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# THE FUNDAMENTAL DIMENSIONFUL CONSTANTS

In the last chapter, we made our first inroads toward sketching the order we can see in the events around us, by creating the process of dimensional analysis and identifying the notion of a scaling relation. Our methods were somewhat haphazard, in that we cobbled tools together as we discovered we could make them, with little attention to what we were seeing in the tools themselves. For instance, we ``discovered'' that it is possible to make the abstraction to the notion of dimension from the fact that we found we were able to make certain kinds of substitutions, of one ruler for another ruler, or one clock for another clock, that kept all the important functions of rulers and clocks the same, independently of any of the details of how they were made or how they worked internally.

Having in this way ``discovered'' that it was possble to make measurements, and having given names to ``the quantity that was measured'' (as in, ``length'', ``mass'', ``time'', etc), in other words having named the kind of substitution that constituted the measurement, we found that there were scaling relations among the results of various measurements. For instance, because all things fall toward earth ``in the same way'', there is always a certain relation between the distance they are allowed to fall and the time they take to cross it, while there is a different but still precise relation between the length of a clock pendulum and the time it takes to complete a swing.

The introduction of the possibility of measurement and the resulting observation that there are scaling relations are useful observations, but one can imagine that their power could have ended there, short of our full process of dimensional analysis, because there could conceivably have been any number of measureable ``properties'' (or kinds of interchangeability, or ``dimensions'') none of which had anything to do with any other. Certainly there is no reason to suspect otherwise, especially given our process of ``discovering'' dimensions essentially by accident, by finding that certain kinds of substitutions can be made transparently to the effect we care about.

Yet what we found was far simpler and more powerful than what could have been imagined. Notice that all of the dimensional properties that we discovered, no matter how or through what process, always turned out to be reducible (in other words, equivalent) to some combination of measurements of three of them, such as mass, length and time. There is no magic in these particular three; we could as well have reduced to comparisons of mass, speed and time, or many other possibilities. The important point is that a properly chosen set of three is enough. This result is remarkable in two ways. It is obviously interesting that three very simple dimensions, among the earliest abstractions people learned to make, turned out to be complete for reducing others. Since there was no ``method'' for seeking dimensions, it is interesting that nonetheless a complete set was found. The remarkable point in this is that the dimensions were so obvious, even when we didn't understand well what they ``meant''. The second and more remarkable aspect is that, among all of the kinds of dimensionful measurements that have been made since then, no new dimensions have been needed beyond the original three to reduce them. The completeness of this very small set of dimensions for reducing all dimensional comparisons, and therefore all scaling relations that have been observed, made it possible to connect the various scaling relations with the process we have called dimensional analysis, whereby it is possible to predict the scaling behavior, and therefore estimate the magnitude, of dimensionful properties that have not been measured yet, from any arbitrary but sufficient knowledge of dimensionful properties that have, such as the period of a pendulum from its length and the fact that it swings in the earth's gravity, which can be characterized through other means.

In this chapter we set out to account for another remarkable aspect of nature, which is a certain kind of increadible repeatability. At the outset, it is not at all obvious where such repeatability could originate, or that it has anything to do with scaling relations. We will find, though, that they are both expressions of essentially the same thing, and what we have discovered that accounts for the repeatability turns out to suggest a far deeper order than expected, and probably accounts for the existence of scaling relations as well.

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desmith@
Thu Aug 31 12:01:42 CDT 1995