How many dimensions is this?

For generic fractal curves the topology induced on the curve from the surrounding space will typically prevent that curve from being a topological manifold, since locally it won't look like R^n. It will merely be a topological space.

If, however, the (image of) the curve is a topological manifold (with the induced topology), then the (integer) dimension of the manifold will agree with the Minkowski dimension that's explained in the article (and also with the more commonly used Hausdorff dimension[0]).

[0]: https://en.m.wikipedia.org/wiki/Hausdorff_dimension

The article seems to miss the difference between just using low-dimensional manifolds (e.g. any line) and escalating to a higher-dimensional space to distinguish different ones (e.g. various lines in a plane).

Not quite - as I understand it box-counting measures global space-filling, manifolds handle local coordinate structure. Consider that the Earth is locally flat but globally spherical, and a Möbius strip vs cylinder are locally identical but globally different. Related problems, but the tools reveal different aspects of geometry. So I think whether “this is exactly what topological manifolds are for” depends what you’re trying to understand.

There's a load of things I think are things many kids could do much earlier in maths, though my personal feeling may be off. My son seems to be taking very quickly to the feeling of maths so I'm trying to share some parts that I only realised drastically later on.

Rules you learn in maths are often based on much simpler, truer (I'll get back to this), things. I learned loads of "move this denominator up to this side, like an escalator!" and such and only when I was something like 13 I nearly shouted "EQUALS MEANS THE TWO SIDES ARE THE SAME THING" as I realised why so many of these god damn rules existed. It also explained suddenly why some things are OK to do and some aren't (notably taking roots).

The other key thing was finding out that these things are decisions. We can choose whatever rules we want and see what happens, and keep whatever is useful. Lots of the more interesting things I've seen later in life have been "well there's no number you can square and get a negative number. What if there was?" or "What if there was a way to raise to a real number, what would that mean?". There's no magical reason we can't divide by 0 but there's not a neat useful answer really, but you can totally define a setup where you can do that and have fun if you want.

Imaginary numbers and Argand diagrams are things my 6 year old can mess about with. He's still developing reasoning so certain things trip him up about seeing the process for solving a thing (fascinating to see where his comprehension is up to and the leaps that happen, I remember the first time seeing him pick up a chess piece, go to move it and before putting it anywhere say "no, if I do that you'll do X") but lots of things are very accessible. Powers are fun because you can make big things quickly, factorials too. Basic solving of equations I was only taught in secondary school yet I was taught "two bananas cost 30p and one banana and one apple costs 40p, how much is one apple" years before and that's just swapping fruit for letters.

I was always good at maths, but I think it's far more fun now I've understood more about what you can mess about with and how simple parts are at times.

Yeah, it is not entirely obvious. You need to prove that if you fix a number x from range [0,1] then the point at "x * length(Hn)" converges to a point in 2D, for every x, as n goes to infinity (Hn = nth Hilbert curve).

The limit for x is the mapping from x to a point in 2D.

The proof is by observing that the subsequent "jumps" are exponentially smaller.

If you take the length of the entire curve as a number l irrespective of the iteration depth, you can say you are at position 42% of l. Because at iter n, the segments are replaced by a new shape that does not grow beyond the bounds of the segments at n-1, the iterations are 'contractive', and iterating more deeply only makes the x, y location of 'i am at 42%' more precise.

Wdym? Can zoom just fine on desktop.