Fractal Moon

[neufer]

Finally, I write about Lunar Reconnaissance Orbiter Camera photos
By Emily Lakdawalla Aug. 21, 2009 | 13:59 PDT | 20:59 UTC

• Lunar highland terrain at four scales
  
  These four snapshots of cratered terrain in the lunar highlands were cropped
  from an unremarkable section of the "Rille within a rille" image.
  Credit: NASA / GSFC / ASU / montage by Emily Lakdawalla

This [LRO] image contains four tiles. Would you believe that we are looking at the same piece of terrain at four different scales? The upper left image is at 12 meters per pixel; upper right, 6; lower left, 3; lower right, 1.5. Each tile is the lower right quadrant of the preceding one, enlarged 200%. The first three tiles are basically indistinguishable even though they are actually different scales. The last one looks slightly blurrier because, I think, we are at the resolution limit of the camera -- you're seeing the way details spread across more than one pixel.

At the high resolution of LROC, the appearance of some highland lunar cratered terrain is largely independent of scale -- in other words, it's fractal. Crater saturation and the scale-independence of cratered terrain is one of the things Benoit Mandelbrot wrote about in his book The Fractal Geometry of Nature. Of course, nature never quite perfectly matches mathematical models, and determining the ways in which size-frequency distributions of craters depart from a perfect scale-independence -- and what causes those departures -- is one of the basic tools of the lunar geologist.>>

[Dr. Morbius]

Your reference to nature not quite matching mathematical models seems, as it were, like putting the cart before the horse. Shouldn't it be the other way around: the models don't quite match nature? I'm going to take a WAG here and suggest that everything invented by humans already exists in some form in nature? Perhaps our understanding will always be playing 'catch-up".

[Chris Peterson]

Your reference to nature not quite matching mathematical models seems, as it were, like putting the cart before the horse. Shouldn't it be the other way around: the models don't quite match nature? I'm going to take a WAG here and suggest that everything invented by humans already exists in some form in nature? Perhaps our understanding will always be playing 'catch-up".

I see no reason to believe that everything invented already exists in nature. In the realm of mathematics, there are any number of internally consistent systems that have no relationship to nature at all. While fractals don't fall into that category, neither is there anything in nature that behaves in a purely fractal way, across all scales- unlike most mathematical treatments. That's hardly surprising, since the constituents of a physical system change with scale (sand doesn't behave exactly like molecules, nor like boulders).

From the standpoint of logic, I see no difference between saying nature doesn't match the models, and the models don't match nature.