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A large change of scale forces adaptive changes in materials or form. However, within narrow limits—in most cases, up to a factor of 2—creatures can grow according to geometric similarity. That is, they can grow proportionally, so that their shape is preserved. A striking example of such growth by a far greater factor is the chambered nautilus (Nautilus pompilius). Each new chamber that it adds to its shell is larger than, but geometrically similar to, the previous chamber and also similar to the shape of the shell as a whole—an equiangular, or logarithmic, spiral (see Figure 18.12).

Most living things grow over the course of their lives by a factor greater than 2. We’ve seen with Dimetrodon that a big specimen was not just a scaled-up small one. Nor is a human adult simply a scaled-up baby: A baby’s head is relatively much larger than an adult’s, and its arms are disproportionately shorter. In growth from baby to adult, the body does not scale up as a whole. Different parts of the body scale geometrically, each with a different linear scale factor. That is, a baby’s eyes grow to perhaps twice their original size, while the arms grow by a factor of about 4.

Although the laws for growth can be much more complicated than for proportional growth (or even for the allometric growth that we discuss later), more sophisticated mathematics—for example, differential geometry, the geometry of curves and surfaces—permits analysis of complex and interlocking scalings. For a model of the process in which a baby’s head changes shape to grow into an adult head, we can use graph paper: First, we put a picture of the baby’s skull on graph paper. Then we determine how to deform the grid until the pattern matches an adult skull. (See Figure 18.13 and Spotlight 18.4.) The same idea lies at the heart of computerized “morphing,” the process in which the face of one person can be changed smoothly into the face of another, with different scalings for different parts of the face.

Allometric Growth

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If we measure the arm length or head size for humans of different ages and compare these measurements with body height, we observe that humans do not grow proportionally; that is, in a way that maintains geometric similarity. The head of a newborn baby may be one-third of the baby’s length, but an adult’s head is usually close to one-seventh of the individual’s height. The arm, which at birth is one-third as long as the body, is by adulthood closer to two-fifths as long (see Figure 18.14a).

Graphing provides a way to test for differential growth. We plot body height on the horizontal axis and arm length on the vertical axis (see Figure 18.14b). A straight line would indicate proportional growth—that is, according to geometric similarity. We do get a straight line from 9 months (0.75 years) on up. But up to 9 months, we get a curve, which indicates that the ratio of arm length to height does not remain constant over the first year.

Is there an orderly law by which we can relate arm length to height? Let’s plot again, this time using a different scale (see Figure 18.14c). For this base-10 logarithmic scale, we mark off equal units, as usual. But instead of labeling the marked points with 0, 1, 2, 3, and so on, we label them with the corresponding powers of , and so on, which are also called orders of magnitude. This is the procedure that we used to make Figure 18.10 on page 758. In that figure, the fact that the data cling close to a straight line indicates a power-law relationship between metabolic rate and body mass, which would not be obvious if we had graphed using ordinary scales instead of logarithmic scales.

Plotting a point on such a scale is not easy because the point midway between 1 and 10 on the logarithmic scale is not 5.5; rather, it is . Special graph paper marks smaller divisions and makes points easier to plot; paper marked with log scales on both axes is called log-log paper, while semilog paper has a logarithmic scale on just one axis. Many computer plotting packages can also produce logarithmic scales.

We could use a logarithmic scale for either height or arm length, or for both. When logarithmic scales are used for both, as in Figure 18.14c, the data plot closely to a straight line. Looking carefully at the figure, we can discern actually two different straight lines: a steeper one that fits early development (we will see shortly that it has slope 1.2), and a less steep one (with slope 1.0) that fits development after 9 months of age.

The change from one line to another after 9 months indicates a change in pattern of growth. The pattern after 9 months, characterized by the straight line with slope 1, is indeed proportional growth (sometimes called isometric growth). For the pattern before 9 months, the slope is 1.2. The fact that it is greater than 1 means that arm length is increasing relatively faster than height. This early growth also follows a definite pattern, called allometric growth.

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In geometric scaling, area grows according to the square (second power), and volume according to the cube (third power), of length, so they grow allometrically with length.

If we denote arm length by and height by , a straight-line fit on log-log paper corresponds to the algebraic relation

where is the slope of the line and is the point where the graph crosses the vertical axis. If we raise 10 to the power of each side, we get

where . This equation describes a power curve: is a constant multiple of raised to a certain power.

In this chapter, we have explored the limitations on life imposed by dwelling in three dimensions. In Chapters 19 and 20, we will see that dimensionality also imposes surprising limits on artistic creativity in devising patterns.

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