0

I learned recently that lines such as the boundary of some Julia sets and the Hilbert curve have area. I was wondering what the strict definition of such an area would be, and I intuitively came up with the following:

Let $S$ be the set of all points on the line.

Let $R$ be the set of points in some contiguous region containing the line fully inside of it.

The area of the line is the percent of $R$ also in $S$ multiplied by the area of the region.

William Grannis
  • 1,844
  • I don't know about the boundary of a Julia set, but the Hilbert curve is space-filling, and the area is just the usual two-dimensional area. – saulspatz Dec 27 '18 at 18:30
  • The word "line" is not a good choice for the boundary you ask about. Lines are straight. While words like "curve" or "path" avoid implying straightness, your post would probably be better phrased in terms of the boundary or border of a region. – hardmath Dec 27 '18 at 18:39

1 Answers1

1

Your definition suffers from the following(fatal, I would say) flaws:

  1. You rely on the word “percent” to define your “area”. But you clearly don’t mean percent in the usual sense, because the word “percent” is (usually) defined for finite quantities and has something to do with amount out of $100$ if you like.

  2. Your choice of the enclosing region R is not unique. Thus you must give a clear indication of how R is chosen and moreover prove that different choices of R give the same value of area.

The usual definition of area of S is $\mu(S) $where by $\mu$ I denoted Lebesgue measure(I guess you could use other measures as well). You can read more about it on wiki of course.

Sorin Tirc
  • 2,087
  • Thank you. Is my wild stab in the dark at least intuitively correct, if not concrete in details? – William Grannis Dec 27 '18 at 18:46
  • It is only very distantly correct if I may say so :) because when you define the Lebesgue measure of S you also cover S, but, here’s the difference!, you cover S not just with one region R, but with many many(as many as you can) regions R, usually of rectangular shape(so we know their area) and sum the value of these rectangular areas and tale the limit. – Sorin Tirc Dec 27 '18 at 18:52