CalGraph Bob's Calculus

Growth and Form: Real Birds and Bees

Bear with us for a tiny bit of elementary Mathematics to see why there are no person-sized insects nor insect-sized persons, why lobsters need no bones and why there weren't any skyscrapers before the late 19th century.

Insect Elephant Tallest Building

Consider two cubes.

Cube 1x1x1 Cube 2x2x2

The smaller cube has a volume of 1m³ and the combined area (surface area) of all its faces is 6m².

The bigger cube has a volume is 8m³ and its surface area is 24m².

In other words, by doubling all the dimensions, the volume grows by a factor of 8, but the area only by a factor of 4. Yawn! This is Fun Stuff?

Looking at it another way, when dimensions halve, the volume decreases by a factor of 8, but the surface area by a factor of only 4.

Ostrich Elephant

Still awake?
The idea is that small objects (and animals) have a lot more surface area (skin) than larger ones per unit volume (weight).

It's true that not all objects are cubes, but the volume/surface area calculations apply with other shapes such as spheres etc.

Insects can glide through the air and we can't because they have a lot more skin than we do (per unit mass). People can cheat by wearing parachutes which add lots of surface area, but relatively little weight.

Mouse A mammal uses much of its fuel (food) to maintain its internal temperature. A typical field mouse (excluding its tail) is about 1⁄20 as long as an average human. It has therefore 1⁄20³ the weight, but 1⁄20² the amount of skin we do. i.e. Fly It has proportionately 20 times as much skin as we do. It loses so much heat through it skin that it needs to eat one-third of its body weight every day. Imagine if a fairly average human of 75 kilos had to eat 25 kilos (about 125 Big Macs) of food per day!
A smaller mammal would have to eat even a relatively larger amount daily - which is why there are no smaller mammals!

Insects breathe through little holes in their skins called invaginations. If a house fly somehow increased in size to that of an adult human i.e. became 50 times as long, 50 times as wide and 50 times as deep, it would weigh 125000 as much but the area of its skin would be only increase by a factor of 2500. It would have only 2% as much skin and invaginations to take in oxygen for each unit mass as a normal fly, so it would suffocate. Now you know why we see giant insects only in the movies.

Cat Lobster

If, somehow, a land animal (e.g. a cat) had its skeleton removed, it would immediately collapse into a puddle of protoplasm. Lobsters do shed their (exo-) skeletons as they grow, but they, in effect, live in a zero-gravity environment. Thus, despite their size, they, like insects, feel surface forces and hardly notice gravity.

Tower of Piza Tower of Piza

Imagine if every dimension of a building was doubled. The cross-sectional area of its supporting pillars would only increase by a factor of 4, yet its weight would increase by a factor of 8. It would be weaker. Its interior would be darker as the windows would only let in 4 times as much light but the rooms would be 8 times as spacious.

Calculus had not been invented in medieval times. Then architects were not able to calculate stresses very accurately. They used trial and error and sometimes employed condemned prisoners in case their structures collapsed.


Hong Kong skyline


It was only with the advent of steel, which can support far more weight than wood, and electric lighting that it became possible to build skyscrapers.


Bee Eagle


If you have understood and enjoyed this essay, think about why bees can hover, but not eagles, even though the latter have proportionately much longer wings. You should also now realize that it is not because they have Crazy Glue on their feet that spiders can walk on ceilings.

Footnotes

If you are interested in learning of many more ways in which size, gravity and surface forces influence the shapes of objects and animals, you can get much more information from "On Growth and Form" by D'Arcy Wentworth Thompson. Although written almost a century ago, it is still an excellent reference.

Although it can do so much more, CalGraph verified the calculations in this article.

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