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Figure 2.7.
A sketch of a tree from Leonardo da Vinci's
Notebooks
, PL.XXVII [
64
]. Note that da Vinci was
relating the branches of equal generation number to make his association with flowing streams.
and fluvial landscapes have this tree-like structure. The quantification of branching
through the construction of the mathematical laws that govern conduit size can be traced
back to Leonardo da Vinci (1452-1519). In his
Notebooks
da Vinci wrote the following
regarding Figure
2.7
[
64
]:
All the branches of a tree at every stage of its height when put together are equal in thickness to
the trunk [below them]. All the branches of a water [course] at every stage of its course, if they
are of equal rapidity, are equal to the body of the main stream.
Da Vinci also admonished his readers with the following statement:
Let no man who is not a Mathematician read the elements of my work.
This last statement increases in significance when we consider that da Vinci wrote it
nearly two centuries before Galileo, the person who is generally given credit for estab-
lishing the importance of mathematics in modern science, crafted the algebra to describe
his experiments.
The first sentence in the da Vinci quote is clarified in subsequent paragraphs of the
Notebooks
. With the aid of da Vinci's sketch reproduced in Figure
2.7
, this sentence has
been interpreted as follows: if a tree has a trunk of diameter
d
0
that bifurcates into two
limbs of diameters
d
1
and
d
2
, the three diameters are related by
d
0
d
1
d
2
.
=
+
(2.1)
Simple geometrical scaling would yield the diameter exponent
2, which corre-
sponds to “rigid pipes” carrying fluid from one level of the tree to the next, while
retaining a fixed cross-sectional area through successive generations of bifurcation. The
diameter exponent for botanical trees was determined empirically by Murray in 1927
α
=
