III. The Power Function

A way of formalizing these geometric relationships is to use a power function:

y = a x^b

where x and y are the variables you want to relate (such as length or volume), and a and b are parameters that describe the relationship. For example, if you want to relate the volume and radius of a sphere, the power function is

Volume =( 4/3 p) radius³

"x" in this case is radius, y is volume, a takes the value of ( 4/3 p), and b is 3. The table below shows the values of a and b describing various relationships between areas, volume, and linear dimensions of two common geometrical shapes.

shape	y	x	b	a
cube	surface area	length	2	6
cube	volume	length	3	1
sphere	surface area	radius	2	4p
sphere	cross section	radius	2	p
sphere	volume	radius	3	4/3p

Note that b, the scaling exponent, could be predicted easily by the methods you learned earlier. The other parameter, a, depends on the shape of the object in question.

For most of this discussion, we will ignore the value of "a" merely because it is shape-specific. Note that regardless of shape, volumes (for example) are always proportional to a length cubed, and can be expressed as we did earlier:

V L³

You can work out other relationships for yourself if you just remember the rules for working with exponents. For example, if we let V = volume, A = surface area, and L = length, then:

A L² or L A^1/2

V L³ or L V^1/3

therefore, A L² (V^1/3 )² V^2/3

Power functions can be awkward if you don't already know the value of the exponent. For example, the graph below plots the volume of a cube versus the length of one side.

If you didn't already know that the exponent "b" was equal to three, could you deduce it from this plot? Not likely, but here's an easier way. If you plot the same information on log-log graph paper, the relationship appears linear, with the slope equal to the scaling exponent "b"

To make the computation of the slope easier , you can plot log(y) against log(x) as follows:

This is the equivalent of doing a logarithmic transformation of our original equation:

y = a x^b

log (y ) = log (a x^b)

log (y ) = loga + log(x^b)

log y = log a + b log x

This last equation describes the straight line on the graph above with slope b and y-intercept of log a.

You can estimate the scaling exponent (b) by doing a least squares linear regression of log y on log x, but the estimate of b (the slope of the line) is somewhat biased. This bias occurs because least squares linear regression techniques assume that all of the errors in your measurements lie in the y variable and that the x variable is known without error. A better technique is reduced major axis regression, which assumes that the errors are evenly partitioned between the x and y variables. Although reduced major axis regression is rarely included in commercial software packages, by a fortunate fluke of the mathematics, the reduced major axis slope is equal to the least squares slope divided by the correlation coefficient (r). In the exercises you'll do in lab, we'll expect you to report both the least squares and reduced major axis estimates of b.

So far we have used simple geometrical shapes to illustrate the relationships between lengths, areas, and volumes because they have simple equations for each of these measures. You have probably noticed that most organisms have complicated shapes which can not be so easily represented mathematically. However, the proportionalities between lengths, areas, and volumes holds for complex shapes as well as simple ones, as long as they are geometrically similar. That is, if we consider a species of snail that exhibits a constant shape regardless of size, then the surface area of its shell will be proportional to its (height)² , and if you plot log(surface area) vs. log(height), you'll find a straight line with b = 2.