As I am new to understanding fractals, I appreciate their beauty when graphed in cartesian coordinates. My intuitive understanding of the definition of a 'fractal dimension' is the relation of the length of a fractal segment as a proportion of the subdivisions used in making it. In this way. the cartesian 2D coordinate system is calculated to have dimension 2. When we therefore create 2 axes, we can plot a set of points each of which has a real number value in each of these dimensions.
But I imagined a coordinate system not of 2 dimensions, but for example of the dimension of a Koch snowflake, 1.26186. But what does it mean for there to be 1.26186 coordinates for a point? Is this a concept that has been addressed in mathematics? Please do not be hasty to condemn this question as stupid- the concept of the square root of negative one seems equally absurd to me. Yet mathematicians invented a way to deal with that question. And the result was a tool that was miraculous-enabling Fourier transforms and a host of other useful tools-that still want to make my brain explode.
Certainly the concept of length in the fractal dimension has been addressed
Defining distance in fractal dimensions.
So if there is length, then is there not a way to map the fractal coordinate system into the cartesian one so i can visualize it? or would the result be so mind boogling as to exceed comprehensibility- like those drawings of hypercubes in 3 space? We map all the aspects of a hypercube into different 3D cubes and either throw them into a 3D picture together (very messy) or we make a movie of transitioning from one cube to another through time.(very trippy)
When we say $K$ has dimension $1.28$ that simply does not mean that a point of $K$ is described by $1.28$ coordinates.
That answers the question, but of course it raises the second question "Ok, then what does it mean?". For an answer to that look on google, especially Wikipedia, for "Hausdorff dimension", "fractal dimension", and various other sorts of dimension that will be mentioned in those articles.