# Assignment 4 Solution

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## Description

Supply networks and allometry:

1. (3 + 3 points)

This question’s calculation is a specific, exactly-solvable 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 L1 and L2 such that

L1 / A 1 and L2 / 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 L1 and L2.

1. Find an exact form for how the volume of the most efficient distribution network scales with overall area A = L1L2. (Hint: you will have to set up a double integration over the rectangle.)

1. 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.

 x2 = L2 x x 2 1 L1 L2 2 dx1dx2 2 2 1 1 L1 L 1 x1 − 2 2 1 1 2 2 L2 − 2
1. From lectures on Supply Networks: Show that for large V and 0 < ϵ < 1/2

min Vnet / jjxjj1

d;D(V )

Reminders: we defined Li = ci 1V i where

1 = max 2 : : : d:, and c = i ci

2ϵ dx 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 jjxjj = (x21 + x22 + : : : + x2d)1/2.

2

1. (a) For a family of d-dimensional regions, with scaling as per the previous question, determine, to leading order, the scaling of hyper-surface 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 hyper-surface area of a hyperrectangle (or orthotope) would scale.

1. 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

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