  Size varies. Small things are typically more frequent than large things. The
logarithm of frequency often declines linearly with the logarithm of size. That
power law relation forms one of the common patterns of nature. Why does the
complexity of nature reduce to such a simple pattern? Why do things as
different as tree size and enzyme rate follow similarly simple patterns? Here I
analyze such patterns by their invariant properties. For example, a common
pattern should not change when adding a constant value to all observations.
That shift is essentially the renumbering of the points on a ruler without
changing the metric information provided by the ruler. A ruler is shift
invariant only when its scale is properly calibrated to the pattern being
measured. Stretch invariance corresponds to the conservation of the total
amount of something, such as the total biomass and consequently the average
size. Rotational invariance corresponds to pattern that does not depend on the
order in which underlying processes occur, for example, a scale that additively
combines the component processes leading to observed values. I use tree size as
an example to illustrate how the key invariances shape pattern. A simple
interpretation of common pattern follows. That simple interpretation connects
the normal distribution to a wide variety of other common patterns through the
transformations of scale set by the fundamental invariances.
