I fear that this will be a shaggy dog post. One of my students posed an interesting question the other day, and I found myself surprised by the answer. So here’s the quick version of today’s post: What’s the dimension of a tetrix? A tetrix is also known as a Sierpinski tetrahedron, and indeed it is like a three-dimensional version of the Sierpinski triangle, except that, being a fractal, its dimension is less than three. There’s a picture below (which I think of as upside-down), and also a cool java applet version here which you can spin around.
As background, the dimension of a fractal is usually pretty weird — they have so many holes that they’re not quite two or three dimensional. So to calculate its dimension, we have to think about exact what dimensions are. If you take a square or a triangle, for example, and double the length of the outside edge then you can fit 4 (which is 22) of the original pieces in the new object. And if you triple the length of the outside edge, you can fit 9 (that is, 32) of the original object in there.
In other words, assuming the original side length is 1, if the new outside edge has length S and you can fit N of the original objects in there, it turns out that . The 2 in the exponent is related to the dimension being two.
Now look at three-dimensional objects. If you double the length of the outside edge, you can fit little cubes in there. And if you triple the length of the outside edge, you can fit cubes in there.
Once again, treating the original cube as being 1×1×1, if the new outside edge has length S and you can fit N of the original objects in there, it turns out that . And this time the 3 in the exponent is because it is three-dimensional. In general, if D is the dimension, then . This description of dimension is known as the Hausdorff dimension.
So what about fractals? Let’s look at the famous Sierpinski Triangle, below on the left. It’s made from smaller versions of itself, but because it has a big old hole in the middle, you only put three Sierpinski triangles together to form a larger version. That’s easier to see in the colored version on the right.
In other words, if then and we’re faced with . Taking the natural log (or any log) of both sides gives , but that’s just , giving a dimension of . So even though the Sierpinski Triangle lives in the two-dimensional world, it has so many holes in it that it’s almost half a dimension shy of that.
Now look at the Sierpinski Pyramid. It looks like the Pyramids of Giza, except with lots of holes.
You make a larger version by putting together five smaller versions: if then . And that leads to . As before, taking the natural log (or any log) of both sides gives so .
Which brings us back to the tetrix/Sierpienski tetrahedron. Larger versions are formed by putting together four smaller versions (one on each corner). So what’s the dimension?
Like I said, a bit of a shaggy dog post. But I still found it harder to wrap my brain around than the dimension of 2.32 in the pyramid. And it made me wonder: what’s the smallest possible dimension for a fractal that lives in the third dimension?