Space filling curves

a paradox resolved by 3d printing

Discuss the language of the universe.

Moderators: Calilasseia, ADParker

Space filling curves

#1  Postby newolder » Sep 28, 2016 11:41 am

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...
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
User avatar
newolder
THREAD STARTER
 
Name: Albert Ross
Posts: 7310
Age: 1
Male

Country: Feudal Estate number 9
Print view this post

Ads by Google


Re: Space filling curves

#2  Postby CdesignProponentsist » Sep 28, 2016 10:50 pm

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 :D

Image
"Things don't need to be true, as long as they are believed" - Alexander Nix, CEO Cambridge Analytica
User avatar
CdesignProponentsist
 
Posts: 12683
Age: 53
Male

Country: California
United States (us)
Print view this post

Re: Space filling curves

#3  Postby Rumraket » Sep 29, 2016 6:54 am

This thread is not what was advertised. :smug:
Half-Life 3 - I want to believe
User avatar
Rumraket
 
Posts: 13215
Age: 40

Print view this post

Re: Space filling curves

#4  Postby CdesignProponentsist » Sep 29, 2016 7:34 pm

Rumraket wrote:This thread is not what was advertised. :smug:


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

Image
"Things don't need to be true, as long as they are believed" - Alexander Nix, CEO Cambridge Analytica
User avatar
CdesignProponentsist
 
Posts: 12683
Age: 53
Male

Country: California
United States (us)
Print view this post

Re: Space filling curves

#5  Postby igorfrankensteen » Sep 29, 2016 10:38 pm

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.
User avatar
igorfrankensteen
 
Name: michael e munson
Posts: 2114
Age: 67
Male

Country: United States
United States (us)
Print view this post

Re: Space filling curves

#6  Postby CdesignProponentsist » Sep 29, 2016 11:19 pm

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.
"Things don't need to be true, as long as they are believed" - Alexander Nix, CEO Cambridge Analytica
User avatar
CdesignProponentsist
 
Posts: 12683
Age: 53
Male

Country: California
United States (us)
Print view this post

Re: Space filling curves

#7  Postby igorfrankensteen » Oct 01, 2016 4:24 pm

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.
User avatar
igorfrankensteen
 
Name: michael e munson
Posts: 2114
Age: 67
Male

Country: United States
United States (us)
Print view this post

Ads by Google


Re: Space filling curves

#8  Postby newolder » Oct 01, 2016 4:27 pm

^ 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 am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
User avatar
newolder
THREAD STARTER
 
Name: Albert Ross
Posts: 7310
Age: 1
Male

Country: Feudal Estate number 9
Print view this post

Re: Space filling curves

#9  Postby Evolving » Oct 01, 2016 4:48 pm

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.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Space filling curves

#10  Postby igorfrankensteen » Oct 02, 2016 12:31 am

I guess it depends on how you define "fill" and "space."
User avatar
igorfrankensteen
 
Name: michael e munson
Posts: 2114
Age: 67
Male

Country: United States
United States (us)
Print view this post

Re: Space filling curves

#11  Postby Evolving » Oct 02, 2016 8:59 am

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.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post

Re: Space filling curves

#12  Postby igorfrankensteen » Oct 02, 2016 1:21 pm

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.
User avatar
igorfrankensteen
 
Name: michael e munson
Posts: 2114
Age: 67
Male

Country: United States
United States (us)
Print view this post

Re: Space filling curves

#13  Postby Evolving » Oct 02, 2016 1:29 pm

:thumbup:

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.
How extremely stupid not to have thought of that - T.H. Huxley
User avatar
Evolving
 
Name: Serafina Pekkala
Posts: 11993
Female

Country: Luxembourg
Luxembourg (lu)
Print view this post


Return to Mathematics

Who is online

Users viewing this topic: No registered users and 1 guest