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

The derivation of the 3/4 scaling law

So, our new formula for "b" is:

Based on some of the principles above (and some we haven't presented yet) - the authors suggest that size (and by size, they really mean the volume of the transport network) is related to three main factors: 1) the number of times the network branches from the top level to the final level (N), 2) how the branches decrease in radius from level to level ( a value they call γ), and 3) how the branches decrease in length from level to level (a value they call β). They then show that size (volume) is proportional to the quantity (γβ²)^-N. So using that information, we can do some further manipulations and come up with a new equation:

Finally, the authors show us that γ is equal to n^-1/3 and β is equal to n^-1/2, so we can do one final manipulation:

Whew!!!! We finally arrived at our final destination - that "b" should be equal to 3/4!!!