New Look and Space Filling Curves

In honor of the new year, the OU Math Club blog has gotten a new look.  Most of it is self explanatory, but you might be wondering about the image at the top.  Maybe it will help to see the whole picture:

Blue Orange Frames II by Don Relyea

Blue Orange Frames II by Don Relyea

The header is an excerpt from this work of art by Don Relyea.

What is it?  It is an artistic interpretation of Hilbert’s curve.  First considered by Hilbert in 1891, it’s an early example of a fractal type construction in mathematics.  It is defined as the limit of curves H_{1}, H_{2}, H_{3}, \dots.  The first few are given below:

The first few approximations to the Hilbert Curve

The first few approximations to the Hilbert Curve

Hilbert’s curve is the limit of this construction.  Most amazingly, this is an example of a space filling curve.  That is, a continuous function with domain [0,1] and range the entire unit square!

The strange consequence is that there is the same number of points in a square as on just one edge of that same square!  Strange but true.  As you might remember from your discrete math class, even though the integers are a subset of the rational numbers, there are the same number elements in each set (ie. they have the same cardinality).  This is another example of the same strange phenomenon.

Here is another example of Hilbert curve art by Don Relyea.

Blue Recursive by Don Relyea

Blue Recursive by Don Relyea

P.S.  Remarkably, one of our favorite comics, xkcd, rediscovered Hilbert’s curve when the artist created a map of the internet:

xkcd, the internet, and Hilbert

xkcd, the internet, and Hilbert

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5 thoughts on “New Look and Space Filling Curves

  1. This just shows how much art involves math — although idk how much Medieval artists thought about math!

    • Don,

      I’m sure the math department would be delighted to be home to one of your pieces! And thanks for stopping by the blog to say hi!

  2. He’s got even more “mathematical” and “algorithm” art on his website – it’s amazing! I would love to have one of the Space Filling Curve pieces… I’d like to know how you can buy his work – I didn’t find any info about it on the website.

  3. Pingback: Opalka 1965/ 1-∞ | OU Math Club

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