Yery nice; thanks!

]]>1) self-similarity,

2) “Fine or detailed structure at arbitrarily small scales”,

3) “Irregularity locally and globally that is not easily described in traditional Euclidean geometric language”,

4) “Simple and “perhaps recursive” definitions”.

I don’t think the third criterion is really met here. In any case, the figures seem to have Hausdorff dimension one. ]]>

In other words, Burak, I think your picture is as follows. Being a simple closed curve, a circle divides the plane into two regions, one having finite area, one not. If the circle represents a set, then we normally think of the elements of the set as lying within the region that has finite area. But we could just as well think of the elements of the set as lying in the other region. In one sense of the word, each of the two regions determined by the circle is *finite*, because it has a boundary that keeps some points away from it.

Under this conception, the circle representing zero or the empty set will have nothing *outside* it, because the outside holds the elements of the set. The circle representing one will lie inside the circle representing zero. But then to represent two, we need two circles: one to lie “inside” Circle One, thus “containing” it; and one lying inside Circle Zero (thus containing it), but *not* inside Circle One. These two new circles *together* represent Two; and so on.

Your picture will give us representations of all of the natural numbers, within a finite region (namely the original circle, representing zero). This is a good feature. Unfortunately I do not see how the set omega of all of these natural numbers can itself be represented; do you?

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