**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

View 15 previous comments

- +John Baez +Simon Willerton

If the current definition really does force dimension to vary entirely smoothly, causing a disk to pass through a 1D-regime when it clearly only ever is 0D or 2D, I feel like it might not be ideal yet.

Perhaps the dimensionality could be modelled in a way similar to energy eigenstates?

Like, 0D would be the groundstate and 2D would be the 2nd excited state and a disk would have a smooth transition from 0D to 2D without ever passing through 1D.

At least for a disk it seems to me like a "superposition between 0D and 2D" would make a fair bit of sense.

It might suffice to only take classical values though - i.e. the state vectors only assume linear combinations where all the coefficients are real numbers between 0 and 1 and in total they sum to 1. - Mainly because I am unsure what complex linear combinations (i.e. ones where there is interference) would even mean.

But classical linear combinations seem pretty sensible to me. They could even properly describe sets where point cluster in fairly clear curves which all run in parallel, clustering in surfaces. A situation where all three aspects of dimensionality are pretty clearly visible at the same time. This would require a linear combination featuring all three dimensionality-states.Mar 20, 2015 - Mar 20, 2015
- Always great posts JB, I'd be interested in your take on Laurent Nottale's work.Mar 20, 2015
- Nottale's work seems like nonsense to me. The Wikipedia article is remarkably uncritical:

https://en.wikipedia.org/wiki/Scale_relativityMar 20, 2015 - From a distance this seems like a refinement of the notion of doubling dimension that is popular in computational geometry

https://en.wikipedia.org/wiki/Doubling_space

A tangentially related (dots, and extreme scales) comment: I was recently looking into this cool pen that records your writing electronically (in addition to the ink-on-paper). The trick is that you need to write on paper that is pre-printed with an extremely high resolution matrix of dots that are perturbed from a fine grid in such a way that the pen can identify its location on the page uniquely by an inspection of the dots in a small neighbourhood. They supply pdfs so that you can print the paper yourself. I found it amusing to zoom in to over 700%:

http://www.livescribe.com/en-us/support/wifi-smartpen/howto/print_dot_paper.html

The dot location system is discussed, if not described, on Wikipedia:

https://en.wikipedia.org/wiki/AnotoMar 21, 2015 - +Ramsay Dyer - when you zoom in closer, it won't look like doubling dimension. :-) For one thing, it's explicitly scale-dependent. I actually think it should be both scale-dependent and location-dependent.

But I hadn't known about doublig dimension - thanks!Mar 23, 2015

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