3d Fractals - AGAIN

[Achernar]

Also, this has nothing to do with fractals, but I think the Klein bottle is technically considered to be a 2-D object. It’s a 2-D manifold anyway. What people mean when they say that it’s a 4-D object is that to get a good idea of its structure, you can’t embed it in three dimensions. This is analagous to not being able to embed a 1-D knot in two dimensions. The next-highest space you could embed it in, then, would be 4-D space.

However, I’m wondering now if it would be possible to embed it 3.5-D space. Hmmm…

[Jabba]

A fractal is a shape whose Hausdorff-Besicovitch dimension strictly exceeds its topological dimension.

[Jabba]

The link I gave doesn’t work now. It did when I checked it the first time.
This site gives some information.
So does this one.

[Achernar]

From a page in your first link, Jabba:

Mandelbrot offers the following definition of a fractal: A fractal is a set for which the Hausdorff-Besicovitch dimension exceeds its topological dimension. This definition, although correct and precise, is too restrictive, since it excludes many fractals that are useful in physics.

An alternative definition is given by Mandelbrot: A fractal is a shape made of parts similar to the whole. This definition uses the concept of self-similarity. A set is called strictly self-similar if it can be broken into arbitrary small pieces, each of which is a small replica of the entire set.

Of course I was referring to Mandelbrot’s second definition.