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The best description of Fractal Dimension that I am presently aware of is the one presented in Mandelbrot's book: The Fractal Geometry of Nature.

You start off some time in the 19th or early 20th century, when cartographers were trying to work out the length of the coastline of Britain. Despite cartography being so mature of a discipline that we can launch rockets and photograph the Earth from space for the first time and find basically zero surprises compared to what we've already mapped by crawling across the surface like microbes on a watermelon, here we are with a dozen survey teams all reporting lengths for the same portions of British coastline off by factors of 2-5. I mean, it's simply preposterous!

Hell, cartographers from Portugal are reporting coastal lengths for their country — with impeccable methodology, mind you — greater than Spanish cartographers find for the entire Iberian peninsula.

Well, somebody did a meta-analysis and found that reported coastal lengths not only correlate directly with what atomic measurement scale the surveyors used (EG: over how short of a distance do you stop trying to count the winding details), but the correlation was exponential and it followed different exponential constants for different coastlines. For example, shrinking how short a measuring stick you use to measure the coastline of West Britain by N will give you a total length that is longer by about N^1.25, regardless the starting value of your yardstick or the value you choose for N.

Mathematically, this means that if you keep shrinking your yardstick and count every bay, every outcropping of rock, every pebble, every molecule dividing a time-perfect snapshot of sea from land, the total length that you measure will not converge onto any attractor representing the "real" length of the coastline.. it will instead predictably diverge to infinity.

But we get the same effect if we try to measure the "length" of a square area, say 1 foot square. You can try splitting it into square inches, by lining them up in a row and seeing that they measure 144 inches long. Or you can divide smaller into square half-inches.. but now they get to be 288 inches long. And splitting more finely by N always nets you a "length" that is N^2 yardsticks "longer".

So, any mathematician would just patiently explain to somebody trying to find such a length that there isn't one because they're trying to measure magnitude in the wrong number of dimensions, and that the exponential constant they are running against is the number of dimensions they should measure with to get a reliable and finite result.

That said, one can theoretically measure the coastline of Britain and converge to a finite result so long as they are constantly considering inch^1.25 's, but of probably more use is the understanding that the 1.25 gives us a reliable measure of how "rough" the coastline is: how much extra length one gets from studying another successive factor of detail. :)

All surfaces that remain "rough" or bumpy no matter how far you zoom in can be said to have fractal dimension. From "dusts" of points like the cantor set (log(2)/log(3) ≈ 0.631) between dimensions 0 and 1 .. infinitely complicated collections of elements each dimension 0 to coastlines like the Koch Snowflake (log(3)/log(4) ≈ 1.2619) or foams like the Seirpinski Triangle (log(3)/log(2) ≈ 1.585) between 1 and 2.. infinitely complicated collections of (or kinks in) elements each dimension 1, to surfaces like any given land area on Earth, or foams like the Menger Sponge (log(20)/log(3) ≈ 2.727) with dimensions between 2 and 3 represented by infinitely varied kinks and folds in 2d elements or continued aspiration of 3d elements until all 3d volume is lost. (obviously cantor set and sierpinski triangle can equally be described as aspiration of larger-dimensional solids as well! ;D)

Fractional dimensionality can obviously be extended farther, and even measurably in our own universe one can posit that the gravitational warping of spacetime around infinitely varied mass distribution gives us slightly greater than 4 space+time dimensions prior to even leaving the bounds of mundane general relativity: EG, any attempted measurement of volume * duration of any portion of the universe is doomed to diverge to infinite values by some constant as your measuring stick to account for smaller and smaller curvatures around smaller and smaller gravity wells keeps shrinking.

But in addition to cylindrical and spherical coordinate systems (themselves just elliptical dimensions combined with euclidean ones) it is fun to consider more exotic additions like hyperbolic dimensions (Yeah, you can cross hyperbolic dimensions with Euclidian ones in the same space) or fractional dimensionality or add more Minkowski dimensions because you did remember that we already have one of those, right? Well heck, we can even take that one away and make it Euclidian instead. xD

But yeah, it's true that "adding more Euclidean spatial dimensions to our 3E+1M reality" is a fun thought exercise, and that the result of adding more E is the same as adding more elements to a vector for our linear algebra formulas to nom upon. And there are a ton of fun alternative to consider as well. :)