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!!!
Copyright University of Maryland, 2007
You may link to this site for educational purposes.
Please do not copy without permission
requests/questions/feedback email: firstname.lastname@example.org