Fractal charm: Space filling curves

If you watched my video about Hilbert's space-filling curve… …you might be curious to see what a few other space-filling curves look like. For each one, see if you can figure out what the pattern of construction is. This one is the Peano curve. It is the original space-filling curve. But of course, there's no reason we should limit ourselves to filling in squares. Here's a simple triangle-filling curve I defined in a style reflective of a Hilbert curve. This one has the most delightful name, thanks to mathmetician/programmer Bill Gosper: "Flow Snake" What makes this one particularly interesting is that the boundary itself is a fractal. It might become a surprise how some well-known fractals can be described with curves. This is another famous fractal. The "Koch Snowflake" Let's finish things off by seeing how to turn this into a space-filling curve First, look at how one section of this curve is made.

This pattern of four lines is the "seed" With each iteration, every straight line is replaced with an appropriately small copy of the seed Let's see what happens as we change the angle in this seed A sharper angle results in a richer curve A more obtuse angle gives a sparser curve And as the angle approaches 0… We have a new space-filling curve..

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