### A higher power -- units again

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Those of you who have been following this blog for a while
know I have been somewhat obsessed with units. You know, those things we
physicists like students to stick on to nice clean mathematical variables to
make them messier and more complicated? And that students would much prefer to
ignore? Well, if you’ve been reading from the beginning you also are beginning
to get an inkling of why I care about units so much. My evolving view of
science is rooted by the phrase, “Science is not about how the world works;
it’s about how we can think about how the world works.” This puts science
philosophically as a bridge between cognitive science and the epistemology of
reality – a “how do we know” about what we presume actually exists. And this is
where my interests and research have been living for some years now.

My previous posts on units (Units and stoichiometry, Cutting mathematicians some slack, and Teaching units) begin to make the case that
units are a nice example of the bridge between cognitive modeling and math; that
we use our everyday experience with distance, time, and mass to allow us to do
some mathematically very sophisticated things at an introductory level without
actually bringing in that math. (What units are doing is tracking the
irreducible representations of a transformation group in our equations!) But
the implications of this are that our “map of the real world onto a
mathematical system” (see Units and stoichiometry) becomes limited. There are
things we can do with the math that don’t make sense physically. For example,
space, time, and mass get mapped onto the real number line by choosing
coordinate systems and operational definitions, but while space and time can be
negative as well as positive, mass cannot.

I also made the point that multiplication is allowed on the
real line, but as we multiply we begin to construct new “conceptual correlates”
such as area and volume, but run out quickly. We have no conceptual correlates
with L

^{4}, though we might build four-dimensional volumes by analogy.
Well, I’ve found two nice examples of good physical uses of length to a higher power in my recent work looking for applications of physics
to medicine and biology. These illustrate nicely how these combinations correlate
the math cognitively to the physical world in a different way from the
straightforward sense we have of area or volume.

Of course the simplest and prototypical example is the square
of time. Although we have no conceptual correlate of “square time”, when we
define an acceleration we are chaining the idea of rate. The first time we
apply the idea of rate of change to position we get a velocity: length divided
by time has dimensions L/T. When we do it again, we get an acceleration: velocity
divided by time has dimensions, (L/T)/T, which by the rules of math is L/T

^{2}. So while we don’t have a concept of square time, we do have a concept of “per unit time per unit time”.
This conceptual chaining is very reminiscent of the way
modern linguists and semanticists build up complex and abstract concepts, by
chaining of metaphor (Lakoff and Johnson [1]) and by cognitive blending
(Fauconnier and Turner [2]).

My biological explorations have led me to two different
examples of distance to the fourth and sixth powers – hypervolumes. Figuring
out the conceptual correlates for these unit combinations is interesting.

The first example is medical. Various glands in your body
secrete a variety of chemicals. The health of those glands can be probed by
checking how much of that chemical is circulating in your blood. A simple blood
test can measure the amount of chemical found in a particular sample and
calculate a density by dividing the mass found by the size of the sample. On my
annual blood tests lots of these are measured in micrograms per milliliter
(µg/ml). (Unfortunately for those of us who like to push the conventions of SI
units, often in micrograms per deciliter. But we shouldn’t be surprised. Blood
pressure is still measured in “millimeters of mercury”!) Now if you want to
know about the health of the gland, to find out how effective the cells in the
gland are in producing the chemical you might want to divide the density of the
chemical found in the blood by the volume of the gland – measurable with
a sonogram. The result is a “mass per volume per volume” – reported as
(µg/ml)/ml but recognizable to a physicist as M/L

^{6}. In this case, we make sense of a sixth power of a length as a “per volume per volume” -- a density of a density.
A second interesting example is the Hagen-Poisseuille equation – “Ohm’s law for the pipe.” The pressure drop along a length of pipe
that has a continuous steady state flow is equal to the rate of flow times the
resistance; this is analogous to the more familiar electrical law that the
voltage drop across a resistor is equal to the electric current times the
resistance. This law has a lot of important biological implications in situations ranging from the motion of sap in trees to the motion of the blood in animals.

In the familiar Ohm’s law rule, the resistance is inversely
proportional to the area of the resistance perpendicular to the flow. This
physics of this is that Ohm’s law is basically about the balance of forces: the
electric force due to the potential drop pushes the charge through the resistor
against the drag, which is proportional to the velocity. Since the charges are
moving at a uniform velocity (on the average) there is no net force. The fact
that the electrical push balances the drag is Ohm’s law. The area arises
because the drag is proportional to the velocity and we want to express the law
in terms of electric current – basically velocity times area (times charge
density). We introduce the inverse area to change velocity into current.

In the Hagen-Poisseuille law, we have a similar bit of
physics: the forces due to the pressure drop pushes the fluid through the pipe
against the viscous drag, which is proportional to the velocity. Since the
fluid moves at a constant velocity, there is no net force. (There is an
additional complication in this case since the fluid doesn’t flow at the same
rate of speed in all parts of the pipe, moving fastest at the center of the
pipe, but we’ll ignore that here.) The fact that these two forces balance is
the H-P law. One factor of the area arises because the drag is proportional to
the velocity and we want to express the law in terms of current – velocity
times area (if we use volume current; add a factor of mass density if we use
matter current). We introduce the inverse area to change velocity into current.
But we also want to use pressure rather than force. In this case (not in the
electrical case), pressure is related to force by a factor of area. This
introduces a second factor of area into the resistance of the H-P equation. (To
see this with equations, check out our text on it for the NEXUS Physics class.) This corresponds to the fourth power of the radius and can have powerful medical implications as well. (See our homework problem, Hold the mayo.)

So in the HP equation, we get a factor of L

^{4}. In our previous example, the higher powers of length (the square of a volume) came because we were using two different volumes – sort of a double density. In this case, both of our areas are the same but they have two sources. One from the force of the fluid on itself into pressure, the other from converting the fluid velocity into current.
In both of these examples the conceptual correlate of the
complex unit is a product of different conceptual objects – two different
volumes in the density of density example, and the same area for two different
purposes in the HP case. These examples suggest to me that when we have
dimensions that don’t have a direct conceptual correlate with a physical
concept, understanding the conceptual blends that lead to the combined dimensional
structure can help us make better sense of why a complex quantity looks the way it
does.

[1] G. Lakoff and M. Johnson,

*Metaphors We Live By*(U. of Chicago Press, Chicago, 1980).
[2] G. Fauconnier and M. Turner,

*The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities*(Basic Books, 2003)*.*