Imagine the whole space of existence as an infinitely small "dot" (zero dimensions). Seeing as though size, shape or any other attribute exists only as a relative comparison to the rest of existence, without placing that dot in a "bigger" space so as to compare it, the dot may either have no size, or infinite size. This goes for a line and a plane as well, but what about a cube? A cube is representative of true space, but... 3-dimensional space? No. Why would we think that... because a cube has 3 lines connecting to each point? What about a sphere? Infinite points, infinite lines... infinite dimensions? Of course infinite dimensions! Or zero dimensions! A Point, line, plane, cube, sphere... these are not real, these are asymptotic illusions; simply a way of structuring chaos, of organizing, or squishing chaotic space into understandable, logical "shapes" (dot, line, plane, cube).
Your argument will probably now be: 'but what about the Cartesian coordinate system'? 'Given any point, I can define movement to any other point using 3 straight-lines, 2 at 90 degree angles to each other, and the third perpendicular to the conceptual plane that the first 2 create.'
Thinking about this argument logically, i.e. simple and structured, it makes "sense". But here's the catch, in the form of a question; do "straight-lines" exist in reality, or only in simple/logical theory?
Here is a quote from Thich Nhat Hanh's "no death, no fear":
"In the darkness one person holds a torch and draws a circle of fire. Another person, standing a little bit away, has the impression that there is a fire circle. But there is no circle, there are only dots of fire that succeed each other and give the impression of a fire circle. This is like the illusion of fixed identity and the illusion of permanence. The fire circle is an idea. It cannot be applied to reality, and it cannot describe the true reality."
"...the illusion of fixed identity, the illusion of permanence."
The idea of dimensions came from "simple", and logical "shape" concepts. Why did these ideas about simple shapes get applied to the far more complex surroundings? Because of the global awareness/intelligence at the time; i.e. 'what more complex surroundings?' personal quote: "All aspects of existence exist as themselves and as the opposite of themselves, how different the opposites are is determined by how closely the focus is placed on something unique." With the use of more and more powerful telescopes and math looking to observe the nature of the "unique/structured" building blocks of the universe, what do we see? "Chaos".
We now accept the existence of "fractal dimensions", but I think a large amount of us still don't get it. Think back two paragraphs; now see everything around you as being made up of that single dot, see it vibrating throughout the whole universe at "maximum" (light?) speed, making "everything", all at the same instant (give or take the plank time). Now see that that dot is really just made up, itself, of another infinitely (infinity-sub-one, if you will) small dot. Now see that this goes on down forever, and... up forever. 3-D space as it is known is really just... space... with different... "densities".