Zooming in

If you only see a dot, click on the picture!

As you zoom in, this will stretch out to become a line segment.  As you zoom in further, you see its thickness.  It's really a long thin rectangle.

But wait longer and you see it's really a field of dots.  And zooming into any one of these dots, this process repeats... forever!

Each long thin rectangle is 10,000 times longer than the next smaller one.

So, you're looking at a very complicated set of points in the plane, whose dimension seems to depend on how closely you zoom in. 

In this example, created by +Simon Willerton, the dimension keeps cycling: 0, 1, 2, 0, 1, 2, ...  But you can make examples that do other things.

The moral?  Mathematicians have various ways of defining the dimension of a set of points in the plane, or even more general sets.  A point, or a finite set of points, is 0-dimensional.  A line, or a smooth curve, is 1-dimensional.  A solid rectangle, or a disk, is 2-dimensional. 

But sometimes it's more complicated!  There are fractals whose dimension is not an integer... at least if we use the right definition of 'dimension'.  The old Lebesgue dimension is always an integer, but the Hausdorff-Besicovich dimension or Minkowski dimension can be fractional, or even irrational. 

And there are also sets whose dimension seems to depend on how closely you look at them!  That's what we have here.

Simon is working on a theory of scale-dependent dimension, to make this precise.  He's writing a series of blog articles on it - and the first is here:

https://golem.ph.utexas.edu/category/2015/03/a_scaledependent_notion_of_dim.html

There's a lot of nice math here, but a lot of open questions... which is good if you're a mathematician!  More puzzles to work on!

For the hard-core details, go here:

•+Simon Willerton, Spread: a measure of the size of metric spaces, http://arxiv.org/abs/1209.2300.

#spnetwork arXiv:1209.2300 #fractal #dimension #geometry