Space filling curves filling with water
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Published 2023-06-30
*literally
Space filling curves are fractals that are one dimensional but they fill 2 dimensional (or 3dimesional space). And you know I can't resist making 2D transparent versions of things.
My water solving mazes video: • Space filling curves filling with water
3Blue1Brown's video about the Hilbert curve: • Hilbert's Curve: Is infinite math use...
AlphaPhoenix's video about water and electricity solving mazes: • How does electricity find the "Path o...
Henry Segerman talking about Hilbert curves: • Hilbert Curve
Henry Segerman talking about Hilbert curves on Numberphile: • Space-Filling Curves - Numberphile
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My video about why white things are white: • Why white things are white
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All Comments (21)
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"In reality, it's impossible to show infinitely many knobbles." -Steve Mould
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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".
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Im expecting a shirt saying “ It’s impossible to show infinitely many knobles ! “
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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.
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It’s amazing to me that you’ve turned “water runs between two transparent sheets” into a genre
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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...
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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.
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You did so much work for so many rapid-fire visual "proofs" and I honestly really appreciate the immediacy of it.
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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?
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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.
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2:24 “in reality it is impossible to show infinitely many knobbles, and you can quote me on that” -Steve Mould
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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.
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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.
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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.
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03:00 never heard more satisfying "yup"
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…I’m at a party rn but watching this instead of talking to people
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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?
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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.
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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.
