A curve that reaches every point in the unit square is known as a space-filling curve. One of the most famous space-filling curves is the Hilbert curve, which arises from the iterative procedure shown above.
I am just slightly confused how the curve in the next stage in the Hilbert curve procedure is produced by using 4 copies of the curve from the previous stage.
As you state, the 2 by 2 grid has a curve with 3 unit line segments and the 4 by 4 grid has a curve with 15 unit line segments. Wouldn't that mean that the 4 by 4 curve is made out of 5, not 4, copies of the previous 2 by 2 curve?
The 4 by 4 curve is indeed made out of 4 copies of the 2 by 2 curve, but we need to use 3 additional segments to connect those copies to each other. So the 15 breaks up as (4*3)+3, and not as 5*3. Thanks for your comment!
Great post!
I am just slightly confused how the curve in the next stage in the Hilbert curve procedure is produced by using 4 copies of the curve from the previous stage.
As you state, the 2 by 2 grid has a curve with 3 unit line segments and the 4 by 4 grid has a curve with 15 unit line segments. Wouldn't that mean that the 4 by 4 curve is made out of 5, not 4, copies of the previous 2 by 2 curve?
The 4 by 4 curve is indeed made out of 4 copies of the 2 by 2 curve, but we need to use 3 additional segments to connect those copies to each other. So the 15 breaks up as (4*3)+3, and not as 5*3. Thanks for your comment!
Ah okay, thanks!