I'm trying to read and understand the article in which Hilbert gave an illustration of a space filling curve, namely "Ueber die stetige Abbildung einer Linie auf ein Flächenstück". It's only a short 2 page paper, but me being not the best mathematician, I'm not able to fill in the gaps left in the explanation.
This is the three figures available in the paper, below, I give my translation of the relevant piece of text.
"We take a line segment of length 1 and divide it up into 4 pieces of equal length. We then take a unit square and divide it by means of two perpendicular lines into 4 equal quadrants marked 1,2,3,4 (Fig 1). Next, we divide each piece on the line segment into four equal pieces, yeilding 16 pieces; at the same time, we also divide each of the quadrants into equal quadrants, and write the number $1, 2, 3,\ldots, 16$ in the 16 resulting quadrants, where the ordering of the quadrants is to be chosen so that each quadrant shares a side with it's predecessor (Fig 2). If we take this process further - Fig. 3 illustrates the next step - it can easily be seen that to each point on the line, we can assign a singular point in a quadrant. All we need to do is to mark the piece of the segment that contains the point. The quadrants with the same numbers lie necessarily within each other and enclose in the limit (or "on its border", orig. "in der Grenze") a point of the unit square."
Now, I have several issues with this, namely:
- I do not understand how the first bold part gives an unambigous definition of the curve. Even if I follow the transitions between quadrants from fig 1 and fig 2, I can still come up with a different ordering (curve) for fig 3 (assuming the recursive nature of the definition), see the red line on the picture below. Notice that it's not even symmetrical (although it could be if I ordered the bottom right (last four) quadrants differently). Where did I go wrong here? (I understand there are other ways to define the curve, such as L-Systems, I'm just curious about this specific definition)
- The second thing I do not understand is the second bold part. I can se how he maps intervals on the segment to quadrants, and that in the limit, the quadrants become points, as do intervals on the line. Intuitively, this is clear. However, what I do not understand is the part about quadrants with the same numbers being contained within each other; I'm, however, not all that sure about my translation being correct here.
Any other explanations welcome! I do have a little math background, but I am not a mathematician. I'd just like to convince myself about the correctness of the definition; pardon the inevitable lack of rigor.
PPS: Here's the Springer link to Hilbert's paper (may be behind a pay wall)