The Biggest Curve of All
This thing doesn't look like a curve. It's called the Menger sponge
. It's a fractal built in an obvious way: take a cube and keep poking square holes in it in all 3 directions, smaller and smaller, with each new generation of holes being 1/3 as big in each direction as the last.
Fractals have dimensions that can be fractional, or even irrational numbers. But there are different kinds
of fractal dimension. If you visit my blog you'll see how to compute the 'Minkowski dimension' of the Menger sponge:http://blogs.ams.org/visualinsight/2014/03/01/menger-sponge/
This dimension is between 2 and 3, which makes sense, since the Menger sponge looks kind of 3-dimensional... but it has so many holes in it that it has no volume at all!
However, I also explain another kind of dimension, the Lebesgue covering dimension
, which takes only integer values. For the Menger sponge it's equal to 1... so this set is a curve
according to a certain (perhaps rather odd) definition of that word.
More ordinary curves also typically have Lebesgue covering dimension 1. But the Menger sponge is special because every curve fits inside it!
That's what I mean by saying it's the 'biggest curve of all'. It's not literally enormous, but we can take any curve - in the sense I've defined - and, treating it as made of flexible rubber, stuff it inside the Menger sponge.
More precisely, using more math jargon: every compact metric space of Lebesgue covering dimension 1 is homeomorphic to a subset of the Menger sponge.
This image was created by 'Niabot' and licensed under the Creative Commons Attribution 3.0 Unported license at Wikimedia Commons. #fractals