# Right size for the job

Everybody knows that humans are animals with big brains. But it may not be so obvious that physically we match up pretty well to the rest of the animal kingdom - not in single skills, but as all-round athletes. In an "all-animal Olympics", humans may just top the medal table - not with many golds, but a lot of silvers and bronzes.

We can beat an elephant over 100m and a leopard over a mile. We can jump higher and further than an antelope, lift more weight than a chimpanzee and swim better than most land animals (see lesson ideas). We also have great stamina. Kalahari Bushmen hunt game by running after it until it drops from exhaustion.

But we must avoid meaningless comparisons. Especially, we should not imagine what animals could do if they were different sizes. Shape is often determined by size and some animals simply can't be a different size. You may have heard the saying, "if a flea was as big as a man it could jump over a house". This gives an image of the jumping prowess of a flea, but it is nonsense in the real world. If a flea was as big as a man it would be stuck to the ground, with its legs broken by the weight of its hugely over-sized body.

The reason is the scaling effect of volume and area over length. When length doubles, area multiplies by four and volume by eight. Take a real flea, and an imaginary flea, the same shape but 100 times its size (mathematically, the two fleas are "similar": two objects are "similar" if they are exactly the same shape, but different sizes). The imaginary flea will be 100 times taller. The volume of its body, and hence its weight, will be one million times as great (100 x 100 x 100). This weight must be supported by the legs, whose sectional area (the area you would see if you sliced a leg horizontally and looked down on it from above) will have multiplied by 10,000 (100 x 100). So the pressure on the legs has been multiplied by 100. They will surely collapse.

A flea the size of a man just doesn't work. We can imagine how humans might fare in an animal Olympics. We are big enough to move quickly on land and on sea and to carry heavy loads while we do it. But we are not too big to jump off the ground (an elephant can't jump). We are big enough to swim well. Smaller land animals struggle in water because they have a high surface area in relation to their size, which makes for a lot of drag in the water.

Biologist JBS Haldane wrote a fascinating paper on the meaning of pure size in biology, called "On being the right size". We humans may be the all-rounders of the animal kingdom because we are "the right size".

Nicholas Derby teaches maths at Beaconsfield School, Buckinghamshire

* JBS Haldane died in 1964, so his paper is still in copyright, but is published freely on the web. Find it at http:irl.cs.ucla.edu papersright-size.html

LESSON IDEAS

KS4 Why does an elephant have thick legs? Treat an elephant as a cuboid on four cylindrical pillars, and a human as a cylinder on two cylindrical pillars. Estimate their volume. Work out the weight of the elephant's cuboid shape by assuming its body density (weight per unit volume) is the same as a man's. Estimate the total cross-sectional area of two human and four elephant legs, and work out the pressure exerted on each square centimetre of bone. Then assume Finn McCool the giant is five times as big as a man. Work out the pressure exerted on his thigh bones.

Calculate the volume of an ovoid, or "prolate spheroid" (a solid obtained by rotating an ellipse around its long axis). Its volume is equal to 43pa2c, where "a" is half the length of the long axis and "c" is half the length of the short axis. (What happens when a = c?) Calculate the volume of a sample egg? What is the volume of an egg that's three times longer than the sample egg?

KS4 Can a man beat a cheetah over a mile? Produce a wall-chart comparing performance. Make a scale of times over various distances, and heights and lengths for jumping. Place pictures of animals and a human on the scale.

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