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1Scaling and Order-of-

Magnitude Estimates

1.1Introduction

Why can’t an insect be the size of a dog. Some skinny stretched-out cells

in your spinal cord are a meter tall — why does nature display no single

cells that are not just a meter tall, but a meter wide, and a meter thick as

well. Believe it or not, these are questions that can be answered fairly easily

without knowing much more about physics than you already do. The only

mathematical technique you really need is the humble conversion, applied

to area and volume.

Area and volume

Area can be defined by saying that we can copy the shape of interest

onto graph paper with 1 cm x 1 cm squares and count the number of

squares inside. Fractions of squares can be estimated by eye. We then say the

area equals the number of squares, in units of square cm. Although this

might seem less “pure” than computing areas using formulae like A=

p

r

2

for

a circle or A=wh/2 for a triangle, those formulae are not useful as definitions

of area because they cannot be applied to irregularly shaped areas.

Units of square cm are more commonly written as cm

2

in science. Of

course, the unit of measurement symbolized by “cm” is not an algebra

symbol standing for a number that can be literally multiplied by itself. But

it is advantageous to write the units of area that way and treat the units as if

they were algebra symbols. For instance, if you have a rectangle with an area

of 6 m

2

and a width of 2 m, then calculating its length as (6 m

2

)/(2 m)=3 m

gives a result that makes sense both numerically and in terms of units. This

algebra-style treatment of the units also ensures that our methods of

Amoebas this size are seldom

encountered.

Life would be very different if

you were the size of an insect.