The edge grows forever. The area stops at 1.6.

What you're seeing

The rule is the whole exhibit. Take any straight edge. Cut it into three equal parts, throw away the middle part, and replace it with the two sides of a triangular bump pointing outward. One straight line becomes four shorter lines. Now do the same to all four. Then to all sixteen. Keep going.

Look at what happens to the two numbers under the picture. Each level replaces every side with four sides that are each a third as long. Four thirds is more than one, so the edge gets longer by the same factor every single time — and a number that multiplies by 1.333 forever has no ceiling. Name any length you like. Level 21 passes it.

The area does something completely different. Each level adds a fresh ring of bumps, but the bumps are getting small much faster than they are getting numerous. Level 1 adds three triangles. Level 2 adds twelve triangles that are each a ninth the size. The amounts being added shrink by four ninths each time, and a pile like that has a total. The area climbs, slows, and settles on exactly eight fifths of the triangle you started with. It gets closer forever and never arrives.

The dim circle is the honest version of that argument. It passes through the three corners of the original triangle. Every bump tip you will ever add lands exactly on it and none ever crosses it, so the whole shape is penned inside a disc of fixed size — no matter how long its edge becomes.

Now turn the bump height down to zero. The rule still cuts every side into three, so there are just as many pieces, but the edge stops growing entirely. Any height above zero and it diverges again. That is the switch: not detail, not the number of pieces — the bump keeping its shape as it shrinks. Press let each bump flatten to see the other side of it: bumps that get relatively shallower at every level give you a curve with genuine structure at every scale and a perfectly finite edge.