Weird curves bound normal areas

Curves of infinite length may enclose finite areas.

When have you last used the word circumference to refer to the set of points equividistant from a given point? Would not you rather call it a circle like everyone else? But then what is your term for the set of points enclosed by the circle? Disk? Personally, I consider both circumference and disk somewhat formal and prefer circle to either provided of course such usage does not result in ambiguity.

Descriptive identification of a region with its boundary is quite natural. Indeed one unequivocally defines the other. At least it was natural to me. So that when I first read about fractal curves which not only appeared weird but were mostly of unexpected (often beautiful) shapes I perceived as weird also the regions the curves enclosed. Would not you consider a region peculiar if its boundary were a strange looking curve?

The first unexpected feature of fractal curves is that none of them has finite length. As everyone will learn sooner or later, in order to describe them it's customary to introduce some kind of dimension (Hausdorff Besicovitch or similarity dimension.) Rarely it's mentioned that for every dimension it's possible to define a measure - a set function, i.e. a function that associates a real value with some sets, similarly to how length applies to curves and area to sets. This function, a Hausdorff Besicovitch measure md, is such that a curve C has dimension d0 if and only if md(C) = ∞ for d < d0 and md(C) = 0 for d > d0. The function m1 is the regular length so that for fractal curves we always have m1(C) = ∞. m2 is the same as area. For all but the space filling curves C, m2(C) = 0.

Now, what would be your answer to the following question

A plane curve of (Hausdorff Besicovitch) dimension 1 < d < 2 encloses a region G. What is the dimension of G?

Is it surprising to learn that the dimension of G is 2? Old, plain two. In other words, a region enclosed by a fractal curve has finite area. Let's imagine the snowflake curve built on the three sides of an equilateral triangle. The area of an equilateral triangle with side a is given by 3a2/4. Therefore, on every step, we add (1/3)2 of its area on each of the three sides. If the original triangle had area 1, the resulting Star of David would enclose a region of area 1 + 3·(1/3)2 = 4/3 and consist of 12 segments 1/3 in length each. (Do I want to say Star of David would have the area of 4/3 and consist of 12 segments 1/3 in length each.) On the next step therefore we'll have to add 12 small triangles each with side 1/9. This will increase the enclosed area by 12·(1/9)2. The resulting curve will consist of 48 segments of length 1/9. The next step will increase the area by 48·(1/27)2 and so on. Thus our construction generates a geometric series with a factor of 4/9 whose sum, as well known, is finite.

Where did 4/9 come from? The number of segments from one step to another grows by a factor of 4. Their common length, on the other hand, decreases by a factor of three. So that the area of the triangles built on these segments decreases by a factor of 3·3 = 9.

Some curves like, for example, Sierpinski gasket look like sets of substance. However, Sierpinski gasket has dimension log(3)/log(2) and according to the above mentioned property of the Hausdorff Besicovitch dimension has area of 0. It's a sheer boundary with no internal points but a lot of self intersection. However, accepting that the set it bounds is the part of the original triangle that is obtained after removal of the gasket, then the area of this set is the same as that of the original triangle.

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