Scaling is the study of dynamic changes in the characteristics of a pattern in nature. Scaling deals with the changes in a pattern’s form and function as it grows. Scaling gives us a perspective on how and why a pattern changes both structurally and functionally within the context of its environment.

Consider a giraffe. Whatever caused its long neck to evolve, it is clear that other parts of the animal's anatomy and physiology had to also change. In particular, its heart had to scale up in size to handle the increased blood flow and blood pressure necessary to get blood up the neck into the now higher head. Without interrelated scaling of both form and function within his anatomy and physiology, the giraffe would have died of a heart attack and never evolved into its present form.

Because patterns in nature are a series of connected relationships, they tend to be highly complex. The reasons for this complexity, according to Geoffrey West, are:

• Patterns are highly complex phenomena that can usually be described only as "course grain" behavior.
• There is a huge number of sub-agents exhibiting self-organization that produce emergent properties at a system level.
• Patterns are based on historical contingencies.

The study of pattern scaling charcteristics gives us an opportunity to describe and compare complex patterns -- at least on a "course grain" level. The reason is that the dynamic characteristics (such as the comparison of body mass and metabolism) of a huge number of patterns in nature follow what is know as a power law. A power law is a simple mathematical model that is capable of portraying the dynamics of many phenomena.

If a pattern's changes are directly proportional on a one-to-one basis, the growth is said to be isometric (same sized). The shape of an isometric object remains the same as it's size grows larger. If the change in different parts or aspects of a pattern differ with overall size change, the growth is said to be allometric (different sized). Allometry is far more common in nature than is isometry. Isometric growth is a linear phenomena. Allometric growth is non-linear and follows a power law model most of the time.

An adult is not simply a baby scaled up. Different parts of the body scale up at different rates and with different scale factors. If a human's head were to grow at the same rate as the rest of the human body, the adult head would be enormous. Patterns within patterns such as body parts or the timing of reproductive events do not necessarily change in direct (isometric) proportion to body size, and the ways in which they change relative to body size can often provide insights into organisms' construction and behavior. Large organisms are often not linearly magnified small organisms.

Despite the complexity of and differences between nature's patterns, many of their dynamic processes are described by the same power law exponent. For example, the metabolic rate of organisms -- from bacteria to blue whales -- scales as the body mass increases to the power of 0.75. This universal power law scaling exponent is known as Kleiber's Law.

Similar scaling rules hold for other physiological variables. Aorta size versus mass has a scaling exponent of 3/8 for all mammals, birds, and fish. Heart rate versus body mass scales to the power law exponent of -3/4. The life span versus mass scaling exponent is 1/4 and the genome length versus cell mass scales to the power of 1/4. There seems to be a universal pattern of scaling laws where the number "4" is ubiquitous (e.g. 1/4, 3/4, 3/8, etc.). (Look at a wonderful description of scaling here.)

Despite the high complexity of organisms, a large number of their characteristics scale in a simple way. The question, then. is why? The answer seems to lie with the fact that all patterns are connected. Those connections are vital to the delivery of resources to all of the agents (cells, organs, etc.) within the patterns. These connections are manifested in the form of hierarchal branching networks at all scales -- from large organs to individual cells. What the mathematical power law model is portraying is the scaling characteristics of these natural hierarchal networks that lie within nature's patterns.

Organisms like the giraffe, humans, and plants have evolved scaled hierarchical branching networks that terminate in size-invariant units, such as lungs, capillaries, leaves, mitochondria, and oxidase molecules. Natural selection seems to have maximized metabolic capacity by maximizing the scaling of exchange surface areas. The surface area of human lung's alveoli is the size of a tennis court. Yet, by being arranged in a fractal pattern, this large surface area fits into the relatively small volume of the lung inside the chest cavity. Again, an example of a pattern in nature where it's underlying network is an interplay of form and function within the context of the environment. This pattern also affords internal efficiency by minimizing the scaling of oxygen and blood transport distances and times. These design principles are ubiquitous in nature.

Research and material on the subject of scaling is interesting and extensive. There is far too much good material to include in a single web page.  Instead, the “Useful References” link provided below offers a great deal of detail on the subject.

This document is a draft of work in progress. Please post your comments, thoughts, and suggestions:

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