Space-Filling Curves - Numberphile
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7 жыл бұрынHenry Segerman shows us some 3D-printed space-filling curves, including the Hilbert Curve and Dragon Curve.
More links & stuff in full description below ↓↓↓
Check Henry's book about 3D printing math: amzn.to/2cWhY3R
More Henry videos: bit.ly/Segerman_Videos
Dragon Curve videos: bit.ly/Dragon_Curve
A little snippet we cut from this video: • Infinite Staircase - N...
A paper on this topic by Henry Segerman and Geoffrey Irving: bit.ly/Fractal_Paper
Leave a comment about this video on Brady's subreddit: redd.it/54k7h6
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@CunnininnuC 7 жыл бұрын
"it gets more squiggly" is the best way i've ever heard anyone describe space-filling curves
@arrrda__ 4 жыл бұрын
how many description have you heard before? lol
@saschaschneider9157 Жыл бұрын
I think that's the official scientific term for it. (Or at least it shoul be.) 😂
@future62 7 жыл бұрын
Video could also be titled 'How Ramen Noodles Are Packed'
@DustinRodriguez1_0 7 жыл бұрын
I was going to 3D print a space-filling curve/gasket... but the cost of an infinite amount of filament put me off.
@sergey1519 5 жыл бұрын
Dustin Rodriguez if you will make it 1/3 of the height for every next step you will finish it with finite filament
@pixiedust1383 4 жыл бұрын
sergey technically, if the ratio of the size of the next level down is between 0.999 and 0.001 times smaller then he’d only need an finite filament.
@Anklejbiter 4 жыл бұрын
@@pixiedust1383 Technically, if the ratio of the size of the next level down is between 0 and 1 times smaller then he’d only need an finite filament.
@monasimp87 2 жыл бұрын
@@pixiedust1383 technically if he doesn’t build the next layer he would only need a finite amount of filament
@villanelo1987 7 жыл бұрын
6:12 "Do you recognize this shape?" Well, of course I do, that's the Tri Force.
@otakuribo 7 жыл бұрын
You're a Tri Force Hero.
@iota-09 7 жыл бұрын
yeah, i was like "...the triforce? uhm... but why?" then he continued on.
@gilian2587 5 жыл бұрын
That tri-force has cancer or something...
@samuelthecamel 4 жыл бұрын
There's always someone in the comments who says this
@angelmendez-rivera351 3 жыл бұрын
The Tri-Force is just the step n = 1 of the Sierpinski triangle, not the Sierpinski triangle itself. But yeah, close enough.
@CrimVulgar 4 жыл бұрын
When I was choosing a dissertation project in university, space-filling curves were an option, and the handbook had a little explanation of the project, including the diagram: ■
@LazerLord10 7 жыл бұрын
Where are the 3D models for those 3D prints? I'd love to print them out myself!
@JoeJoeTater 7 жыл бұрын
I cut out a step of the Hilbert curve on a laser cutter, with the cut being the curve. It ended up being very springy as well, but rather than one big connected noodle it was a bunch of interlocking leaves. Protip for anyone laser cutting a Hilbert curve: turn up the power higher than you normally would for your material, especially if you're cutting wood. It's a very long cut for the size of the material (obviously, lol). If there are any incomplete cuts, you'll have to go back and sever them yourself. This is a big problem with something like wood, where lots of individual fibers may stay connected.
@bitspacemusic 5 жыл бұрын
You can summarize this video with, "It gets more squiggly".
@Norbal. 4 жыл бұрын
And it's getting squigglier *AND IT'S GETTING SQUIGGLIER*
@hsterts 7 жыл бұрын
"We need to get into the squiglly zone" -ViHart
@LoopyLaloo 7 жыл бұрын
my first thought too XD
@Carlos-ri6xm 7 жыл бұрын
finally someone rational :v:
@antoniolewis1016 7 жыл бұрын
+
@UriGerhard 7 жыл бұрын
squiggly squooty
@Manni5h 7 жыл бұрын
Shrooms
@eideticex 7 жыл бұрын
I used Hilbert curves in one of my terrain renderers to accelerate node lookup in a quad tree that has varying sub-divisions. Thanks to the mathematical properties of the curve I could find a node in memory (a straight line mathematically speaking) without the need to include every single node, just each sub-divided level. It was amazing just how well it's maps into memory even when certain quadrants of the curve are at a different complexity than the ones around it. Made lookups so fast that I could rebuild the tree every frame at 500FPS on a core 2 quad and so small that it fit within the CPU cache. The only downside was that it was mathematically complex to perform all the required functions of a very advanced terrain renderer that could support collision detection as well.
@MacDeth 7 жыл бұрын
Wow! Nice work. I'd like to see a video if you made one! I love hearing about people making neat things like that. Keep it up! :P
@maximusdizon7267 7 жыл бұрын
dude what?
@suave319 7 жыл бұрын
I didnt fully understand your comment, but I know it turned me on.
@CaseyShontz 6 жыл бұрын
Alan Hunter I read the first sentence as a long string of math words. The rest I could about half understand
@zh84 7 жыл бұрын
Back in the 1980s I saw a demonstration of an experimental TV system that scanned the screen in a Hilbert curve. The idea was that you could switch between resolutions, or show film of the "wrong" resolution on a different screen, quite easily. I worked out that it would be nearly impossible to adapt to colour. I have never managed to track down any other references to it. Has anyone else heard of this?
@aldobernaltvbernal8745 5 жыл бұрын
I need to find this
@NoriMori1992 5 жыл бұрын
Any luck?
@ushasingh6204 4 жыл бұрын
3b1b has a similar video on this
@LeonidasKaragiannis 4 жыл бұрын
Anyone?
@axalarat90 4 жыл бұрын
I found a paper from 1982 that is possibly related. The paper is "Using peano curves for bilevel display of continuous tone images", written by Ian H. Witten and Radford M. Neal.
@feroxcious 7 жыл бұрын
"And you were surprised by that?" OWWW SASS LEVEL UP
@williamcompitello2302 Жыл бұрын
You can't deny that mathematicians are hella passionate with their craft. You can see it in their eyes.
@finisanerd 7 жыл бұрын
The audio editing for the sped-up drawing section near the start is incredible.
@tetraedri_1834 7 жыл бұрын
I'd like to see some buildings shaped like space filling curves dragged through time. I think they look cool!
@komnishura 7 жыл бұрын
It is amazing to see how mathematics with just "simple" rules can create something mindboggeling as this.
@curtiswfranks Жыл бұрын
These three-dimensional extrusions remind me of corals and brain folds. This never boggled my mind until one of uni profs gave us a long lecture on how weird it is that a filled in square is/can be a curve and, therefore, that curves are not what we think that they are and can even be a bit tricky to define. This was part of a larher harangue on how every definition admits pathologies.
@Barnaclebeard 7 жыл бұрын
Gorgeous models.
@ganaraminukshuk0 7 жыл бұрын
Interesting timing considering I was watching all sorts of videos about the subject the past weekend.
@funicubing7340 7 жыл бұрын
love this channel keep up the good work!
@joshuagrahm3607 7 жыл бұрын
You guys do much better at selling math than any teacher I've ever had.
@janis.berzins 7 жыл бұрын
Reminds me of how brains are so densely packed in skull, props nature for figuring space filling curves out without KZfaq!
@SkyeMpuremagic 9 ай бұрын
This is so awesome!
7 жыл бұрын
Yay, fractals again! \o/ One of my favourite maths topics.
@ytsas45488 7 жыл бұрын
3Blue1Brown!
@luizmeier 7 жыл бұрын
Yep.
@ProfessorEisenoxid 7 жыл бұрын
+Aaron Cruz The one and only!
@UMosNyu 7 жыл бұрын
3:53 is also a nice part of 3Blue1Brown. The "hearing-pictures" video.
@david21686 7 жыл бұрын
How does this video relate to 3Blue1Brown at all?
@samekong5519 7 жыл бұрын
he released a video a while back illustrating space filling curves and a pretty cool use for them
@paqx3534 7 жыл бұрын
Hilbert curves are available as an algorithmic auto fill option on most 3d printing slicing software, as well
@GuyWithAnAmazingHat 7 жыл бұрын
The squiggly curve reminds me of the brain, are cranial folds similar to space filling curves?
@zh84 7 жыл бұрын
Yes. It is a nearly space filling surface. The surface of the brain has some desirable property, so the brain folds to create as much surface as possible. There is something about this somewhere in Mandelbrot's "The Fractal Geometry of Nature", a book I admit I have never finished.
@jmiquelmb 7 жыл бұрын
I read something a log time ago about this. I believe it said that reptiles and fish tend to have smoother brains, while the smarter mammals tend to have foldings on the surface.
@MrMartinSchou 7 жыл бұрын
Yeah, once he showed us the pink model, that gave me the same feeling.
@AvinashtheIyerHaHaLOL 7 жыл бұрын
The main reason our brains are folded is that we can have more surface area in our brain, but the volume doesn't grow in our head.
@salmjak 7 жыл бұрын
Folds create greater surface area per volume. You can see this phenomenon in several parts of our bodies, e.g. our intestines to increase absorption, in our brain to increase the amount of gray substance (the actual nerve cells are at the surface of our brain and the inside is filled with axons) and mitochondria also have folds in their inner membrane to increase sites for ATP (energy) production. It's an interesting observation, maybe the nerve cell layout follows a space filling curve since the actual filling in the brain are mostly axons.
@elenasanz977 7 жыл бұрын
WOW! Amazing work...
@Kabitu1 7 жыл бұрын
The way he draws grids with fractal crosses instead of continuous lines makes me irrationally mad.
@somitomi 7 жыл бұрын
How dare he make a fractal grid in a video, that has fractal-like curves in it! Scandal! Okay, actually it bugged me too.
@simoncarlile5190 7 жыл бұрын
That last squiggly thing reminded me of a brain. So that got me thinking about neuromorphic computing. Maybe neuromorphic chips could be modeled after space-filling curves?
@Markobass96 7 жыл бұрын
Thought of the same thing
@RFC3514 7 жыл бұрын
Chips aren't a single linear string of components, there are many things going on in parallel, and many components that need to be connected to multiple things at the same time. Newer designs tend to make better use of three dimensions (ex., having some functional components serving as links between layers, instead of just having a bunch of stacked layers with passive connections between them), but they are fundamentally different from a linear space-filling curve.
@Fanofjambi 7 жыл бұрын
I really don't have the entry knowledge required for some of these less intuitive more formulated ways of expressing mathematical concepts but I still enjoy these videos as it is like I'm using what I see to solve what was coming 5 seconds ago if that makes sense.
@debadityabhattacharya2405 7 жыл бұрын
Windows XP screen saver, REDISCOVERED.
@craig3.0 7 жыл бұрын
I'm noticing a pattern here. At some point in like every 5th numberphile video in recent memory, whoever Brady's filming will just pull out several 3d printed models of what they're talking about. Something tells me that around 6 months ago, the math department got a 3d printer and that behind the scenes, everyone is still super excited about it. I bet if you watch all of these videos in the order they were released, you'll see those models slowly get more and more intricate as the professors get better at using 3d CAD software.
@Maharani1991 7 жыл бұрын
Incredible.
@ilidenstrmrege987 7 жыл бұрын
if we learned something like this in school, math classes would have been much more enjoyable
@trashedlife1 2 жыл бұрын
So much to learn
@venkateshbabu5623 6 жыл бұрын
Looks like a beautiful corals reef.
@The.Talent 7 жыл бұрын
I appreciate the fact that sometimes in mathematics it's cool to take a step back and just say "hey look, it's a clone trooper", or "oh wow. It's wiggly", and forget all the math stuff for a moment. I really like he that.
@valsvoicevault 3 жыл бұрын
A better way to describe a space-filling curve, than to say “If you go infinitely far, it suddenly fills space”, is to say “Each iteration makes the curve longer, so its limit after indefinitely many steps goes through every possible point”
@luiginotcool Жыл бұрын
This seems right but it’s not! Even though it’s length diverges, it doesn’t mean it will hit every point. If you think about all the points in the rational number space (Q^2) then an infinitely long curve could hit all of the points (because there are infinitely many rationals) but still miss infinitely many irrational points!
@goo_user 7 жыл бұрын
3Blue1Brown has an amazing video on this!
@claycon 7 жыл бұрын
Wow! Those 3D prints are so cool ^_^
@Lulink013 7 жыл бұрын
6:10 "Recognize this?" Oh yeah that's the trifo- "NOPE IT'S A SIERPINSKI TRIANGLE!!" k then...
@bootblacking 7 жыл бұрын
Two types of nerds...
@valemaxema73 7 жыл бұрын
I was just going to comment that!
@SurajGrewal 7 жыл бұрын
never played Zelda but had same thoughts
@Feroxylos 7 жыл бұрын
you're missing out on one of the best Gaming Franchise in my opinion
@plokijum 7 жыл бұрын
Disappointment.
@sixhundredandfive7123 2 жыл бұрын
The tactile senses tell us more than the math show.
@IsickPuke 7 жыл бұрын
Can you post links to the 3D models so we can print / view them ourselves? Or is that accessible in the book or something?
@orioleaszme3415 5 жыл бұрын
I really liked this video
@1019wc1019 7 жыл бұрын
this reminds me of the wooden puzzles that are 1 string of smaller cubes connected at 90 degrees or 180 and you have to rotate them around to fit back into a cube
@debtcollector5632 4 жыл бұрын
i enjoyed this
@under_score3829 7 жыл бұрын
Fractals? Finally! Fractals are my childhood.
@samhuntington96 7 жыл бұрын
It's things like the sculptures in this video that make me want a 3D-printer.
@lancethompson37 7 жыл бұрын
Henry: "Recognize this shape? that would be the Stravinsky Triangle." Me: "Eh no mate, that's the Triforce."
@ricardo.mazeto 7 жыл бұрын
I designed some 3D models like this in Blender some months ago. Didn't know all the math behind it back then.
@franciscodiaz6432 7 жыл бұрын
Ahhh i want/need to print those curves
@ProfessorEisenoxid 7 жыл бұрын
These curve-objects would be some nice stamps..
@Rialagma 7 жыл бұрын
This reminds me a lot of one of Vihart's videos about doodling
@VicvicW 7 жыл бұрын
Up a squiggle, down a squiggle, Up a squiggle, down a squiggleUp a squiggle, down a squiggle.
@angeldude101 7 жыл бұрын
Up a squiggle, down a squiggle, up a squiggle, down. Woop!
@zoranhacker 7 жыл бұрын
not related but: snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek snek
@otakuribo 7 жыл бұрын
+zoranhacker 🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍🐍💕
@AscendingApsolut 7 жыл бұрын
+Elliot Grey you forgot 6 snek-s. Here they are: 🐍🐍🐍🐍🐍🐍 !
@jonahdewing6919 7 жыл бұрын
I need to make this... telling my STEM class about this tomorrow
@Fiyaaaahh 7 жыл бұрын
So many applications and yet he only talks about them for a few seconds. They're so much more than just visually interesting shapes.
@jeshudastidar 7 жыл бұрын
Have an awesome day!
@JorWat25 7 жыл бұрын
"If you enjoy Henry's videos here on Numberphile, you're really gonna love his new book about visualizing mathematics with 3D printing." Ooh, sounds interesting! Let's see that link: "Kindle: $45.52, Hardcover: $63.06" Never mind...
@henryseg 7 жыл бұрын
Yeah, I wish it were less expensive - the publisher's decision, not mine.
@Polarbuzzaroo_ 7 жыл бұрын
Trash is cheap as it is available in masses, but quality is rare and has it's price in order to get it at all.
@somitomi 7 жыл бұрын
Quick question: can I order a kind of uncomfortable bracelet somewhere?
@somitomi 7 жыл бұрын
Henry Segerman Whoa, thanks for the quick reply.
@theartistflores 7 жыл бұрын
Is it possible to get the link? I cannot see Henry's comment (might have been erased?).
@7chanconn7 7 жыл бұрын
Reminds me of 3Blue1Brown's video
@xelvoz3934 7 жыл бұрын
Funny how I had a course on this in my computer science class only two days ago. We had a recursive, beautifully made, solution for the Dragon Curve in Python using Turtle Graphics.
@venkateshbabu5623 6 жыл бұрын
What are equation of extra dimensions. The powers of random numbers when multiplied by wave functions gives shape as it splits. And that takes various regular form as it progresses with sheets of partial structures.
@Toad_Hugger 7 жыл бұрын
Vi Hart's video about this was much more entertaining XD
@DrGerbils 7 жыл бұрын
But probably much less informative.
@glenneric1 2 жыл бұрын
It's interesting that there are so many points that it could never hit. For instance it could only ever approach the center point of the main square with infinite recursion.
@markfarrelly1623 7 жыл бұрын
thanks
@holdmybeer 7 жыл бұрын
3d printing is badass
@Jame5Mead 7 жыл бұрын
Boom, you've got everything
@bumpty9830 5 жыл бұрын
Extra credit (5 points divided by number of terms in the solution): Compute the lowest spring stiffness across a long diagonal of the Nth iteration of a Hilbert curve in M dimensions with filament bending stiffness k.
@bluebluerson3493 2 жыл бұрын
Thanks!
@NorthRoyalton 2 жыл бұрын
Cool kid
@RnBoy15 7 жыл бұрын
Can someone tell me what's the best approach to model in such amazing 3d shapes? how do you connect the iterations?
@congchuatocmay4837 Жыл бұрын
Matters Computational is a book and it is free. For example it shows you how to construct Hilbert curves in code/
@nO_d3N1AL 7 жыл бұрын
Never thought a cube could be made form a single curve
@susydanelon 7 жыл бұрын
Henry, the space filling curves reminds me of fractal antennas
@fonno_ 7 жыл бұрын
Reminds me of this one pipe screensaver I had on an old Windows computer :^)
@matthewpeterson5835 7 жыл бұрын
I want that shirt!
@espen990 7 жыл бұрын
that's an amazing 3D printer
@ShilohJanowick 7 жыл бұрын
I saw (and met) this guy at the Museum of Math in New York. He was giving a talk about 4 dimensional shadows.
@rolfvankleef8439 7 жыл бұрын
Spaaaaaace-Filling Curves - Numberphile - thanks to XKCD
@saschaschneider9157 Жыл бұрын
Fractals are really weird. I mean, there's this thing called Menger's sponge, and it's like Sierpiński's triangle, but with squares and actually in three dimensions, so not really squares, but cubes. And what amazes me about it is that with every step, the surface area increases and the volume decreases. So at the limit it will have infinite surface area and absolutely no volume? I'm very fond of fractals and I'm a programmer, so I've been playing around with a few of them for some time. It was really fun. In the end, I think, I never fully understood them.
@mrmotl1 Жыл бұрын
The string maybe infinitely small, but the impressionable media of which is the defined memory is going to have some size and scale. Therefore for these infinite strings to be meaningful and usable they have to be impressed with some distortional capacity and this is where the use is implied. You have to understand the infinitely small strings are just the ideal impressionable capacity to store information, but they need to be impressed and therefore stored into some complex wave function as opposed to the ideal sine.
@Starblind11 6 жыл бұрын
The Dragon Curve is printed at the start of every chapter of one of the Jurassic Park books, can't recall now if it's the original or The Lost World
@monkeyman2497 7 жыл бұрын
"Recognize this shape?" Ooo, the triforce! "It's the sepinski triangle." ...
@alcesmir 7 жыл бұрын
The Sierpinski triangle is the triforce for a brief second step. In fact, Zelda 1 introduces only separate pieces of triforce (wisdom and power), Sierpinski triangle at step 1. Zelda 2 introduced the triforce of courage, but it took until Zelda 3 (a link to the past) for the pieces to appear as a Sierpinski triangle at step 2. Sadly, there has yet to be a game that goes for another iteration, maybe 9 pieces is a bit much to handle.
@inos151 7 жыл бұрын
omg, i was just thinking about that last night
@godsadog 7 жыл бұрын
4:34 this looks like De Sitter space! Prof. Susskind talked about it on several videos!
@kozator3 7 жыл бұрын
OMG! Please give us 3D models of those objects!
@alexissalguero6875 7 жыл бұрын
Can you fill space in the 4th dimension? is it possible to create a Hilbert curve or Peano curve in 4D? if possible please make a vid of this!!!!
@kordellcurl7559 7 жыл бұрын
Yes it is possible
@alexissalguero6875 7 жыл бұрын
How will the animation look like?
@crnobijeli13 7 жыл бұрын
That also means you can map every single point in space and time on a 1d line, even accounting for the expansion of spacetime.
@geurgeury 7 жыл бұрын
+crnobijeli13 That makes no sense. Stop trying to look intelligent.
@geurgeury 7 жыл бұрын
You can't animate it, but there is a map between a line and every n-th dimensional space.
@inflivia 6 жыл бұрын
Henry Segerman has a really nice voice
@DarkAngelEU 7 жыл бұрын
Imagining these objects to be buildings, would be so awesome :D
@huphtur 7 жыл бұрын
Some of Frank Gehry's buildings come pretty close...
@Lucas-yf1qb 7 жыл бұрын
...?...
@DarkAngelEU 7 жыл бұрын
***** They're curvy but not like this ;)
@angelinawang4866 7 жыл бұрын
The Gilbert curve! Yay geometry!
@phpn99 7 жыл бұрын
These things would make great heatsinks
@4.0.4 7 жыл бұрын
Lol, a Menger Sponge heatsink would look cool! Very... space filling though ;)
@spencer3389 7 жыл бұрын
Would that even work? I mean, you are pushing air over an infinite amount of surface area.
@ancbi 7 жыл бұрын
it's a very interesting idea! but i could imagine 2 reason why it would be a bad heat sink. 1) long conduction path. heat needs to conduct through solid before it gets to the surface touching cold air on "the other side". and since this curve fills space locally first, the solid path to get to "the other side" is a lot further than it could be. 2) from engineering point of view it would be harder to produce than simpler shape.
@phpn99 7 жыл бұрын
Two solutions: a) As mentioned you can tune the relative thickness of the elements as they leave the heat source; b) You can use selective laser sintering technology to build the parts quickly, with as much intricacy as needed. Other than this in my opinion the problems will be more: Optimal shape for airflow, and overall compactness of design. I think that grille-type radiators could use the Peano space-filling curve, for instance.
@4.0.4 7 жыл бұрын
I don't know what I'm talking about, but if the copper from a CPU heatsink is at around 60-90°C, there's a lot of heat that can be transferred to air (through larger surface area) even if it takes longer to "travel through the heatsink", right? (also, would heat radiation, as in the one that still happens in vacuum, begin to play a role?)
@azoshin 6 жыл бұрын
The infinity limit of the curve in the Hilbert curve is composed of countably many finite curves or does it have uncountable many finite curve. In other words how many steps (countably infinite or uncountable infinite) are required to hit all the points?
@smuecke 7 жыл бұрын
I'd buy all of those 3D thingies :D
@argenteus8314 7 жыл бұрын
Just a thought, but could these sorts of shapes be useful in materials engineering? A structure similar to that springy cube but on a molecular level seems like it could exhibit interesting properties if it could ever be synthesized.
@vinightshade6026 7 жыл бұрын
This would make awesome origami
@laemmeelagi 7 жыл бұрын
0:16 we're gonna build a curve, and Mexico's gonna pay for it
@TiagoTiagoT 7 жыл бұрын
My favorite is the Flowsnake
@tarunsingh2171 7 жыл бұрын
Reminds me of the Euler trail!
@MartinDxt 7 жыл бұрын
Just Imagine that in a 1000 years this Videos as exciting as it is right here right now will be considered boring basic math by most of people as they learn it in preschool
@Mephisto28890 6 жыл бұрын
7:26 i'd like to see the (electro-)magnetic field of that thing. Could you cast that out of magnetic material or make an electric circuit like that?
@FreeFireFull 7 жыл бұрын
My favourite space-filling curve is the Sierpiński Curve, a bit sad that it didn't get shown.
@FASTFASTmusic 7 жыл бұрын
If you took a cross section at an arbitrary point on those sculptures, would there be a formula or pattern between the levels?
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