In honor of the White Hexagonal Fluffy Stuff, here's a nice fractal carpet made of hexagons that illustrates how an infinite number of copies of a shape can converge on a larger fixed-area version of the same shape.
This one's generated from about five and a half thousand hexagons, but obviously, you can keep going infinitely far.
The construction rule's simple. You start with one hexagon (with sides of length "one"), and then add half-size hexagons to any free corners. Then repeat, infinitely (with sides of length "one half", "one quarter", and so on).
What the process converges on is a larger completely-filled hexagon with sides of length "three", so the final area is exactly nine times the original.
If you wanted to get even more recursive, you could try copying the entire hexagonal shape into every hexagon that you used to draw it (and then repeat that). Which would look rather cool. But take rather longer to calculate.