35
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.