Poking Holes

F

ractals are really awesome.  I realize that sounds just a little hyperbolic, but that doesn't make it untrue.  As a teenager, I had a Julia Set "centerfold" on my wall (courtesy of OMNI Magazine) - I suppose a Grateful Dead poster would have had a similar look while being a lot more cool, but I was a tiny bit nerdy and had no appreciation for Jerry Garcia's talent at the time. As beautiful as Julia was though, she wasn't my first fractal ...

Imagine a square: three units by three units. If you divide that into nine squares (like a tic-tac-toe board) and poke a square unit out of the middle, what's left over is eight square units. Now imagine dividing each of those squares into nine smaller squares and poking out the middle - like the illustration at the right. Continue that process long enough and you’ll find yourself poking A LOT of very tiny holes. You’ll also have created one of my favorite fractals - the Sierpinski Carpet (which I always thought looked more like a rug), a rather interesting little shape named after an exceptional Polish mathematician: Waclaw Sierpinski.

I’ll admit, the Carpet isn’t as curvy or exotic as some of the “sexier” fractals, but it’s still pretty quirky and, once you get to know it, kind of fun to hang out with. It even knows a pretty interesting trick – but more about that in a minute.

At some point early in our study of geometry, we learned about polygons and how to find their area and perimeter. Later, we learned that a square has a smaller perimeter/area ratio than any other rectangle (although we tend to forget those kinds of things), and that a circle will always have a smaller perimeter/area ratio than any polygon1. But what shape has the greatest perimeter/area ratio? The answer to that question is a lot more subjective than you might think, but my friend the Sierpinski Carpet is a pretty good candidate.

Calculating the area is kind of a process ... it starts by adding up those holes. Our first hole was one square unit, then we made eight smaller holes that are each 1/9 of one square unit. Next, 64 even smaller holes that are each 1/9 of 1/9 (1/81) of one square unit. After ten sets of holes the white space would be almost 6.23 square units. The eleventh set of tiny little holes (more than 1 billion of them) only add another 0.31 square units to that white space. The numbers get smaller and smaller each time, but they keep coming ... etcetera, etc ...

= =

This type of expression - where the ratio between successive terms is constant - is known as a geometric series. This particular series is also "convergent," meaning that the sum of the successive terms approaches a limit. In this case, the sum of the area of all those holes converges on nine. Our original area (before we poked any holes) was also nine so, if we kept at it forever, the area of that carpet would, mathematically speaking, approach zero - which shouldn't come as a surprise, but it usually does.

The perimeter is a little more complicated. Our original square had a perimeter of 12 units (four sides, three units each). Poking that first hole in the middle added four units (four sides; one unit each). The next set of holes (eight of them) had sides measuring 1/3 of a unit - multiplied by four, that makes 32/3 additional units. After that we made 64 holes, with sides measuring 1/9 of a unit - for another 256/9 units. As you can imagine, those figures are going to add up pretty fast ...

= =

 Unlike the geometric series that led us to the combined area of those holes, the one that describes the growth of the perimeter doesn't converge: the amount we add at each step just keeps getting larger. Provided we kept it up forever, the perimeter of that little carpet would just keep working its way toward infinity.

The chart below shows the total perimeter and remaining area after each of the first ten successive set of holes:

Total HolesTotal PerimeterRemaining Area
Start (no holes):129
1:168
9:26.677.11
73:55.116.32
585:130.635.62
4,681:333.234.99
37,449:872.634.44
299,593:2,311.003.95
2,396,745:6,146.673.51
19,173,961:16,375.123.12
153,391,689:43,650.992.77

So, eventually, what we would have (if we could actually GET there) is an infinite perimeter surrounding an area of zero! Hyperbole or not, that is awesome.

  1. No, circles are not polygons []
Posted in Mathematics