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I've seen a lot of high level videos on fractals, etc, and how they might apply to the real world. So a tree is branches with branches with branches, and our blood vessels branch and then branch again. Same with snowflakes. I think there are analogies to software, as another example.

But I think there's more to the story that maybe isn't told in these high level presentations:

  • Blood vessels can only branch so far down, until they get to the limits of one cell. Of course there can be other structures inside the cell.
  • When I look at natural things the subunits aren't orderly. One tree looks different than the next.
  • Or a coastal shoreline in the real world doesn't have the exact same patterns as you zoom in - they're similar, but not mere copies of each other.
  • Or if you think about countries, states, cities, companies, families, etc., the organization at each level shares common traits with the parent and child units, but are not identical in form and function.

Obviously in some cases randomness playing a role, place subunits in various places.

Or in the case of capillaries vs. the pathway inside cells, the medium has changed.

My questions:

  • Is there a name or measurement for the "imperfections" in the real world as you zoom in or zoom out to another level of detail.
  • Is there a name for when an organizational structure is generally preserved between levels, but the actual "implementation" or medium is different? So a city has roads, whereas a neighborhood park has only pedestrian sidewalks, and then inside of a building you actually have hallways.

This is one of those things I'm not sure what I'd "google" for. Having the terms makes all the difference.

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I guess I'd go with "approximate fractal." –  Qiaochu Yuan Jul 11 '12 at 19:50
4  
The phrase that's traditionally used for things like coastlines (and was in the title of Mandelbrot's original paper) is Statistical Self-Similarity; it expresses the concept that while the precise shape may not be the same at different scales, many of the broad properties are the same. That might be a reasonable starting point for your searches... –  Steven Stadnicki Jul 11 '12 at 21:00
    
Thanks @StevenStadnicki, that sounds reasonable, I'll see where it leads. –  Mark Bennett Aug 10 '12 at 16:31

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