The point I would like to make is that the very existence of such diagrams
is evidence for evolution. That is, the fact that it is possible to make
such diagrams is consistent with the idea of molecular evolution, and would
be a spectacularly implausible coincidence in the absence of evolution.
To understand this, let us define the _ultrametric_ property: Given a
notion of distance, we say that it is ultrametric if all triangles are
either
-- equilateral, or
-- narrow isoceles (i.e. two equal long sides plus one short side).
That is, all triangles have two sides the same, and the third side is either
the same or shorter.
As you well know, ordinary Euclidean geometry is not ultrametric. As
another example, physics ideas are not ultrametric; you cannot impose any
reasonable tree structure on the ideas of time, distance, mass, velocity,
acceleration, force, momentum, energy, torque, area, volume, et cetera.
As yet another example, Hugh H. mentioned color space (color wheels and
color solids). There is no ultrametricity in color space; given two
colors, you can generally define another color between the two, and closer
to one than the other (so the triangle is not isoceles, and even if it
were accidentally isoceles it wouldn't have a narrow base).
In the case of trees, the natural metric is how far back do you need to
go to find the nearest common ancestor. It is possible (albeit unlikely)
for three species to have a single nearest common ancestor, in which case
the triangle XYZ is equilateral.
/-------------X
/
A --------------Y
\
\-------------Z
More commonly, there is a branching and then another branching, so that
the triangle XYZ is narrow isoceles:
/--------------X
/
A
\ /------Y
\------B
\------Z
Specifically: in the diagram above, the XY distance is long, the XZ
distance is equally long, and the YZ distance is short.
We can very precisely quantify "how far back" by counting the number of
mutations in the DNA; less precisely that corresponds to _time_ back in
history.
If you select three points at random from (say) color space, they will by
accident appear to be approximately ultrametric some small fraction of the
time, less than 50% of the time. In contrast, if you select three species
at random and calculate the evolutionary distances between them, they will
be ultrametric to high precision with very high probability.
We have data on thousands of species. That makes well over 100 million
triangles you can check. I encourage you to check them. Check as many as
you like.
I interpret these observations as follows: At the top level, either
h1a) The underlying physical process is ultrametric, or
h1b) it is not; the data just looks ultrametric by accident.
Hypothesis (h1b) can be rejected with high confidence; the uncertainty
is something like one part in 2^100,000,000 or better ... which is pretty
close to a sure thing, close enough for me.
This is not a matter of opinion; you can write a computer program to
perform the check impartially.
At the second level, either
h2a) The process is ultrametric because it involves mutations, inheritance,
and selection, or
h2b) you need to come up with an alternative hypothesis.
I've never seen a plausible alternative hypothesis. (I don't count unscientific
theories such as unfalsifiable conspiracy theories.)
===========
The concept of an ultrametric space is quite general. It shows up in computer
science and in physics, including the physics of spin glasses.
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