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[
51
] to be insensitive to the kind of botanical tree and to have a value of 2.59 rather than
2. Equation (
2.1
) is referred to as Murray's law in the botany literature.
The significance of Murray's law was not lost on D'Arcy Thompson. In the second
edition of his work
On Growth and Form
[
83
], Thompson argues that the geomet-
ric properties of biological webs can often be the limiting factor in the development
and final function of an organism. This is stated in his general
principle of similitude
,
which was a generalization of Galileo's observations regarding size discussed in the last
chapter. Thompson goes on to argue that the design principle for biological systems is
that of energy minimization. Minimization principles have been applied in related stud-
ies with mass and entropy sometimes replacing energy, as we did in the first chapter.
The significance of the idea of energy minimization is explained in a subsequent chapter
when we discuss dynamics.
The second sentence in the da Vinci quote is just as suggestive as the first. In modern
language we interpret it as meaning that the flow of a river remains constant as trib-
utaries emerge along the river's course. This equality of flow must hold in order for
the water to continue moving in one direction instead of stopping and reversing course
at the mouth of a tributary. Using the above pipe model and minimizing the energy
with respect to the pipe radius yields
3in(
2.1
). Therefore the value of the diam-
eter exponent obtained empirically by Murray falls between the theoretical limits of
geometric self-similarity and hydrodynamic conservation of mass, 2
α
=
3. Here we
consider how a geometric structure, once formed, can be analyzed in terms of scaling,
and consequently how scaling can become a design principle for morphogenesis.
≤
α
≤
2.1.1
Two kinds of scaling
Suppose we have a tree, with the generations of the dichotomous branching indexed by
n
. Consider some property of the tree denoted by
z
n
in the
n
th generation, to which we
impose a constant scaling between successive generations
z
n
+
1
=
λ
z
n
,
(2.2)
λ
where
is a constant. Note that the quantity being scaled could be the length, diameter,
area or volume of the branch and that each of these has been used by different scien-
tists at different times in modeling different trees. The solution to this equation can be
expressed in terms of the size of the quantity at the first generation
n
=
0 by iterating
(
2.2
) to obtain
n
z
0
.
z
n
=
λ
(2.3)
Consequently, we can write the solution to (
2.2
) as the exponential
e
γ
n
z
0
,
z
n
=
(2.4)
where the growth rate is given in terms of the scaling parameter
γ
=
ln
λ.
(2.5)
