Lines have no thickness so one would imagine curves don't fill space. But this guy argues in the limit of 'passing through every point', curves can fill spaces...

And it's a half-interesting use of a 3-d printer...

When I was much younger, I stumbled on the Dragon Curve while playing around with some graph paper and tape. Found out later that it was already a thing

This thread is not what was advertised.

Rumraket wrote:This thread is not what was advertised.

I'm guessing you were expecting something more akin to...

I'm not seeing the "space filling." Rather misleading title, whot?

Some of the printed representations do look like excellent models to visualize both problem solving, and the nature of human reasonings going out of control an feeding in on themselves. Very magical looking, but in the end, a fun idea turns into a scratchy, uncomfortable bracelet.

igorfrankensteen wrote:I'm not seeing the "space filling." Rather misleading title, whot?

They are called space filling because as length increases with each iteration, the area they fill remains constant and when taken to infinite iterations occupies all points within that space.

You can also say that they completely fill space in a finite number of iterations where space is quantized. Like a sheet of graph paper where you fill in every single square in the sheet while following the curve.

CdesignProponentsist wrote:

igorfrankensteen wrote:I'm not seeing the "space filling." Rather misleading title, whot?

They are called space filling because as length increases with each iteration, the area they fill remains constant and when taken to infinite iterations occupies all points within that space.

You can also say that they completely fill space in a finite number of iterations where space is quantized. Like a sheet of graph paper where you fill in every single square in the sheet while following the curve.

Well, yeah, you could say that, but you also have to completely ignore the opening stipulation that

Lines have no thickness.

It seems to me, and please tell me if I'm wrong, that even if you put an infinite number of elements into a limited space, if they have no thickness, the space will still appear to be empty.

It could lead to some fun ideas on making quantum filters of some sort though.

^ The fill only occurs in the infinite limit when the curve passes through every (zero dimensional) point in the space. I think that's the idea the maths-guy is trying to convey - but I'm no mathematician.

I think it's almost trivial. How do you prove that a curve doesn't fill a space? By identifying a point in the space (at least one) that it doesn't pass through. If there is no such point, clearly the curve fills the space.

I guess it depends on how you define "fill" and "space."

Everything in mathematics always depends on the definitions.

I think the natural way here would be like this. The "space" S would be the set of all points that are enclosed within a defined boundary: for instance the set of all points (x, y, z), where x, y and z have their usual Cartesian meanings, and x2 + y2 is no greater than 1 and z is at least zero and no more than 5. That would be a cylindrical space.

Similarly the "curve" C is the set of all points that are in the domain of a certain continuous function.

If every point that is an element of S is also an element of C, then clearly C "fills" S.

Thanks, I can certainly grasp that. I still remain delighted and amused that we can truly say that a space is "filled" with something which has no dimensions itself. Or only one, in the case of lines.

It's fun. In my own little mental world, it actually helps me to know that seeming paradoxes exist, because that knowledge helps me to be more patient with OTHER seeming paradoxes.

It's no different, in principle, from the continuum of - say - the interval zero to one being "filled" with infinitely many points, each of which has no spatial extension.