. It is the largest passenger airliner in the world. Due to its size, many airports had to modify and improve facilities to accommodate it...

: the lift and control surfaces (wings, rudders and elevators) are relatively big compared to the fuselage of the airplane. In e.g. a Boeing 737
Boeing 737
The Boeing 737 is a short- to medium-range, twin-engine narrow-body jet airliner. Originally developed as a shorter, lower-cost twin-engine airliner derived from Boeing's 707 and 727, the 737 has developed into a family of nine passenger models with a capacity of 85 to 215 passengers...

these relationships seem much more 'proportional', but designing an A380 sized aircraft by merely magnifying the design dimensions of a 737, would result in wings that are too small for the aircraft weight, because of the square-cube rule.
A clipper
Clipper
A clipper was a very fast sailing ship of the 19th century that had three or more masts and a square rig. They were generally narrow for their length, could carry limited bulk freight, small by later 19th century standards, and had a large total sail area...

needs relatively more sail surface than a sloop
Sloop
A sloop is a sail boat with a fore-and-aft rig and a single mast farther forward than the mast of a cutter....

to reach the same speed, meaning there is a higher sail-surface-to-sail-surface ratio between these crafts then there is a weight-to-weight ratio.

Biomechanics

If an animal were scaled up by a considerable amount, its relative muscular strength would be severely reduced, since the cross section of its muscles would increase by the square of the scaling factor while its mass would increase by the cube of the scaling factor. As a result of this, cardiovascular and respiratory functions would be severely burdened.

In the case of flying animals, the 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

Lift (force)

A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

. Air resistance per unit mass is also higher for smaller animals, which is why a small animal like an ant

Ant

Ants are social insects of the family Formicidae and, along with the related wasps and bees, belong to the order Hymenoptera. Ants evolved from wasp-like ancestors in the mid-Cretaceous period between 110 and 130 million years ago and diversified after the rise of flowering plants. More than...

cannot be crushed by falling from any height.

As was elucidated by J. B. S. Haldane

J. B. S. Haldane

John Burdon Sanderson Haldane FRS , known as Jack , was a British-born geneticist and evolutionary biologist. A staunch Marxist, he was critical of Britain's role in the Suez Crisis, and chose to leave Oxford and moved to India and became an Indian citizen...

, large animals do not look like small animals: an elephant cannot be mistaken for a mouse scaled up in size. The bones of an elephant are necessarily proportionately much larger than the bones of a mouse, because they must carry proportionately higher weight. To quote from Haldane's seminal essay On Being the Right Size, "...consider a man 60 feet high...Giant Pope and Giant Pagan in the illustrated Pilgrim's Progress.... These monsters...weighed 1000 times as much as Christian. Every square inch of a giant bone had to support 10 times the weight borne by a square inch of human bone. As the human thigh-bone breaks under about 10 times the human weight, Pope and Pagan would have broken their thighs every time they took a step."

The giant animals seen in horror movies (e.g., Godzilla

Godzilla

is a daikaijū, a Japanese movie monster, first appearing in Ishirō Honda's 1954 film Godzilla. Since then, Godzilla has gone on to become a worldwide pop culture icon starring in 28 films produced by Toho Co., Ltd. The monster has appeared in numerous other media incarnations including video games,...

or King Kong

King Kong

King Kong is a fictional character, a giant movie monster resembling a gorilla, that has appeared in several movies since 1933. These include the groundbreaking 1933 movie, the film remakes of 1976 and 2005, as well as various sequels of the first two films...

) are also 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

Deep-sea gigantism

In zoology, deep-sea gigantism, also known as abyssal gigantism, is the tendency for species of crustaceans, invertebrates and other deep-sea-dwelling animals to display a larger size than their shallow-water counterparts...

, 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.

Engineering

Engineering Examples

Biomechanics

See also