Table of Contents

• Prequiz...
• The Power Function
• 1: Introduction
• 2: What are scaling relationships?
• 3: The power function
• 4: The shape of the power function
• 5: The conversion factor "a"
• 6: The power factor "b"
• Log Transformations
• 7: Introduction
• 8: What are logarithms again?
• 9: Graphing on a log scale
• 10: Visualizing the data
• 11: Getting to a straight line
• 12: The key parameter is "b"
• The value of "b"
• 13: Introduction
• 14: A geometric approach
• 15: Scaling surface area of cubes
• 16: Scaling surface area of cubes
• 17: A "2/3 scaling law"?
• Confronting ideas with data
• 18: Introduction
• 19: An unexpected result
• 20: A "3/4 scaling law"?
• 21: Internal transport systems
• 22: Branching and supply networks
• 23: Deriving the "3/4 scaling law"
• 24: So what's the real value of "b"?
• Review
• 25: Review of Concepts

An unexpected result

So, what did scientists actually find when they went out to measure the parameter "b"? How does body size scale with metabolism? The most famous early paper was published by Max Kleiber in 1947 and showed a surprising result. Body size seemed to scale with metabolism to the 3/4 power, rather than the 2/3 that was expected! This 3/4 power law became known as Kleiber's law. Here is his original graph:

Notice that the data are displayed on a log-scaled graph (of course!). The green points are the data, and they overlie the red line, which shows the 3/4 scaling relationship. The dotted line labeled "surface" indicates the line with a slope of 2/3 and the dash-dot line labeled "weight" indicates the slope if b = 1 (in other words, if the relationship were linear). These data are for mammals only, but further investigation found that this relationship was consistent for several groups over an unbelievable 21 orders of magnitude (from single-celled organisms all the way to elephants).

This 3/4 scaling pattern caused a great deal of confusion. What process was governing this strange result? Scientists struggled with it for years, and it wasn't until 1997 that a credible model was proposed.