Why this pattern shows up everywhere in nature || Voronoi Cell Pattern
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- เผยแพร่เมื่อ 20 พ.ค. 2024
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0:00 Voronoi Patterns in nature
0:53 Crystallization
3:03 Proving Cholera is waterborne
4:10 Greatest Circle Problem
6:21 The Kolmogorov-Avrami model
13:30 Brilliant.org/TreforBazett
Voronoi cell patterns are ubiquitous in nature with many applications in engineering, computer science, and economics.
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This shows up in magnetic systems, too. The grain structure in metal crystals has a Vornoi pattern, and the magnetic domains that form often match the grain structure. It’s kinda cool looking!
Oh fascinating!
Since I have an interest in metalworking, I was immediately thinking about metal grains when you mentioned crystal growth. I'm willing to bet that exponential formula for crystallization must be used all the time in metallurgy - like maybe how long to let a metal cool when casting or annealing, in order to control it's properties.
Cool! Are the vornoi cells composed of groups of atoms with identical magnetic spin direction?
I've seen it once before in a rat, and I see it now in men.
Once one gets a taste for its own kind, it can spread through the pack like a wildfire. Mindlessly chomping and biting at their own hinds. Nothing but the taste of flesh on their minds.
You know the thing about a rat? It's got lifeless eyes. Black eyes like a doll's eyes. Don't seem to be living at all when it come at ya. Till it bites ya. And then the eyes roll over white. You don't hear nothing but the screaming and the hollering...
I wondered if this occurred immediately, when he said “crystallization” I assumed I was right
Geology professor here. Very nice presentation of the math. However, I must note that most real crystals (well essentially all of them) don't grow in a spherical fashion. Some isotropic crystals grow in a geometry that is not to far from a sphere (e.g., garnet is a dodecahedron, pyrite is a cube) but most crystals are anisotropic with two (or sometimes three) different dimensions for their unit cells (and growth along these different axes may not proceed at the same pace). And if we have more than one type of crystal forming at the same time (typical of igneous rocks) there are other complications because they don't all start crystallizing at the same temperature (this can be time in your illustration if we assume a constant rate of cooling). Nevertheless, the texture of many igneous rocks approximates the Voronoi pattern you show.
Oh cool, thanks for sharing! I didn't get into this in the video, but when I was doing some research you can fairly easily adapt the basic idea for other types of growth than the "euclidean" growth I was showing here.
@@DrTrefor Grains boundaries in metallic materials do show some Voronoi-like tesselation tho, at least in several cases
Take the example of the fountain with cholera; whenever your metric chages, your circles will have different shape. The usual Euclidean distance in space (Rn in general) gives spherical balls, but in a city, you use something closer to a taxi-like metric. Squares come from supp, metric etc. same theory, exactly same proofs for any metric, so, interesting they grow acording to different metric conditions.
It's a nice programming project to use this same concept to build an image to stained glass converter.
Load image, generate a list of random pixel coords, and recolor all other pixels based on the closest seed point.
or even, use edge detection (convolution) to find edges, and then for the voronoi points you chose, how well do it’s edges match the edge detection of the input (if white is an edge and black is not, just multiply them), then jitter the points until it reaches some accuracy or number of cycles. could even change the growth speed (which i think is just a distance multiplier) for each point to improve the accuracy.
interesting that a initially quadratic growth (when none of the circles overlap yet) is well approximated by an exponential
For sure, you might not initially expect that at all!
If you can approximate an exponential with a quadratic, why not the other way around? heh
Behavior near 0: for x in the neighborhood of 0: e^x=exp(x) = 1+x+o(x^2) ~= 1+x , so f(t) = 1-exp(-c*t^2) ~= 1-(1+(-c*t^2)) = c*t^2.
Behavior near INF: however, f(t)=1-exp(c*t^2) never quite reaches 1 (limit is 1 at x->+INF), however, the circles _do_ cover the entire area after a finite amount of time.
As a math degree turned graphic designer I friggin love Voronoi textures. Really great explanation!
Incredible, i came across these textures alot while working with graphics, and wondered what they really are about.
Thanks a million times.
Glad you like them!
I used Voronoi segmentation in my master thesis to quantify certain proteins in the plasma membrane!
Oh very cool!
Trefor, what a fantastic video! Your explanations were so well-developed that I found myself anticipating the next steps even before you presented them. The information you conveyed was clear and insightful, and it made following along a true joy. Thank you very much for providing such valuable content!
Glad you enjoyed it!
In regards to the greatest circle problem, it seems to me that (1) the point which is the center of the largest circle that can fit among the other points, and (2) the point that gives the largest Voronoi cell if added to the other points can be different. One can imagine a situation where the point with the largest circle has lots of points along its boundary which shrink its Voronoi cell volume, while there is another smaller circle that doesn't have as much encroaching points and so can spread out more in those open directions. So if you want to open a store, don't just find the biggest circle, find the point which when added will give the biggest Voronoi cell possible.
While I agree that adding more points along the boundary of a greatest circle may well reduce it's area so it is no longer the greatest in the diagram, the greatest will still occur at one of (perhaps a different one now) vertex.
Nice video. I think you could also make a part 2 where you explain the 2 main methods to compute voronoi diagrams: method 1)for each pixel compute all the distances and pick the shortest. and method 2) solve liniar equations for each pair of circles to find where they met. A lot of people dont put much thought into what goes behind a cool animation like this.
Great suggestion!
Would also be very interested in seeing this!
I remember using the Voronoi cells concept in my Master's Thesis to model 5G network stations, users and social attractors (supermarkets, shops, malls, concerts, and so on). Very interesting system modeling capacity by this simple concept.
Can polygonal mud crack patterns be explained this way? Can one give a mathematical proof of what the tesselation pattern will be (e.g., pentagonal?) for an idealized, perfectly uniform layer of mud that begins to dry? I’ve noticed by experimentation that thinner layers of “mud” yield smaller polygons. The “mud” I used was actually the bit of leftover cocoa powder added to my coffee that I let dry at the bottom of the cup after drinking my coffee.
I made a correction: I meant “mud crack” patterns, not “mud brick” patterns. They tend to be pentagonal tessellations. Not representing growth, they might be unrelated to Voronoi cells; but maybe there’s some mathematical equivalence.
Good. Really good. Your enthusiasm and deep knowledge makes it simple. Great!
This is awesome. My (ongoing) PhD deals heavily with Voronoi tesselation
Oh cool! Feel free to share any particularly cool resources here:)
Nothing to say this time, just wanted to leave a comment encouraging you to keep doing what you're doing :)
I appreciate that!
This is an amazing educational video and deserved much love. Who knew that the crystal growth and sewage system share the same line of math?
1:48 well, it maybe correct for approximation -
but at least for real life bubbles i have read in the surface tension chapter in XIIth standard that
the radius at boundary has something to do with differences in internal pressure of the two bubbles -
so, the "common straight line" case holds only when the two bubbles have near equal radius already.
2:16 2:23 2:33 yeah, this kinda sorta addresses the point -
the growth rate here was equal - that's why the curve of contact is straight
Ya that's a good point. To apply to the milk bubble thing you have to have some additional consideration that the added pressure in the bubbles are relatively close.
Sir, is there any concept or method to connect voronoi and fibonacci. I'm not so good at maths. Need to find a connection between those concepts for a design development.
5:21 "this is always the case" no. for simplicity, if there are only 3 spreading points, and they're all very close to one of the corners, then the optimal spot to put a new point is in a place where it will block the other points from spreading to the large empty area of the square as efficiently as possible, so that new point can fill it instead.
This should be kinda obvious, so there is probably a misunderstanding.
The video is awesome!
Can you please share the application you used for the animation!
I really love the smoothness it has!
Really want to test it out, I have a lot of ideas for it!
amazing video as always!
for a flat 2D surface, can we calculate the average number of sides for a voronoi cell? seems like for the examples in the video it mostly goes from 4-6, i don't see a lot with more sides than that, or triangular cells
and are there cases when the cells all have the same number of sides?
one of your best videos yet, very interesting!
Glad you enjoyed!
Thank you for another interesting lecture!
Glad you enjoyed it!
This concept was one of my first programming projects in python!
Just curious... can you go backwards from the cells to the seed points? That is, given the location of the vertices, can you determine with accuracy the location of all the seeds? It seems like you should be able to, but even looking at the diagram of the greatest circle, it's not clear how you would determine the radius. Then there is also the thought that you could bisect the common line between cells and maybe the intersection points would work - it looks like several would intersect at the seed point but not all of them, so I'm not sure what's up with that.
When I did polyhedral crystal simulation for rare earth magnets, I did Voronoi tessellation. Interesting to see this elsewhere.
Nice video,
Where can i find this simulation of the growing circle code?
1:50 many of these diagrams remind me of tissue cell diagrams in Biology.
I've noticed these structures too occasionally during a salt crystallization project
Ok, we studied stuff that goes on expanding from a bunch of points starting all at the same time.
But what if some of these generation processes are delayed?
In this case I suppose that the lines between areas would be pieces of circonferences.
Would it be an interesting case to study?
maybe we can do interesting math by setting the centers of the greatest circles as the new seed nodes for voronoi cells, then get more circles, and so on
Hi Dr. Bazett!
Does the video and audio start to desync around 9:40?
I think it should work if you reload?
Very interesting! A little question, at around 1:10 when you talk about cristallization, what sort of natural cristallization would you associate with Voronoi Patterns ? I mean they are likely to bump into each others when growing in tight packs, but when growing with enough place they normally have fixed planes and angles anyway because of properties of the molecules clumping together, so we could have two phenomena at workm which could be a bit confusing ?
Great video. Would love a follow up video about Delaunay's Triangulation (dual graph of Voronoi's diagram). It also has a lot of neat properties.
Great suggestion!
This seems, surprisingly useful for planning, Im going to make a mental note about this, maybe check some other themes about spatial math. 10:17 ok, assuming that the radius of all the circles is the same.
Can you make a video on the different tech skills(like matlab,mathematica)an aspiring Math student must learn at university?
[I'm done with exams & have taken up "Mathematics & Computing " :) ]
Reminds me of my dissertation into Rayleigh Benard convection cells.
Cool, I only knew voronoi patterns from procedural generation that is meant to look natural (like procedural textures or world maps)
In natural occurrences of this pattern where it's literally formed by circular growth, not all circles start at the same time and not all parts of the field grow at the same rate (locally different temperature or access to water or nutrients etc.); is there a variant of Voronoi patterns that allow for different strengths or speeds of the circles like that?
It's actually more or less identical. A bigger circle and a smaller circle will both be growing, and then when they intersect a straight line forms. The model I derived at the end needs non-trivial modifications to deal with that, but the basic concept is the same.
Learned this stuff in the 1980s, but clicked on the video to finally get how to pronounce it!
Just bought a brilliant year membership through your link ;)
That's awesome, hope you enjoy!
Any repository for your code?
all this time I thought this was just a Blender node thing
At 9:38 some editing is missing from the video. But it didn't distract from the explanation, it remained clear.
ha if you guys could see what the video is like before I edit it it is about 100 bloopers like that:D Thankfully this is the one thing I can actually correct after the video is up, doing that now!
Also reminds me of the street layouts in old European once-walled city centres.
7:15 I think one more condition you are assuming in this model is that all seeds start growing at the same time.
In Vfx we use voronois logic all the time for making different sorts of procedural textures. i was really curious about who the hell is voronoi
I lost you at the P point®️ but i got that it has been a very interesting explanation 🤓
Side question: Is there an easy way to construct a voronoi diagram by hand? The "growing circles" method shown in the animation doesn't seem to translate in any way I can think of.
Not by hand exactly, but yes there a few algorithms to generate the diagram and I might do that in a future video.
one way is to first construct a Delaunay triangulation, but then the question is how do you contrust Delaunay triangulation. There is O(n log n) algorithm for Delaunay triangulation that uses divide and conquer, and there is also O(n log n) algorithm for constructing Voronoi diagram directly (but it's really hard, and i mean really). If you don't have that much points there are much simpler O(n^2) and O(n^3) algorithms.
and I wouldn't even bother with more than 2 dimensions, if you value your sanity
The lines are equal distence and at right angles from pairs of points
So trace these lines in pencil, then for each point marker in the boundary while erasing lines that aint generated by itself
@@rwarazor You can generate 3d Voronoi models using a tool called Neper. I used that for a research internship.
I was wondering where that setting came from in after effects
also keep in mind that any growth in voronoy formation is a result of predictability in nature - also know as a blueprint that is inside each organic matter's DNA - which leads to the following statement - when a stone breaks into pieces, it also breaks in voronoy formation - which one can assume now that stones were used to be an organic matter that turned into a silicate matter under electromagnetic conditions - basically an organic matter had turned into a rock instead of ashes -
I would definitely like to see a video on more voronoi cell applications. I once saw this concept on the CBS show Numbers and always wanted to learn more about them.
Tonnes more applications, might do more:)
For the great circle problem, Could the summed area of the polygons from the voronoi pattern, of which the circle passes through the seed point of those polygons, also indicate the largest circle possible?
For example, the total area of the polygons, of which the first circle touched their seed points, is less than the total area of the polygons of which the greatest circle touched their seed points.
I’ve realized this is wrong, however, the largest circle on a finite plane can often be a circle whose edge does not pass through a vertex, such as in the case of where all the points are clustered in a corner. Does the criteria of the greatest circle problem require proximity to the other points ??
I believe the vertex theory is too approximated to be used concretely as their is rarely any vertex overlap, as opposed to voronoi polygon area sums, which give a definite answer.
ofc the fucking exponential shows up. Every time.
Voronoi diagram is also closely related to Delaunay triangulation!
But how do you calculate the boundaries efficiently. What if the growth rate is different from cell to cell. Or what if the cell growth depends on the cell's free boundary?
At least in the derivation I did at the end of the video on modelling crystal growth, the model is only as good as it's assumptions. One of those assumptions is that the area is large, which means that effects along the boundary are going to be pretty negligible.
@@DrTrefor True. But this is the classic problem, when you learn something new, you just want to learn even more.🙂 On the other hand I think the video length was perfect. I love numerical simulations, therefor the questions.
So much better when put on 2x speed 👌
Great video, but I'm confused about your explanation for equidistribution - surely thats the same as independance?
They are similar but nonetheless distinct. Equidistribution says that the amount of points in any region is proportional to it's size. So if you look at a patch with 10% of the area, it will have 10% of the points. That isn't guaranteed by independence.
@@DrTrefor
Thanks!
This brings up an obvious follow up though - doesn't equidistribution prevent independance?
In a truly independant variable, there should be some chance (very small admitedly) that all the seed points end up in the same region of the area (quarter for example). Equidistribution does not allow this.
Edit:
Okay, I think I may have got it; each "seed" is independant(they dont affect each other) but the placement of seeds on the plane is equidistributed.
Is this close at least?
@@neshirst-ashuach1881 Your edit remains incomplete because you have yet to clarify what "equidistributed" means in your own words. Here is an example to show that they are distinct concepts: Let X ~ N(0,1), a standard gaussian. Set Y = -X. Then both X and Y are identical in distribution (gaussian), but they are not independent. (That's why people ask for "IID" random variables.)
Wow thanks for this video, doc! For whatever its worth; I bring news from the crusty underbelly that a lot of closed-eye-visuals are Voronoi fields. You lost me in the last fifth of that math with the lambda business I guess I got some studying to do. I better not that's like reading the last chapter of the book first. I'm working through calc right now statistics can go DIE ..t while on a lovely cruise in the Caribbean after a few too many good nights in a row.
nice cup topology shirt
the "where to put a store" is a bad example to use for illustrating the greatest circle problem, because that's not where you want to ideally put your shop in order to attract more people than your competition. Have you ever noticed in real life that whenever there is a type of store somewhere, there is usually a competing store of the same type really close by? maybe even just across the street? Because it turns out that is the best strategy (it's not the ideal strategy; if you both could just agree to spread out and stick to that agreement then that would be better (until a third competitor comes along anyway), not the least for the consumers, but it is the best strategy for actually competing). The usual example used to illustrate why this is, is a stretch of beach with two ice cream vendors (Im sure there are plenty of videos on it). Point is, you dont want to put your store as far away from any competition that you can, in fact quite the opposite.
Do stars form these patterns? D galaxies have these patterns? Using invisible forces I mean. How about living cells? I have not seen this in human cells, and there seems to be a scale involved. Dragon fly wings have small scale no such pattern, but larger scale they do. Are voronoi area surfaces found in electromagnetic effects!
wow, i managed to predict that there will be e to the power of pi*(something with t) in formula
Everyone wants to be convex nobody wants to be concave, in the end we all straight
Is this why Voronoi Textures could texture anything in the digital world??
Oh its just growing circles. That makes it a bit easier to implement.
This pattern is in the Mars Victoria crater.
1:10 - You can't trick me, I saw that you moved the points around.
Nobody missed that
Hydrology has entered the chat.
It’s just an approximation to lessen the computational load on the simulation.
I think it’s even in universal boundaries in a multiverse
the trees look cursed
its because polygon is the bestagon
Why would you not just say 'equidistant' instead of equal distance. Such a great word
Hehe. But theory of games told (and you can see it in real life) what super-market will not be spreaded by Voronoy, but will be stay side by side. )))
For example if you see McDonalds - look around to find Burger King )))
I winder how it would look like if if the speed of growth in different directions would be uneven
Immersion Exposure Therapy for Trypophobia
There's a giant hexagon on Saturn. Is that another example, or is it fundamentally different
I had to look up how to pronounce Voronoi after hearing how confidently you pronounced it wrong
Also, you don't actually demonstrate that the great circle problem is the same as the closest supermarket idea you introduced it with. That circle certainly isn't the region where people will go to yours instead of your competitors
Equidistributed, as you describe it, contradicts independence. Just say uniformly distributed. Every point is equally likely to be the seed.
John Snow knew nothing. ;)
At 9.15: I hear exp-music sounding from a distance. Let’s see..
At 11:50 - there it is! :)
they probably grew those polygonal structures.
Just like a lower frequency that's all it's coming to
the way he pronounces Voronoi
voronoi mispronounciation counter: 13
🍿
You assumed that each crystal seed is EQUALLY strong. What if some seeds are STRONGER than others? Then it would not be equal distance.
This is not a brag bc I don't think it's something to brag about? Just funny
Anyways I think I came up with Voronoi cells on my own when I was like 8 just trying to decide the borders between countries I had drawn :)
Because hexagon is the bestagon
I thought this was normal and intuitive
Cell lab:
It's because hexagons are the bestagons.
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Just as a side note: his surname was actually Voronyi, since he was born in Ukraine and he and all his family were pretty much Ukrainians (one of his sons actually fought against r*ssian invaders in 1918 and was later a famous surgeon and one of his daughter was a teacher of Ukrainian language). Also he did not even work in r*ssia, most of his scientific work was conducted in Warsaw. So basically one more famous person r*ssians stole from Ukraine (that's not even talking about those who had to relocate to USA or Canada or Europe due to r*ssians constantly doing their best to make Ukraine a hellhole to live in).
why did you censor russian
@@asherasher9249 same thought lol
I'm sorry, am I the only one who thinks you provided literally zero support for the idea that the cholera epidemic mapping has anything to do with Voronoi diagrams? Is this assertion just because the boundary of the region contains nothing but line segments?
Not for nothing, but a Voronoi diagram is always composed of convex polygons, and that pink thing aint.
They took other water holes and streets, then made a map of which road is closer to which water hole. Give it a werid shape because it's on the bounds of the road.
@@CheeseLordAlmightytheOneGod which has nothing to do with voronoi diagrams.
@@htomerif it fills up the closet point to a site of nuclration aka the water well
@@CheeseLordAlmightytheOneGod Its depressing how even though this video was flawed it provided pretty good information but people like watched it and understood literally none of it.
You could have watched 20 minutes of a goose honking and come out the other end of it knowing exactly the same as you did with this video.
GUOL.
Cya.
@htomerif your haven't every been invited to a party ever, now have you‽
Stop spreading misinformation, please research your videos more carefully
en.wikipedia.org/wiki/Georgy_Voronoy
thanks