Description
Supply networks and allometry:

(3 + 3 points)
This question’s calculation is a specific, exactlysolvable case of the general result that you’ll will attack (with optional relish and other condiments) in the following question.
Consider a set of rectangular areas with side lengths L_{1} and L_{2} such that
L_{1} / A 1 and L_{2} / A 2 where A is area and _{1} + _{2} = 1. Assume _{1} > _{2} and that ϵ = 0.
1
Now imagine that material has to be distributed from a central source in each of these areas to sinks distributed with density (A), and that these sinks draw the same amount of material per unit time independent of L_{1} and L_{2}.

Find an exact form for how the volume of the most efficient distribution network scales with overall area A = L_{1}L_{2}. (Hint: you will have to set up a double integration over the rectangle.)

If network volume must remain a constant fraction of overall area, determine the maximal scaling of sink density with A.
Extra hints:

Integrate over triangles as follows.

You need to only perform calculations for one triangle.

x_{2}
= ^{L}2 _{x}
x
2
1
L_{1}
L_{2}
2
dx_{1}dx_{2}
2
2
1
1
^{L}1
L
1
x_{1}
− _{2}
2
1
1
2
2
^{L}2
^{−} 2

From lectures on Supply Networks: Show that for large V and 0 < ϵ < 1/2
∫
min V_{net} / jj⃗xjj^{1}
Ω_{d;D}(V )
Reminders: we defined L_{i} = c_{i} ^{1}V i where
∏
_{1} = _{max} _{2} : : : _{d}:, and c = _{i }c_{i}
2ϵ _{d}_{⃗}_{x} _{V }1+ max(1 2ϵ)
_{1} + _{2} + : : : + _{d} = 1,
1 is a shape factor.
Assume the first k lengths scale in the same way with _{1 }= : : : = _{k} = _{max}, and
write jj⃗xjj = (x^{2}_{1} + x^{2}_{2} + : : : + x^{2}_{d})^{1/2}.
2

(a) For a family of ddimensional regions, with scaling as per the previous question, determine, to leading order, the scaling of hypersurface area S with volume V . In other words, find the exponent in S / V as V ! 1.
Assume that nothing peculiar happens with the shapes (as we have always implicitly done), in that there is no fractal roughening.
Hint: As a start, figure out how the circumference for the rectangles in question 1 scales with area A. For d dimensions, think about how the hypersurface area of a hyperrectangle (or orthotope) would scale.

For general d, what is the minimum and maximum possible values of and for what values of the _{i} does these extrema occur?
The goal and a connection to energy metabolism:
3