Is every shape with an infinitely long circumference, but finite area, a fractal?

Question

I'll tighten up on the question, wjsst22:

Is every shape with an infinitely long circumference, but which can fit inside a circle of finite radius, a fractal?

Answer 1

A fractal has certain qualitative properties such as self-similarity (see Wikipedia article) but does not have a precise mathematical definition.

If S1 and S2 are different fractals with the finite area property then can join "half of S1" and "half of S2" together to get a new shape, S. S will also have the infinite circumference/finite area property, but will not be self similar. The S1 part will not be similar to the S2 part.

I am sure that you can construct a shape like the snow flake curve, except that instead of triangular bits you use parts of polygons all with different numbers of sides. That shape will not be self-similar.

But, because there is no strict definition for "self-similar" there can be no strict proof or disproof of your conjecture.

Answer 2

an infinitely tall pyramid but with a finite base area..no problem or an infinitely convoluted splodge shape inside a circle...more interesting though is the horned sphere where a sphere has two sides (inside.outside) but imagine a sphere with two horns (still 2 sides) but if the horns are trying to join up and every time they get close the tips both split into two..if this happens infinitely then another space totally separate from the inside, outside is defined inside the splitting horns..maybe it's called topography.. it was on an open university thing.. weird or what

Answer 3

Absolutely not. Take the shape bounded by the x-axis, the y-axis, and the line y=1/x^2. The area is finite (although I forget what it converges to... pi/6 maybe), but the boundary is infinitely long.

Answer 4

Not 100% of that definition. I seemed to remember it being made up of images of itself. Anyway, according to Wikipedia.....


http://en.wikipedia.org/wiki/Fractal

Answer 5

in the conjectural world maybe-in the real world no

i hope that this helps

Answer 6

Yes.