I am trying to understand a particular version of the "Peano curve." Although definitions will vary from person to person, for the purposes of this question I am talking about the one obtained from the process in the picture below.
More specificically, I am concerned with a curve which maps the unit interval surjectively onto the unit square $[0,1] \times [0,1]$.
Now, I find the above figure a little confusing because there are no axes or points labeled, thus it is hard to tell what points the curves touch.
For the first curve pictured, I would describe it by taking a straight path from $(0,0)$ to $(0,1)$, then from $(0,1)$ to $(1/2,1)$, then from $(1/2,1)$ to $(1/2,0)$, then from $(1/2,0)$ to $(1,0)$, and finally from $(1,0)$ to $(1,1).$
I am having trouble describing the second curve, however. I would start by taking a straight paths from $(0,0) \to (0,1/3) \to(1/6,1/3) \to(1/6,0) \to(1/3,0) \to (1/3,1/3)$. So far so good? Now, I think I'm supposed to continue by going $(1/3,1/3) \to (1/3,2/3) \to(1/6,2/3) \to (1/6,1/3) \to (0,1/3) \to (0,2/3)$. But wait a second! This curve overlaps! This is different from the picture!
So, if the picture is wrong, how does one correct it? I'm also unsure where I'm supposed to go once I reach $(1/3,1)$. Do I move back down (even though I just came that way) or do I move to the right?