CollIB wrote:[...] the fractal dimension is not really a dimension, [...]
Why not? That's one very good way of understanding what a fractional dimension is.
Think of it this way: the surface of a balloon, for example, can be the basis of a 2D universe. In the same way, the surface of a 4D balloon can be the basis of a 3D universe. In fact, it's possible to interpret our universe as the surface of some kind of 4D construct.
Now what if, instead of a balloon, you have, say, a central sphere surrounded by 20 smaller spheres touching its surface, and the surface of all of the spheres combined is the basis of a 2D universe? This 2D universe will have some strange properties: most of the space behaves like usual 2D space, but where the smaller spheres touch the central sphere, there is a point where space has extra-2D properties (motion at the points of contact has extra directions of motion impossible elsewhere -- you can either remain on the sphere you were on, or "transition" between the smaller and larger spheres). So these points of contact behave like "magic portals" connecting what appears to be detached 2D worlds.
But what if these smaller spheres have their own smaller spheres surrounding them? Then instead of 20 "portals", you have 400 portals sprinkled throughout the resulting 2D space. And instead of a central sphere, what if it was composed of 13 smaller spheres, one at the center, 12 surrounding it, with the original 20 smaller spheres touching 3 of the 12 spheres? Then you have a lot more portals, and the universe now consists of small chunks of "normal" 2D space with lots of portals interconnecting between them.
But why stop there? Let's say each smaller sphere reduces to a collection of yet smaller touching spheres. Do this recursively until you get the equivalent of a 3D fractal with a snowflake-like shape. As the recursion level gets bigger, the number of "portals" increase and the area of each "normal" chunk of 2D space gets smaller. At the fractal level, the number of "portals" become infinite, and the area of each "normal" chunk of 2D space becomes 0. So now the universe you have becomes a strange non-Euclidean world consisting virtually entirely of "portals", with infinitesimal "2D areas" in between.
That's what a fractional dimensional space looks like. The dimension of this universe will be some kind of fraction between 2D and 3D. The "portals" are strange, because their interconnectivity is greater than 2D, but less than 3D. Shapes in this universe consists of subsets of the fractal snowflake's surface, just like 2D shapes are essentially subsets of the entire 2D space. But they are truly strange, because their internal interconnectivity is more than 2D, yet less than 3D. In a normal 2D space, a normal (non-fractal) 2D object's interior consists of points connected to other points all around it in a uniform 2D way (this is mathematically called the "neighbourhood" of the point). In a normal 3D universe, the interior of a non-fractal 3D object consists of points connected to other points that surround it uniformly in a 3D way. But in the fractal snowflake, its internal points have very strange neighbourhoods. They are like the 2D neighbourhood of a 2D object, yet there are an infinite number of "portals" that give each point a connectivity that's more than 2D. Yet the neighbourhood isn't 3D either; the "portals" don't connect to a full 3D volume around the point, but to something less than 3D. So the interior of these shapes ride that weird limbo between 2D and 3D.
The boundaries of these shapes are also very strange. They are more than 1D, because each point on the boundary also has a neighbourhood consisting of an infinite number of "portals", but some of the portals are in the shape and some are outside. So each point connects to more than 2 neighbouring points, yet they are not fully surrounded in a 2D sense, but an infinite number of pieces are "missing" from the 2D interconnectivity. So the boundary is somewhere between 1D and 2D.
Because this fractional dimension essentially consists of "portals" (the "normal" pieces of space are all zero area), geometry in this universe behaves very strangely, and nothing like the kind of geometry we're familiar with. It's essentially a universe where space itself is composed of a subatomic network of an infinite number of "portals", and macroscopic objects exist as points connected over a subset of this network. Creatures in this universe exist across an infinite number of portals simultaneously, and they interact with each other through an infinite chain of "empty" portals in between them.
Very strange.
