Steve mould, i think I found your long lost twin sibling!! His name is "david castello-lopes", and he is a french journalist/comedian/youtuber. The resemblance is very much striking
for a clear resin 3d print, like optically clear, you can polish it to 800 grit, then use a clear glossy spray paint. It'll make it optically clear. i've made lenses out of SLA 3d prints
In engineering there is the concept of the "sacrificial anode" where if a structure will be attacked by a lot of corrosion, a focal point is provided to divert the damage from the main structure. Steve wisely knows any maths videos will attract a lot of pedantic corrections. Hence the use of a "sacrificial mathematician".
Seeing the “water runs”… plus the marble run earlier… makes me think of a crossover that I didn’t know I wanted… who’s the guy that does the marble races? Jelly’s or something? It’s been a minute…
A fractal doesn't need to be self-similar at different scales. That's just how we construct a lot of them. That fractional dimension property you described is the important one, and that's achieved in anything that has infinite amounts of detail as you zoom in.
I never understood fractal dimensions for over 5 years and you just explained it perfectly in less than a minute. Just, wow. You're amazing at explaining complex topics.
Dimension is essentially a scaling factor, with higher dimensional things scaling more from the same change in length as lower things (like how doubling the radius of a ball will have more of an impact on its weight than its surface area). Things with fractional dimension are just things that scale at a rate between the rate of the integer dimensions. For example, the area of a Sierpinski triangle scaled up by two does something in between doubling (like it would if it was a line segment) and quadrupling (like it would if it were a full triangle). That's how it made sense to me when I was explained it, so in case it makes sense to you, here you go. It also I think prepares you very well to see the math behind it, where a formula for scaling in terms of dimension is then solved for dimension and used as a definition extension.
@@ludvig3242 no. I heard about fractal dimensions for the first time 5 years ago, didn't understand it. For the longest time, when I watched science videos and the topic came up I didn't understand it. Only finally got it when I saw Steve Mould explain it 6 months ago in this video, right before I made that comment.
That visualization is pretty awesome! I want to see the same thing at like... room scale... with clear pipes. It would be AWFUL to assemble lol. with the locality and stretchyness of the hilbert curve your prints demonstrated REALLY well, I'd imagine it'd be near impossible to hold rigid too...
You could use braces, there are enough parallel parts that can be linked to spo them from wobbling away. It takes a bit away from the concept bit not too much.
Worth noting: Fractals don't necessarily need to be self-similar. Also space-filling curves are definitionally fractal because it's Minkowski dimension exceeds its topological dimension. The trouble with the labyrinth is that you would need to show its limiting behavior actually fills space, which is doubtful but who knows?
Thank you, I was actually here to make the same point. The issue is that people's intuitive idea of a fractal (self-similarity) and the definition in terms of the Hausdorff dimension (which in the case of space filling curves should agree with Minkowski dimension) aren't the same.
the issue is that there is no sequence of finer and finer labyrinths that have a limit. the path the next iteration takes is wildly different, in the hilbert curve, the 20th iteration is very similar to the 19th, in that the point that is, say, 1/pi along the curve barely moved (and same for all other numbers 0 to 1)
"definitionally fractal because it's Minkowski dimension exceeds its topological dimension" - what is this supposed to mean to us ordinary plebs who don't talk math?
@@terdragontra8900 Is the limit of the space filling curve {lim n->∞ (x_n(t), y_n(t))} the set of limits of points on the curve, which is literally the whole area? Or is it {(x,y)|∃N∀n>N, (x,y) ∈ {x_n(t), y_n(t)}} (so just the 1/2^n points)? Or is it the closure of the second thing, which would also be the whole area?
@@ArawnOfAnnwn Minkowski dimension is fairly straightforward in idea: en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension The other is isn't so straightforward but in the case of curves like these think, if you zoomed in close enough what dimension would it look like. This is not what it is, it's about refinements of covering sets or some nonsense like that. But the shapes in the video are well behaved so you can think of it like that Here's the idea: en.wikipedia.org/wiki/Lebesgue_covering_dimension
I work in un underground mine where we use something called a "Belt Storage Magazine". Long story short it's a way to store conveyor belt that can be extended or retracted without taking up more space. The path the conveyor belt takes through the magazine is almost identical to the Celtic Labyrinth, although it's shaped differently.
I think something that'd be really cool would be to have one of the smaller cubes filled with a colored epoxy. I don't exactly know how well it'd set up, but it might be worth a shot. Then you might even be able to polish it by covering it in a thin layer of epoxy. Nice desk piece.
The celtic labyrinth can be defined by a substituion rule, but you have to stretch things a bit. Take two copies of a celtic labyrinth, stretch one out and open it up into an inverted U, and wrap the other curve with it - leaving a gap up the middle so you can join the two. To put it the other way, for a celtic labyrinth, there's clearly two layers to it (an outer and an inner), and each layer has the labyrinth structure.
If you filled the 3d version with 2+ liquids with different densities (and colours) and closed the loop, could you flip it around and watch the liquids re-arrange themselves? Would be a 10/10 desk toy.
This got me wondering: Shouldn't you be able to construct the cube's inlet and outlet in such a way that you could print more cubes and connect them into a bigger and bigger cube that still fulfills the property and could pipe water all the way through?
You could probably print them without inlets and outlets, and then drill as necessary to glue together bigger cubes. Might get time consuming (64 cubies to do it twice, 512 for 3, etc.), and it would possibly stop working after a while.
Potentially... Unlike the sides of a Koch Snowflake, where each iteration is made out of exact copies of the previous generation, the Hilbert curve (and the generalized 3D versions) are copies of the original *plus* extra lines to connect those copies. Those connectors don't always connect at the same angle (sometimes at a right angle on both sides, sometimes only on one), so you'd have to include paths for all possible connectors and then somehow plug the ones you aren't using
FYI fractals are not necessarily self-similar. Shoreline borders are probably the more well-known example of fractal lines that aren't self-similar. All that matters is that they have infinite detail that never smooths away when you zoom in. The self-referential ones are more famous just because it's the easier way to describe a mathematical object with that property.
Are we going to simply ignore the fact that Steve already said what the word "fractal" means, and it is not what you are saying? If you think your words are more correct, you're going to need a reference.
@@u1zha The Onion Curve is a good example of a non-fractal space-filling curve. See "Onion Curve: A Space Filling Curve with Near-Optimal Clustering" (2018) by Xu, Nguyen, and Tirthapura.
The reason the Celtic labyrinth doesn't count as a space-filling curve is that you can't define a limit version of it, or in other words, you can't define a sequence of increasingly filling Celtic labyrinths who end up stabilizing into a limit version. The Hilbert curve is special in that it does stabilize, if you go at the 1/3rd of the point in the curve for instance, each iteration of adding an additional U shape is going to go closer and closer to a limit point, it converges. For each iteration of the Celtic curve, which consists of adding a new line inbetween each existing line, the 1/3rd mark is going to go all over the place. By the mathematical version of a limit, the limit of the Celtic labyrinth curve doesn't exist, it diverges. Or in other words, it's mostly related to the "good low quality" property of the Hilbert curve, which mean low quality versions of the curve are a good approximation of its highest quality version - the limit curve.
you missed what Matt Parker was saying. While it is true that the approach used to demonstrate fractals as space filling will not work on the Celtic labrynth, there is no way of knowing at this time if it is space filling or if it is not. This is because the absence of a method of proof does not itself constitute a valid disproof.
@@Thesupremeone34you misunderstood the comment. It's not that the curve is not a fractal, which we already knew about, and that's what Matt Parker said, it is in fact that the limit of the curves doesn't exist, so you cannot even ask the question in the first place, there's no infinite labyrinth to check if it fills space in any known or unknown way because the infinite labyrinth is not well defined
@@Thesupremeone34no. what matt said was wrong. the labyrinth curve is an infinite family of curves with increasingly more parallel lines, which does not converge to a well defined curve. every space filling curve needs to be a fractal
@@toniokettner4821 what Matt said wasn't strictly wrong, he just didn't stop enough to notice the answer was a definite no because of convergence reasons, so he just erred in the side of caution and said "you cannot prove it with known methods", which is technically true, the best kind of true
The 3d version, like the 2d ones, would have an interesting property in the limit, wouldn't it? The water line through it would have an infinite length, but limited volume. Since you can't make the water go through faster than the speed of light, it would take an infinite amount of time to fill even a tiny cube with a syringe of the same volume, no matter how much pressure you can exert on the plunger.
Very interesting, this is shown in the multiple videos, there is a gradual decrease in pressure from start to finish. If it was infinitely long, there has to be infinite pressure at the start too. So your ending statement of "no matter how much pressure you can exert" is just plainly wrong. the pressure can be represented as a integral of length of the line.
@@acters124 The problem is that no amount of force can accelerate the water beyond the speed of light. The water would have to travel along the path which is infinitely long, so it could never reach the end (or anywhere else in the cube for that matter) in a finite amount of time.
To have an infinite line you would need a tube with no width unfortunately all liquide I know ha e a minimum width for exemple I doubt water could get thinner than an h2o molecule. Therefor none 0 with tube means finite tube means finite time to fill
@@glorrin You are correct that this scenario could not happen in the real world, but original post was about the limit case, a true space filling curve with no thickness. It doesn't really make physical sense, but it's interesting to think about.
Another curve you would really appreciate is the dragon curve. It is construced by "folding" a line segment (e.g. a thin strip of paper) in two, folding that in two, etcetera, each time dividing the total length in two. After that, you unfold it again, keeping 90 degree angles at the "creases". The result is a space filling curve. Maybe not mathematically, but at least visually.
I am always impressed at how well you make it feel like you are just discovering in the moment the content you're presenting, even though I know you must've spent weeks or months preparing it. It brings me along with you as a viewer, and creates a literally wonderful experience! Really great stuff, as always!
That liquid 3D Hilbertish curve came out really nicely! I think that the labyrinth sequence doesn’t limit to a “curve” (in the sense of a continuous map from the line to a 2d disk). As you go further into the sequence, it swings back and forth around the circle faster and faster. So even if you did get a sensible answer for “where is the limit curve at time t?”, I don’t think the answer would continuously vary as t increases.
Interestingly, there are continuous functions from the curve to [0,1], although mutliple values on the curve will get mapped to the same point on [0,1]. Since it isn't injective, it is jective!
I was so excited when Steve started talking about someone who was good at explaining maths! I was thinking, "we're going to get Matt Parker! It's another two for one!" Then, Steve said, "Matt Parker, do you know anyone good at explaining maths?" I actually laughed out loud!
As someone who recently got into resin printing, I learned that you can remove the cloudiness by brushing fresh resin onto the surface and use a uv light to cure it from there, or just not washing the print and go straight to curing helps too.
In my field of digital cartography, space filling curves are critical for efficiently retrieving two dimensional information like longitude and latitude from a one-dimensional data source (like a hard drive or RAM.) For years we've been interlacing the binary digits of long/lats to create a one-dimensional number. When you plot the original two dimensional points in the order defined by that one-dimensional interlaced value, you get - poof- a space filling curve. The key feature of this technique is that objects close together in their two-dimensional representation tend (with some predictable exceptions) to be close together in their one-dimensional representation! Imagine how important this was for reading data from something like a mechanical disk platter.
Essentially the same technique is sometimes used in distributed particle simulations to map nearby particles to the same compute node. So many applications!
one thing i love using hilbert curves for is visualing IP subnetting as will always be able to be displayed as a square or 1:2 rectangle on a grid vs just a very thin line
This reminds me of certain concepts I tried to learn in maths as a teen. I feel unable to learn it, to process it. It almost feels like the explanation is in a foreign language.
I've recently finished my PhD Thesis and a part of that work was on the Hilbert-curve (and its locality principle). Really nice to see this visualization now
So by making these nice large hydrostatic sculptures, you are actually building up a skill. And we can see you improving in it. It might be worth making a couple videos on how exactly you're creating these mechanism and what have you learned. Makers would appreciate watching it. It's almost like microfluidics where I want to get with my skills eventually...
2:02 I do want to clarify what a fractal: While it is generally thought of as self repeating, that isn't really what they are. They are just all infinitely complex curves. The only reason they are typically though of that way is because self-similar fractals look cooler and are able to be constructed to arbitrary precision without much work or randomness.
One thing which I couldn't quite tell which you were saying but should be cleared up, is that fractals are not necessarily self similar. In fact, it is only a small subset of fractals which are self similar.
I loved making space filling mazes in Microsoft Paintbrush as a kid. My familys 486 computer was slow enough that you could follow the different colors fill up the labyrinths you made. This gives me flashbacks to that :)
I really liked how you cited AlphaPheonix - not just because citations are important but I think you did a phenomenal way of doing it scientifically and concisely; almost like it was in a journal.
glad to see 3d printing talked about in such a nonchalant way. It often feels like its such a niche community for something that should be in the public lexicon the hilbert curve and other curves have a ton of potential as a tool in 3d printing too so this was a great crossover!
This was the first video of yours that i found in my feed. I watched 15 seconds in before subscribing and have been binge watching your content since. Thank you Steve
These are useful for storing 2D or 3D data in computer memory (e.g texture images) Due to the locality, data that is visually close together also appears close in memory improving cache performance.
Your comment caused me to review a few papers. Looks like it's useful for very memory bound GPU tasks (without accounting for encoding time). Row major is still fastest on problems below cache size and on the CPU. Hilbert curves were consistently slower than Morton curves, *the effects were quite dependant on the architecture*. The advantage goes away when interdependence is added, then it spends too much time computing Morton indices instead of crunching the numbers.
I played with the locality aspect in analogue signal data, converting time series to 2D using a "Hilbert transform" then applying a 2D fourier transform, it produced some really weird patterns.
OMG thank you so much for this! I hadn't considered non-fractal space-filling curves, but this brings me a step closer to solving an impossible problem I have with computation space as it relates to fractal complexity. Fundamentally, a fractal-inclusive math must include solutions for non-fractal space-filling curves, otherwise it is incomplete and not truly fractal-inclusive.
But non-fractal space filling curves don't exist though, in fact the inclusiveness goes the other way around. Fractal dimensions can go non-integer and cover go from 1D to 1.5D to 2D and even 3D and everything inbetween, actually most fractals aren't space-filling. But being a fractal is a requirement to be space-filling, no smooth curves can fill up the space, and the Celtic labyrinth doesn't count as it doesn't have a limit that respects the definition of a limit. I recommend you 3Blue1Brown's video about fractal dimensions and self-similar fractals on the matter, it does explain exactly how fractal maths generalise to non fractal shapes and how it affects their dimensions. And also his video about the Hilbert curve that is linked in this video's description as it explains why the Hilbert curve does respect the definition of a limit.
@@jAujAl1 I partially disagree. A logarithmic fractal is a pseudo-fractal, because it's fractal at the central point but not at any other point. when Parker said "your problem is that the 'Mould curve' is not defined by a recursive substitution approach" he's actually mostly right but partially wrong. In actuality, the Mould curve is defined by a recursive substitution approach around the origin, and a simple algorithm everywhere else. It's fundamentally halvable just like a Hilbert Curve is, at the origin, but the results of halving are less than the results of the Hilbert Curve because it's only halving at one point rather than every relevant point. meaning that it has a definable fractal dimension at the origin, and a relative fractal dimension everywhere else. This means that it sits between a pure fractal and a non-fractal. So I would instead say that a pseudo-fractal can be pseud-space-filling. It's not that it can't fill a space, but instead that it doesn't inherently fill a space. It only fills the space it's defined to fill, and it would require extra effort to make it fill a space it's not defined to fill. A true space-filling fractal can fill any space, but I realized from this video that a pseudo-space-filling fractal can only fill the space it's defined as filling. And unfortunately, fractal shapes generalizing to non-fractal shapes doesn't solve my problem. I need true in-betweens. Things that are neither fractal nor non-fractal. Logarithmic spirals and the Mould Curve are appropriate examples. Still, I'll look into that 3Blue1Brown video. I've watched most of his, and I might have missed that one. Thanks for the recommendation.
@@jAujAl1 If you're talking about th-cam.com/video/gB9n2gHsHN4/w-d-xo.html, it doesn't describe a generalization of a fractal to a no-fractal at all. It only describes fractal dimension as a secondary characteristic. It describes a good generalization of fractal dimension. But in only in the same way that a number line describes a good generalization of integers and non-integers. In other words, it's completely useless for what I need. I don't need a simple number line, I need a definitional difference between integers, irrational numbers, and transcendental numbers as it relates to the definition of a fractal. Fundamentally, an Archimedes spiral is space-filling for a circle, but not any other shape. The impossible problem I'm trying to solve is inventing a math that includes Hilbert curves, Archimedes spirals, and squares, as computable shapes. in the same way a number line includes transcendental numbers, irrational numbers, and integers.
@@epigeios Yeah, that's the one I was talking about. Fat chance, it sounded awfully close to my understanding of your problem, and my bad, I remembered it as generalizing the definition for non-fractal, sorry for the mixup. I would argue the "Mould curve" has a pretty imprecise definition of how it is defined at the center, but if you interpret it as an infinitely nested U shape with a sharp turn, it could indeed count as a single-point fractal and be space-filling at the top half of the center point (or around the full center point if you mirror it). I don't think it would count as a non-fractal though, so I think making those properties generalize to actual non-fractals would still be an impossible endeavour, but I hope that won't be a limiting factor for your computational model.
@@jAujAl1 You're right about that, it's not really non-fractal in that way. And the "Mould Curve" isn't exactly what I'm looking for. It's just a step in the right direction. It's in the direction of mixing and matching fractal parts with non-fractal parts to create something different.
Imagine a life sized version of the hilbert curve cube where the diameter of the tunnel is roughly a meter and the distance from the previous turn to the next one is 3 or so meters with ladders on vertical segments. That would be my worst fear being inside that.
I have been so pleased to come to understand so many things I didn’t understand or even know about until I came to watching Steve Mould’s videos. It want the case with this one!
Great video again Steve, you are right about building these rather than modelling, much more pleasing to watch (did anyone else search for all those podcasts? It’s the first time I noticed that POD in buckles image looks like POO, very fitting)
All very interesting Steve. Ive been interested in similar - a heat exchanger. But instead of a single space-filling curve it uses TWO space filling curves that are tightly intermingled yet not touching. Hot fluid goes down one curve (tube) warming the fluid in the other. It would be interesting to know of any fractals for this.
Space-filling curves aren't actually very good for a heat exchanger. It's true that they offer enormous surface area, but they also make it very difficult to push the fluids through, and there will always be spots where different-temperature parts meet (because a space-filling curve can't be injective). Tree structures don't have this problem, but they're not optimal either because you waste half the space in the branches before you actually get to where the heat exchanging happens. The best design for a heat exchanger is a _gyroid_ instead.
I wonder: If you take a material that changes color when current runs through it (So you can see the path the current took), you make a maze out of it and put contacts on each end, would the current solve the maze? Would it take other paths too? For example if the walls of the maze are made of gummy for being non conductive
The excellent spatial locality of the Hilbert curve can be useful in distributed particle simulations. There are too many particles to store on a single machine, so you have to divvy them up across multiple nodes. You want nearby particles to be stored on the same node because it's more expensive to communicate between nodes, and particles need to communicate only with their nearest neighbors. A particle's position along a (3D generalization of the) Hilbert curve can be very quickly computed from its coordinates. Then, you can sort the particles by their Hilbert curve values and assign contiguous chunks to each node. The end result is that the particles on each node tend to be very close to each other.
The "Mould curve" isn't space-filling because it isn't a curve. You've defined a sequence of curves, but their limit doesn't exist. This is also why you need something moderately fancy like the Hilbert curve to fill space. The simplest idea would be a "boustrophedon" -- a curve that just goes back and forth. This doesn't work for the same reason as the labyrinth; you can define such a curve with N back-and-forths, but the limit with infinity back-and-forths doesn't exist.
Interesting! Thank you. Why doesn't the limit exist for the back and forth? I'm guessing it's because of some subtlety about limits that I don't know about.
@@SteveMould It's related to the locality property you mentioned. In order for the limit to be a curve, it needs to be a well defined function from a line segment to the plane, so the mapping of each individual point needs to be well defined, and the function needs to be continuous. You can show that where the points are mapped to jiggles around without ever settling down.
You have to think of it as a parametrised curve - I.e. as if it is filling up with water. For t between 0 and 1, there is a point on each approximating curve for the position at time t. For the Hilbert and Gosper curves, these become better and better approximations to the point on the ‘real’ fractal curve, and that means you define the fractal by saying at time t, the point is the limit of the points on the approximating curves. A pointwise limit of a function, technically. For the non-fractal construction, the points at time t do not converge, so it is not possible to define the limit, so there is no limiting curve to speak of. So the trouble is that for some t, the points at time t don’t settle down as you go through the sequence of curves. They jump around too much.
@@SteveMould I think it's kind of important to also emphasize that this situation is in no way at the edge of what mathematicians can handle it's a very standard type of example for learning about limits of functions
@@SteveMouldmagine you're snaking from curve top to bottom, increasing the number of folds each iteration; the X coordinate of a point at an irrational distance along the line (say 1/π) will constantly be changing as you keep iterating and never settle down and approach a fixed point. It's analogous to an infinite series that oscillates instead of converges. Rational distances will also oscillate and not converhe but will do so in a repeating way, like the point halfway along the line will oscillate between x coordinate of 0->1/2->1-1/2->0->...
I'm not sure if this makes it not space-filling, but one thing I notice about the labyrinth is that it doesn't really use the same approach to filling the two dimensions. It just kind of takes the fact that a line will naturally fill a line and stacks an bunch of those together to fill a plane. Also, and again I don't know if this costs it the space-filling title, but it can't be expanded to a bigger space easily. You can make the Hilbert curve bigger and bigger because you can do the iteration "upwards" (duplicating the whole thing and stitching it together) as well as "downwards" (adding more iterations in the sections you've already drawn to fill space more finely) and you can't really do that with the labyrinth.
"In reality it's impossible to show infinitely many knobbles and you can quote me on that" My good sir are you saying algebraic representations using limit notation doesn't count as "showing"? The mattematicians will heartily disagree
1:20 I can’t believe I was doodling a fractal in my notes at school. I’d start with a big star then on each of the points I’d draw stars on them and I’ll continue the process on each point
For anyone wondering what a practical use for space filling curves would be: Nvidia actually uses (or at least published an article about the application) a type of space filling curve to inversely map the 3D space they are rendering onto the linear memory of their cards. This is done because then points (or in this case objects) which are close to each other in 3D space are also close in memory, which is handy for, e.g, occlusion computations. This is partially achieved via Morton codes which are also a very interesting computer science topic. For anyone who is interested the title of the article is "Thinking Parallel, Part III: Tree Construction on the GPU". Should come up if you search for that.
9:30 it'd be really cool if you filled it up with the liquid with the similar refractive index so that by the time the liquid reaches the middle, some of the outside of the cube looks empty!
@ 6:39 that's a pretty sweet hilbert capacitor design, if one were inclined. I'm curious how E and B fields and waves would interact with it as well if it were made of copper. Bearing in mind it would be two separate pieces of material, possibly joined with a resistive junction or diode. There's something interesting about how it would perform as an EM wave antenna/ detector. Not in the mood to solve Biot Savart or Maxwell for it, but it would be interesting. Also for the 3d mini cube @ 9:29.. I would think using a fluorescing fluid and sealing the caps after filling would make a cool desk toy.. slap some uv leds in the stand and boom..
me at the beginning of the video :"that looks like a fractal structure" I honestly thought I was wrong as I'm kind of obsessed with fractals, and I see them everywhere but no that's indeed somewhat of a fractal !
This guy makes videos so good that i forget what i saw on the thumbnail, then he shows it in the video and its a pleasant surprise!!!! Great video Steve :)
The labyrinth is not a space filling curve, because the finite versions of it don't converge to a curve. In the Hilbert curve, every iteration of the curve restricts a section of the runtime into a smaller and smaller square, where this section also stays in the next iterations. The points in that square are all close to each other and so the limit exists. However, in the labyrinth you have sections which look like many lines stacked on top of each other. In this case, a portion of the runtime can only be restricted to thinner and thinner strips but they always stay the same length, and the same point can end up in completely different positions along the strip even among more and more complicated versions of the curve, which means you're unable to define the limit and so the final space filling curve doesn't exist. You can also think about it this way: In the Hilbert curve you can know for certain where a point like 1/3 goes - it is in the second quarter of the line so it goes in the top left quarter, then it is in the second quarter of that so it goes in the top right quarter of that square and so on and so forth and in the end you can deduce the exact point. On the other hand, if you tried to do the same with the labyrinth you'd find out that every point goes to some line on the plane, and changing the point makes the line sweep through the plane, which is not what is meant by the mathematical concept of a "space filling curve". It is similar to filling a square with a snake that goes right to the end, then turns, then goes left to the end, then turns and so on. In every version of the curve you'll have a finer snake that gets closer to itself and so to every other point in the plane but you won't get a limit that is a space filling curve. Hope I helped!
we use Hilbert curves in game programming, or sometimes Peano curves. Specifically because it has good locality compared to a simple row-by-column 2D array. (realize that all of your computer's memory is viewed as a long 1 dimensional strip, and it is up to the programmer to interpret those simple memory addresses in interesting ways to make more complex structures like a 2D array or 3D matrix). The reason we use Hilbert curves and Peano curves is because it is very easy to calculate any X,Y position from a simple scalar index and back again. #include /* draw a 31x31 ASCII of a Hibert Curve - JDM 2004.06.05 */ char m[31][31];int a[2];void s(unsigned d){m[a[1]][a[0]]='#';a[d&1]+=1-(d&2);} void h(int o,int r,unsigned d){if(o--){d+=r;h(o,-r,d);s(d);s(d);d-=r;h(o,r,d); s(d);s(d);h(o,r,d);d-=r;s(d);s(d);h(o,-r,d);}}int main(){int i;for(i=961;i;)m[ 0][--i]=' ';h(4,1,0);s(0);for(i=0;i
A cool tabletop toy would be the 3D version filled almost completely with colored oil and water, or a single bubble in colored water. So you can play with the different colors inside by turning it or watch the bubble move as you spin it
Really interesting video. In my recent (free) science fiction novel, "Breathe" I had my characters build supercapacitors by using space-filling curves as a boundary between two halves. I illustrated this with a Hilbert curve. As the order of the curve increases, the length of the 2D curve increases. If this is made into a 3D space-filling membrane separating two electrodes, then the surface area increases with the complexity of the curve. Since the storage capacity of a capacitor depends largely upon the surface area of the "plates", this opens the possibility of making supercapacitors with storage capacity far beyond anything we have today. Of course, in my story I didn't go into the details of how that might be made, but interesting, huh? :) I should note that our lungs perform a similar trick, packing a membrane with about the surface area of a tennis court into the small volume of your chest, and still leaving room for your heart, ribs, spine, and aesophagus.
That's a neat idea! Capacitors are generally tubes because a plate-membrane-plate sandwich can be rolled up easily by machines and server a similar, but less visually interesting, space filling role
If anybody wants to see how I use the Hilbert curve as a separator in my novel "Breathe", it is at my website miriam-english dot org where all my stories are free -- genuinely free -- my website has no trackers or scripts, and no advertising.
*literally
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Steve mould, i think I found your long lost twin sibling!! His name is "david castello-lopes", and he is a french journalist/comedian/youtuber. The resemblance is very much striking
for a clear resin 3d print, like optically clear, you can polish it to 800 grit, then use a clear glossy spray paint. It'll make it optically clear.
i've made lenses out of SLA 3d prints
8:54 this surface isn't polished, it's ground very smooth.
@@adri1572 Steve is cuter.
Please make and sell the small 3D Gilbert curve resign cube!!
In engineering there is the concept of the "sacrificial anode" where if a structure will be attacked by a lot of corrosion, a focal point is provided to divert the damage from the main structure. Steve wisely knows any maths videos will attract a lot of pedantic corrections. Hence the use of a "sacrificial mathematician".
These are used on boats if anyone is curious on specific use cases
cool
This proves that maths is applied philosophy
@@mamoopy Wow, I never knew they used sacrificial mathematicians on boats. This explains what happens to most graduates, thank you
Turns out "Matt" was short for "Magnesium galvanic anode" all along 😌
It’s amazing to me that you’ve turned “water runs between two transparent sheets” into a genre
Seeing the “water runs”… plus the marble run earlier… makes me think of a crossover that I didn’t know I wanted… who’s the guy that does the marble races? Jelly’s or something? It’s been a minute…
Just looked it up. Jelle’s! I was close-ish. :)
@@DavidLindes Wintergaten makes those crazy marble machine musical instruments.
asmr community gonna have a new toy to play with
@@DavidLindes🎉🎉🎉🎉🎉🎉🎉🎉
"In reality, it's impossible to show infinitely many knobbles." -Steve Mould
"and you can quote me on that."
- Steve mould
glad this was appreciated
Never went to a nightclub in small town then.
@@gorgenfol You should take this down. It was "you can quote me on that", not "you can quote me on this". But then I'm making the same faux pas.
its nubbles lol
A fractal doesn't need to be self-similar at different scales. That's just how we construct a lot of them. That fractional dimension property you described is the important one, and that's achieved in anything that has infinite amounts of detail as you zoom in.
I never understood fractal dimensions for over 5 years and you just explained it perfectly in less than a minute. Just, wow. You're amazing at explaining complex topics.
If you haven't already seen it, 3Blue1Brown has an excellent explainer on the topic (though it is far longer than a minute).
Dimension is essentially a scaling factor, with higher dimensional things scaling more from the same change in length as lower things (like how doubling the radius of a ball will have more of an impact on its weight than its surface area). Things with fractional dimension are just things that scale at a rate between the rate of the integer dimensions. For example, the area of a Sierpinski triangle scaled up by two does something in between doubling (like it would if it was a line segment) and quadrupling (like it would if it were a full triangle).
That's how it made sense to me when I was explained it, so in case it makes sense to you, here you go. It also I think prepares you very well to see the math behind it, where a formula for scaling in terms of dimension is then solved for dimension and used as a definition extension.
This was a good video but he didn't actually explain fractal dimension, he just gave a couple examples
damn, were you actively researching it for five years or something? lmao
@@ludvig3242 no. I heard about fractal dimensions for the first time 5 years ago, didn't understand it. For the longest time, when I watched science videos and the topic came up I didn't understand it. Only finally got it when I saw Steve Mould explain it 6 months ago in this video, right before I made that comment.
That visualization is pretty awesome! I want to see the same thing at like... room scale... with clear pipes. It would be AWFUL to assemble lol. with the locality and stretchyness of the hilbert curve your prints demonstrated REALLY well, I'd imagine it'd be near impossible to hold rigid too...
Do I detect a nerd sniping? Can't wait for the video!
You could use braces, there are enough parallel parts that can be linked to spo them from wobbling away. It takes a bit away from the concept bit not too much.
Count me in.
@@standupmaths ba dum tss...
would use a very thin exit nozzle, to make sure the water is well pressured, and fills everything as it goes through. 🤔
Worth noting: Fractals don't necessarily need to be self-similar. Also space-filling curves are definitionally fractal because it's Minkowski dimension exceeds its topological dimension. The trouble with the labyrinth is that you would need to show its limiting behavior actually fills space, which is doubtful but who knows?
Thank you, I was actually here to make the same point. The issue is that people's intuitive idea of a fractal (self-similarity) and the definition in terms of the Hausdorff dimension (which in the case of space filling curves should agree with Minkowski dimension) aren't the same.
the issue is that there is no sequence of finer and finer labyrinths that have a limit. the path the next iteration takes is wildly different, in the hilbert curve, the 20th iteration is very similar to the 19th, in that the point that is, say, 1/pi along the curve barely moved (and same for all other numbers 0 to 1)
"definitionally fractal because it's Minkowski dimension exceeds its topological dimension" - what is this supposed to mean to us ordinary plebs who don't talk math?
@@terdragontra8900 Is the limit of the space filling curve {lim n->∞ (x_n(t), y_n(t))} the set of limits of points on the curve, which is literally the whole area? Or is it {(x,y)|∃N∀n>N, (x,y) ∈ {x_n(t), y_n(t)}} (so just the 1/2^n points)? Or is it the closure of the second thing, which would also be the whole area?
@@ArawnOfAnnwn Minkowski dimension is fairly straightforward in idea: en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension
The other is isn't so straightforward but in the case of curves like these think, if you zoomed in close enough what dimension would it look like.
This is not what it is, it's about refinements of covering sets or some nonsense like that. But the shapes in the video are well behaved so you can think of it like that
Here's the idea: en.wikipedia.org/wiki/Lebesgue_covering_dimension
I work in un underground mine where we use something called a "Belt Storage Magazine". Long story short it's a way to store conveyor belt that can be extended or retracted without taking up more space. The path the conveyor belt takes through the magazine is almost identical to the Celtic Labyrinth, although it's shaped differently.
I think this is also similar to the way pythons and boas rest.
Im expecting a shirt saying “ It’s impossible to show infinitely many knobles ! “
I think something that'd be really cool would be to have one of the smaller cubes filled with a colored epoxy. I don't exactly know how well it'd set up, but it might be worth a shot. Then you might even be able to polish it by covering it in a thin layer of epoxy. Nice desk piece.
The celtic labyrinth can be defined by a substituion rule, but you have to stretch things a bit. Take two copies of a celtic labyrinth, stretch one out and open it up into an inverted U, and wrap the other curve with it - leaving a gap up the middle so you can join the two. To put it the other way, for a celtic labyrinth, there's clearly two layers to it (an outer and an inner), and each layer has the labyrinth structure.
If you filled the 3d version with 2+ liquids with different densities (and colours) and closed the loop, could you flip it around and watch the liquids re-arrange themselves? Would be a 10/10 desk toy.
Yes! Please! I got to know!
Green coolant and orange coolant in your radiator.
The business man
should use oil and water so they don't mix.
Yeah I would totally buy that
This got me wondering: Shouldn't you be able to construct the cube's inlet and outlet in such a way that you could print more cubes and connect them into a bigger and bigger cube that still fulfills the property and could pipe water all the way through?
You could probably print them without inlets and outlets, and then drill as necessary to glue together bigger cubes.
Might get time consuming (64 cubies to do it twice, 512 for 3, etc.), and it would possibly stop working after a while.
Potentially... Unlike the sides of a Koch Snowflake, where each iteration is made out of exact copies of the previous generation, the Hilbert curve (and the generalized 3D versions) are copies of the original *plus* extra lines to connect those copies. Those connectors don't always connect at the same angle (sometimes at a right angle on both sides, sometimes only on one), so you'd have to include paths for all possible connectors and then somehow plug the ones you aren't using
It seems like by definition that should work
@@zlacthe problem with drilling and glueing is that you create branches and shortcuts
Oh nice idea!
You did so much work for so many rapid-fire visual "proofs" and I honestly really appreciate the immediacy of it.
0:35 would make a really satisfying loading screen for a game
And if the load takes longer. The loading screen just zooms in the fractal because its infinite
@@jacobschut5857I think it should zoom out
FYI fractals are not necessarily self-similar. Shoreline borders are probably the more well-known example of fractal lines that aren't self-similar. All that matters is that they have infinite detail that never smooths away when you zoom in. The self-referential ones are more famous just because it's the easier way to describe a mathematical object with that property.
Are we going to simply ignore the fact that Steve already said what the word "fractal" means, and it is not what you are saying? If you think your words are more correct, you're going to need a reference.
@@u1zha😂 he’s exactly correct and it would take you a few seconds to learn for yourself.
@@u1zhathe fact that you need someone to reference a fact that disagrees with something you learned from TH-cam is hilarious.
@@u1zhastart with a guy named Mandelbrot and go from there.
@@u1zha The Onion Curve is a good example of a non-fractal space-filling curve. See "Onion Curve: A Space Filling Curve with Near-Optimal Clustering" (2018) by Xu, Nguyen, and Tirthapura.
The reason the Celtic labyrinth doesn't count as a space-filling curve is that you can't define a limit version of it, or in other words, you can't define a sequence of increasingly filling Celtic labyrinths who end up stabilizing into a limit version.
The Hilbert curve is special in that it does stabilize, if you go at the 1/3rd of the point in the curve for instance, each iteration of adding an additional U shape is going to go closer and closer to a limit point, it converges. For each iteration of the Celtic curve, which consists of adding a new line inbetween each existing line, the 1/3rd mark is going to go all over the place. By the mathematical version of a limit, the limit of the Celtic labyrinth curve doesn't exist, it diverges.
Or in other words, it's mostly related to the "good low quality" property of the Hilbert curve, which mean low quality versions of the curve are a good approximation of its highest quality version - the limit curve.
you missed what Matt Parker was saying. While it is true that the approach used to demonstrate fractals as space filling will not work on the Celtic labrynth, there is no way of knowing at this time if it is space filling or if it is not. This is because the absence of a method of proof does not itself constitute a valid disproof.
@@Thesupremeone34you misunderstood the comment. It's not that the curve is not a fractal, which we already knew about, and that's what Matt Parker said, it is in fact that the limit of the curves doesn't exist, so you cannot even ask the question in the first place, there's no infinite labyrinth to check if it fills space in any known or unknown way because the infinite labyrinth is not well defined
@@Thesupremeone34no. what matt said was wrong. the labyrinth curve is an infinite family of curves with increasingly more parallel lines, which does not converge to a well defined curve. every space filling curve needs to be a fractal
Can you prove that the Celtic labyrinth doesn't converge? In my mind I can see it converging so I don't agree with your whole premise.
@@toniokettner4821 what Matt said wasn't strictly wrong, he just didn't stop enough to notice the answer was a definite no because of convergence reasons, so he just erred in the side of caution and said "you cannot prove it with known methods", which is technically true, the best kind of true
The 3d version, like the 2d ones, would have an interesting property in the limit, wouldn't it? The water line through it would have an infinite length, but limited volume. Since you can't make the water go through faster than the speed of light, it would take an infinite amount of time to fill even a tiny cube with a syringe of the same volume, no matter how much pressure you can exert on the plunger.
Very interesting, this is shown in the multiple videos, there is a gradual decrease in pressure from start to finish. If it was infinitely long, there has to be infinite pressure at the start too. So your ending statement of "no matter how much pressure you can exert" is just plainly wrong. the pressure can be represented as a integral of length of the line.
@@acters124 The problem is that no amount of force can accelerate the water beyond the speed of light. The water would have to travel along the path which is infinitely long, so it could never reach the end (or anywhere else in the cube for that matter) in a finite amount of time.
To have an infinite line you would need a tube with no width unfortunately all liquide I know ha e a minimum width for exemple I doubt water could get thinner than an h2o molecule. Therefor none 0 with tube means finite tube means finite time to fill
@@glorrin You are correct that this scenario could not happen in the real world, but original post was about the limit case, a true space filling curve with no thickness. It doesn't really make physical sense, but it's interesting to think about.
Water wouldnt fit on the cavities
Another curve you would really appreciate is the dragon curve. It is construced by "folding" a line segment (e.g. a thin strip of paper) in two, folding that in two, etcetera, each time dividing the total length in two. After that, you unfold it again, keeping 90 degree angles at the "creases". The result is a space filling curve. Maybe not mathematically, but at least visually.
2:24 “in reality it is impossible to show infinitely many knobbles, and you can quote me on that”
-Steve Mould
I am always impressed at how well you make it feel like you are just discovering in the moment the content you're presenting, even though I know you must've spent weeks or months preparing it. It brings me along with you as a viewer, and creates a literally wonderful experience! Really great stuff, as always!
That liquid 3D Hilbertish curve came out really nicely!
I think that the labyrinth sequence doesn’t limit to a “curve” (in the sense of a continuous map from the line to a 2d disk). As you go further into the sequence, it swings back and forth around the circle faster and faster. So even if you did get a sensible answer for “where is the limit curve at time t?”, I don’t think the answer would continuously vary as t increases.
Interestingly, there are continuous functions from the curve to [0,1], although mutliple values on the curve will get mapped to the same point on [0,1].
Since it isn't injective, it is jective!
I was so excited when Steve started talking about someone who was good at explaining maths! I was thinking, "we're going to get Matt Parker! It's another two for one!" Then, Steve said, "Matt Parker, do you know anyone good at explaining maths?" I actually laughed out loud!
Same here! 😊
…I’m at a party rn but watching this instead of talking to people
Based
03:00 never heard more satisfying "yup"
As someone who recently got into resin printing, I learned that you can remove the cloudiness by brushing fresh resin onto the surface and use a uv light to cure it from there, or just not washing the print and go straight to curing helps too.
I work with epoxies.
Just spray it with clear lacquer of pretty much any type.
In my field of digital cartography, space filling curves are critical for efficiently retrieving two dimensional information like longitude and latitude from a one-dimensional data source (like a hard drive or RAM.) For years we've been interlacing the binary digits of long/lats to create a one-dimensional number. When you plot the original two dimensional points in the order defined by that one-dimensional interlaced value, you get - poof- a space filling curve. The key feature of this technique is that objects close together in their two-dimensional representation tend (with some predictable exceptions) to be close together in their one-dimensional representation! Imagine how important this was for reading data from something like a mechanical disk platter.
Essentially the same technique is sometimes used in distributed particle simulations to map nearby particles to the same compute node. So many applications!
one thing i love using hilbert curves for is visualing IP subnetting as will always be able to be displayed as a square or 1:2 rectangle on a grid vs just a very thin line
Karnough maps as well. Traversing a truth table with only a single bit or state changing each turn.
Until, IPV6...
i could barely understand anything in this video but i still found it all really cool, glad i watched all of it
This reminds me of certain concepts I tried to learn in maths as a teen. I feel unable to learn it, to process it. It almost feels like the explanation is in a foreign language.
I've recently finished my PhD Thesis and a part of that work was on the Hilbert-curve (and its locality principle).
Really nice to see this visualization now
I love the dynamic between Steve and Matt. I knew exactly who Steve was calling, and I knew exactly what question he was going to ask.
So by making these nice large hydrostatic sculptures, you are actually building up a skill. And we can see you improving in it. It might be worth making a couple videos on how exactly you're creating these mechanism and what have you learned. Makers would appreciate watching it.
It's almost like microfluidics where I want to get with my skills eventually...
Thank you so much for the simplest explanation of fractals that I'd personally heard so far.
2:02 I do want to clarify what a fractal: While it is generally thought of as self repeating, that isn't really what they are. They are just all infinitely complex curves. The only reason they are typically though of that way is because self-similar fractals look cooler and are able to be constructed to arbitrary precision without much work or randomness.
One thing which I couldn't quite tell which you were saying but should be cleared up, is that fractals are not necessarily self similar. In fact, it is only a small subset of fractals which are self similar.
I loved making space filling mazes in Microsoft Paintbrush as a kid. My familys 486 computer was slow enough that you could follow the different colors fill up the labyrinths you made.
This gives me flashbacks to that :)
Hilbert curve is one of my favorite top fill patterns for 3d printing. Works really well with silky filaments
Fancy seeing you on here as I'm catching up on some YT I've missed due to life! :D
Did anyone else uncontrollably get up and stick a fork in the wall outlet at 2:15?
I really liked how you cited AlphaPheonix - not just because citations are important but I think you did a phenomenal way of doing it scientifically and concisely; almost like it was in a journal.
Steve, you never cease to deliver really solid content with such a simple, approachable way of showcasing your thoughts process. You rock, dude.
glad to see 3d printing talked about in such a nonchalant way. It often feels like its such a niche community for something that should be in the public lexicon
the hilbert curve and other curves have a ton of potential as a tool in 3d printing too so this was a great crossover!
This was the first video of yours that i found in my feed. I watched 15 seconds in before subscribing and have been binge watching your content since. Thank you Steve
I feel like video games with loading screens that have a progress bar could benefit from something like this instead of a straight line.
These are useful for storing 2D or 3D data in computer memory (e.g texture images) Due to the locality, data that is visually close together also appears close in memory improving cache performance.
Your comment caused me to review a few papers. Looks like it's useful for very memory bound GPU tasks (without accounting for encoding time). Row major is still fastest on problems below cache size and on the CPU. Hilbert curves were consistently slower than Morton curves, *the effects were quite dependant on the architecture*.
The advantage goes away when interdependence is added, then it spends too much time computing Morton indices instead of crunching the numbers.
I played with the locality aspect in analogue signal data, converting time series to 2D using a "Hilbert transform" then applying a 2D fourier transform, it produced some really weird patterns.
It always makes me happy to see content creators cooperate to get even better results. 👍
I love it when educational TH-camrs start referencing and replying to each other's videos and a single topic becomes an internet-spanning obsession.
OMG thank you so much for this!
I hadn't considered non-fractal space-filling curves, but this brings me a step closer to solving an impossible problem I have with computation space as it relates to fractal complexity.
Fundamentally, a fractal-inclusive math must include solutions for non-fractal space-filling curves, otherwise it is incomplete and not truly fractal-inclusive.
But non-fractal space filling curves don't exist though, in fact the inclusiveness goes the other way around. Fractal dimensions can go non-integer and cover go from 1D to 1.5D to 2D and even 3D and everything inbetween, actually most fractals aren't space-filling. But being a fractal is a requirement to be space-filling, no smooth curves can fill up the space, and the Celtic labyrinth doesn't count as it doesn't have a limit that respects the definition of a limit.
I recommend you 3Blue1Brown's video about fractal dimensions and self-similar fractals on the matter, it does explain exactly how fractal maths generalise to non fractal shapes and how it affects their dimensions. And also his video about the Hilbert curve that is linked in this video's description as it explains why the Hilbert curve does respect the definition of a limit.
@@jAujAl1 I partially disagree. A logarithmic fractal is a pseudo-fractal, because it's fractal at the central point but not at any other point.
when Parker said "your problem is that the 'Mould curve' is not defined by a recursive substitution approach" he's actually mostly right but partially wrong. In actuality, the Mould curve is defined by a recursive substitution approach around the origin, and a simple algorithm everywhere else. It's fundamentally halvable just like a Hilbert Curve is, at the origin, but the results of halving are less than the results of the Hilbert Curve because it's only halving at one point rather than every relevant point. meaning that it has a definable fractal dimension at the origin, and a relative fractal dimension everywhere else.
This means that it sits between a pure fractal and a non-fractal.
So I would instead say that a pseudo-fractal can be pseud-space-filling. It's not that it can't fill a space, but instead that it doesn't inherently fill a space. It only fills the space it's defined to fill, and it would require extra effort to make it fill a space it's not defined to fill. A true space-filling fractal can fill any space, but I realized from this video that a pseudo-space-filling fractal can only fill the space it's defined as filling.
And unfortunately, fractal shapes generalizing to non-fractal shapes doesn't solve my problem. I need true in-betweens. Things that are neither fractal nor non-fractal. Logarithmic spirals and the Mould Curve are appropriate examples.
Still, I'll look into that 3Blue1Brown video. I've watched most of his, and I might have missed that one. Thanks for the recommendation.
@@jAujAl1 If you're talking about th-cam.com/video/gB9n2gHsHN4/w-d-xo.html, it doesn't describe a generalization of a fractal to a no-fractal at all. It only describes fractal dimension as a secondary characteristic. It describes a good generalization of fractal dimension. But in only in the same way that a number line describes a good generalization of integers and non-integers.
In other words, it's completely useless for what I need. I don't need a simple number line, I need a definitional difference between integers, irrational numbers, and transcendental numbers as it relates to the definition of a fractal.
Fundamentally, an Archimedes spiral is space-filling for a circle, but not any other shape. The impossible problem I'm trying to solve is inventing a math that includes Hilbert curves, Archimedes spirals, and squares, as computable shapes. in the same way a number line includes transcendental numbers, irrational numbers, and integers.
@@epigeios Yeah, that's the one I was talking about. Fat chance, it sounded awfully close to my understanding of your problem, and my bad, I remembered it as generalizing the definition for non-fractal, sorry for the mixup.
I would argue the "Mould curve" has a pretty imprecise definition of how it is defined at the center, but if you interpret it as an infinitely nested U shape with a sharp turn, it could indeed count as a single-point fractal and be space-filling at the top half of the center point (or around the full center point if you mirror it). I don't think it would count as a non-fractal though, so I think making those properties generalize to actual non-fractals would still be an impossible endeavour, but I hope that won't be a limiting factor for your computational model.
@@jAujAl1 You're right about that, it's not really non-fractal in that way. And the "Mould Curve" isn't exactly what I'm looking for. It's just a step in the right direction. It's in the direction of mixing and matching fractal parts with non-fractal parts to create something different.
Space filling curves (mostly Morton) are used extensively in computer graphics to get better cache locality.
How so?
@@JebFromWarmDays Inverse the morton transform from 2D/3D to 1D so spatially close values also tend to be closer in linear address space.
Imagine a life sized version of the hilbert curve cube where the diameter of the tunnel is roughly a meter and the distance from the previous turn to the next one is 3 or so meters with ladders on vertical segments. That would be my worst fear being inside that.
The back rooms: Hilbert edition.
As mazes go, it's a real easy one! 😎
Assuming you are able bodied enough to climb up/down ladders just move forward there are no branches.
I have been so pleased to come to understand so many things I didn’t understand or even know about until I came to watching Steve Mould’s videos.
It want the case with this one!
Great video again Steve, you are right about building these rather than modelling, much more pleasing to watch (did anyone else search for all those podcasts? It’s the first time I noticed that POD in buckles image looks like POO, very fitting)
All very interesting Steve. Ive been interested in similar - a heat exchanger. But instead of a single space-filling curve it uses TWO space filling curves that are tightly intermingled yet not touching. Hot fluid goes down one curve (tube) warming the fluid in the other. It would be interesting to know of any fractals for this.
Branching structures such as tree roots and lungs, both optimized for surface area exchange, also nehprites in the kidneys
Space-filling curves aren't actually very good for a heat exchanger. It's true that they offer enormous surface area, but they also make it very difficult to push the fluids through, and there will always be spots where different-temperature parts meet (because a space-filling curve can't be injective). Tree structures don't have this problem, but they're not optimal either because you waste half the space in the branches before you actually get to where the heat exchanging happens. The best design for a heat exchanger is a _gyroid_ instead.
Note: a fractal isn’t necessarily self-similar, the defining quality is that not matter how far you zoom you still see detailed structures
I wonder: If you take a material that changes color when current runs through it (So you can see the path the current took), you make a maze out of it and put contacts on each end, would the current solve the maze? Would it take other paths too? For example if the walls of the maze are made of gummy for being non conductive
The excellent spatial locality of the Hilbert curve can be useful in distributed particle simulations. There are too many particles to store on a single machine, so you have to divvy them up across multiple nodes. You want nearby particles to be stored on the same node because it's more expensive to communicate between nodes, and particles need to communicate only with their nearest neighbors. A particle's position along a (3D generalization of the) Hilbert curve can be very quickly computed from its coordinates. Then, you can sort the particles by their Hilbert curve values and assign contiguous chunks to each node. The end result is that the particles on each node tend to be very close to each other.
9:06 I finally understand why Latex clothing looks dull and cloudy until polished with accurate science.
This title broke my brain 🧠
skill issue😊
Only reason I'm here. Need resolution
2:02 A certain video from 3Blue1Brown comes to mind named "Fractals are typically not self-similar"
The "Mould curve" isn't space-filling because it isn't a curve. You've defined a sequence of curves, but their limit doesn't exist.
This is also why you need something moderately fancy like the Hilbert curve to fill space. The simplest idea would be a "boustrophedon" -- a curve that just goes back and forth. This doesn't work for the same reason as the labyrinth; you can define such a curve with N back-and-forths, but the limit with infinity back-and-forths doesn't exist.
Interesting! Thank you. Why doesn't the limit exist for the back and forth? I'm guessing it's because of some subtlety about limits that I don't know about.
@@SteveMould It's related to the locality property you mentioned. In order for the limit to be a curve, it needs to be a well defined function from a line segment to the plane, so the mapping of each individual point needs to be well defined, and the function needs to be continuous. You can show that where the points are mapped to jiggles around without ever settling down.
You have to think of it as a parametrised curve - I.e. as if it is filling up with water. For t between 0 and 1, there is a point on each approximating curve for the position at time t. For the Hilbert and Gosper curves, these become better and better approximations to the point on the ‘real’ fractal curve, and that means you define the fractal by saying at time t, the point is the limit of the points on the approximating curves. A pointwise limit of a function, technically. For the non-fractal construction, the points at time t do not converge, so it is not possible to define the limit, so there is no limiting curve to speak of.
So the trouble is that for some t, the points at time t don’t settle down as you go through the sequence of curves. They jump around too much.
@@SteveMould I think it's kind of important to also emphasize that this situation is in no way at the edge of what mathematicians can handle it's a very standard type of example for learning about limits of functions
@@SteveMouldmagine you're snaking from curve top to bottom, increasing the number of folds each iteration; the X coordinate of a point at an irrational distance along the line (say 1/π) will constantly be changing as you keep iterating and never settle down and approach a fixed point. It's analogous to an infinite series that oscillates instead of converges. Rational distances will also oscillate and not converhe but will do so in a repeating way, like the point halfway along the line will oscillate between x coordinate of 0->1/2->1-1/2->0->...
“Mathematics is struggling to keep up with Steve” that’s the start of a chuck Noris style line and best on a shirt.
I'm not sure if this makes it not space-filling, but one thing I notice about the labyrinth is that it doesn't really use the same approach to filling the two dimensions. It just kind of takes the fact that a line will naturally fill a line and stacks an bunch of those together to fill a plane. Also, and again I don't know if this costs it the space-filling title, but it can't be expanded to a bigger space easily. You can make the Hilbert curve bigger and bigger because you can do the iteration "upwards" (duplicating the whole thing and stitching it together) as well as "downwards" (adding more iterations in the sections you've already drawn to fill space more finely) and you can't really do that with the labyrinth.
2:35 i tried to follow the line and i think there is an error in the lower left corner
?????
Imagine if you or a tiny creature were trapped inside and had to escape in time.
The water would carry them to the exit tho.
3:25 "higher order U's"
You missed a wonderful chance to call them "double U's"
Well you are first in steve moulds video and i subbed you!
And you are in _luck_
This video is far beyond my level of understanding but got this in my recommended feed and was entertained by the visuals and soothing voice at least.
I'm using "knobbles" forever now. Thank you Steve! 😊
9:30 damn, it's so satisfying
"In reality it's impossible to show infinitely many knobbles and you can quote me on that"
My good sir are you saying algebraic representations using limit notation doesn't count as "showing"? The mattematicians will heartily disagree
1:46 in and I'm lost
0:01 in and I'm lost
Basically fractals are a fraction of a dimension, so something like 0.5
1:20 I can’t believe I was doodling a fractal in my notes at school. I’d start with a big star then on each of the points I’d draw stars on them and I’ll continue the process on each point
For anyone wondering what a practical use for space filling curves would be:
Nvidia actually uses (or at least published an article about the application) a type of space filling curve to inversely map the 3D space they are rendering onto the linear memory of their cards. This is done because then points (or in this case objects) which are close to each other in 3D space are also close in memory, which is handy for, e.g, occlusion computations. This is partially achieved via Morton codes which are also a very interesting computer science topic.
For anyone who is interested the title of the article is "Thinking Parallel, Part III: Tree Construction on the GPU". Should come up if you search for that.
your videos help bridge the gap between confusion and understanding!
I mean this in the kindest way possible: I love falling asleep to your videos haha, they’re so soothing
This helps me under pulmonary mechanics a little better! Thank you!
9:30 it'd be really cool if you filled it up with the liquid with the similar refractive index so that by the time the liquid reaches the middle, some of the outside of the cube looks empty!
TIL that refractive index thing. amazing stuff!
I love how many things available on the internet that you could make just from a single idea.
I am entirely too high for this, but that won’t stop me today.
Props for shouting out AlphaPhoenix. Genuinely great content and I'm glad to see him growing as a creator so quickly.
@ 6:39 that's a pretty sweet hilbert capacitor design, if one were inclined. I'm curious how E and B fields and waves would interact with it as well if it were made of copper. Bearing in mind it would be two separate pieces of material, possibly joined with a resistive junction or diode. There's something interesting about how it would perform as an EM wave antenna/ detector. Not in the mood to solve Biot Savart or Maxwell for it, but it would be interesting. Also for the 3d mini cube @ 9:29.. I would think using a fluorescing fluid and sealing the caps after filling would make a cool desk toy.. slap some uv leds in the stand and boom..
9:28 the most satisfying piece of footage i've seen in a while
Most educational video on fractals I ever saw. Thank you, the hilbert curve when stretched into 3d reminds me of our brain structure
Very astute observation
@@AndieZ4U2 Thanks
I love fractals and I really like this idea of filling them up with water!
me at the beginning of the video :"that looks like a fractal structure"
I honestly thought I was wrong as I'm kind of obsessed with fractals, and I see them everywhere but no that's indeed somewhat of a fractal !
Duuude this is possibly one of the coolest things I’ve seen
What if you had it set up where at the exit it sucks the air out and the entrance pours in water, would the water immediately go to the end
Love Fractals, love 3d printing, love shapeways, love this video 🙌
Darknet Diaries!!!! I love that podcast and Jack's narration, writing and interviews!!
“YUÛUP, YŪUP, YÚŪUP, YUÚP”
~Steve Mould 2023
Bruh, this video is bright as hell. I need Steve Mould dark mode.
Love the difference between air and liquid displacement through the loop.
You're literally the best that explaining things that aren't physical with physical things, if that makes sense
This guy makes videos so good that i forget what i saw on the thumbnail, then he shows it in the video and its a pleasant surprise!!!! Great video Steve :)
The labyrinth is not a space filling curve, because the finite versions of it don't converge to a curve. In the Hilbert curve, every iteration of the curve restricts a section of the runtime into a smaller and smaller square, where this section also stays in the next iterations. The points in that square are all close to each other and so the limit exists. However, in the labyrinth you have sections which look like many lines stacked on top of each other. In this case, a portion of the runtime can only be restricted to thinner and thinner strips but they always stay the same length, and the same point can end up in completely different positions along the strip even among more and more complicated versions of the curve, which means you're unable to define the limit and so the final space filling curve doesn't exist. You can also think about it this way: In the Hilbert curve you can know for certain where a point like 1/3 goes - it is in the second quarter of the line so it goes in the top left quarter, then it is in the second quarter of that so it goes in the top right quarter of that square and so on and so forth and in the end you can deduce the exact point. On the other hand, if you tried to do the same with the labyrinth you'd find out that every point goes to some line on the plane, and changing the point makes the line sweep through the plane, which is not what is meant by the mathematical concept of a "space filling curve". It is similar to filling a square with a snake that goes right to the end, then turns, then goes left to the end, then turns and so on. In every version of the curve you'll have a finer snake that gets closer to itself and so to every other point in the plane but you won't get a limit that is a space filling curve.
Hope I helped!
Replying to @roeesi-personal:
DARZHNARZHERLARLEFREFLARZSHERBLE?
DARZHNARZHERLARLEFREFLARZSHERBLE?
we use Hilbert curves in game programming, or sometimes Peano curves. Specifically because it has good locality compared to a simple row-by-column 2D array. (realize that all of your computer's memory is viewed as a long 1 dimensional strip, and it is up to the programmer to interpret those simple memory addresses in interesting ways to make more complex structures like a 2D array or 3D matrix).
The reason we use Hilbert curves and Peano curves is because it is very easy to calculate any X,Y position from a simple scalar index and back again.
#include /* draw a 31x31 ASCII of a Hibert Curve - JDM 2004.06.05 */
char m[31][31];int a[2];void s(unsigned d){m[a[1]][a[0]]='#';a[d&1]+=1-(d&2);}
void h(int o,int r,unsigned d){if(o--){d+=r;h(o,-r,d);s(d);s(d);d-=r;h(o,r,d);
s(d);s(d);h(o,r,d);d-=r;s(d);s(d);h(o,-r,d);}}int main(){int i;for(i=961;i;)m[
0][--i]=' ';h(4,1,0);s(0);for(i=0;i
A cool tabletop toy would be the 3D version filled almost completely with colored oil and water, or a single bubble in colored water. So you can play with the different colors inside by turning it or watch the bubble move as you spin it
Really interesting video. In my recent (free) science fiction novel, "Breathe" I had my characters build supercapacitors by using space-filling curves as a boundary between two halves. I illustrated this with a Hilbert curve. As the order of the curve increases, the length of the 2D curve increases. If this is made into a 3D space-filling membrane separating two electrodes, then the surface area increases with the complexity of the curve. Since the storage capacity of a capacitor depends largely upon the surface area of the "plates", this opens the possibility of making supercapacitors with storage capacity far beyond anything we have today. Of course, in my story I didn't go into the details of how that might be made, but interesting, huh? :)
I should note that our lungs perform a similar trick, packing a membrane with about the surface area of a tennis court into the small volume of your chest, and still leaving room for your heart, ribs, spine, and aesophagus.
That's a neat idea! Capacitors are generally tubes because a plate-membrane-plate sandwich can be rolled up easily by machines and server a similar, but less visually interesting, space filling role
If anybody wants to see how I use the Hilbert curve as a separator in my novel "Breathe", it is at my website miriam-english dot org where all my stories are free -- genuinely free -- my website has no trackers or scripts, and no advertising.