A Space Filling Curvature

These are the first few iterations of the Hilbert curve, a space filling curve : a curve whose range contains the entire 2D plane, this can be extended to a 3D space or more with n dimensions, however it gets harder and for your understanding, we'll stick to a 2D plane.

Starting with the construction on the left of the upper picture, which is the first iteration, each following iteration is built using the previous one, by copying the pattern on the upper left and upper right, and rotating it on the bottom left and bottom right. Joining the patterns together will give you the next iteration.

As it's quite hard to translate the iterations through words I strongly suggest you try to create the curve for yourself, underneath are the 3 first iterations of the curve, red first, blue second and black third.

The fifth iteration can be found underneath, the color of the curve also changes as you progress along the line. The picture is from 3 blue 1 Brown, an awesome math channel on YouTube.

So why talk about the curve ?

Apart from its aesthetics, this curve is used in computer science because of one of its properties, it conserves locality quite well meaning that two points on the 2D plane will also be, in general, quite close on the order created by the curve, for example, the range of IP addresses used by computers can be mapped into a picture using the Hilbert curve.

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