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 L⁴, 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/T². 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/L⁶. 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 L⁴. 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).