Why this pattern shows up everywhere in nature || Voronoi Cell Pattern

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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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3 июл 2024

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Комментарии : 155

@lanog40 Год назад

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!

@DrTrefor Год назад

Oh fascinating!

@jordan4192 Год назад

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.

@Piocoto123 Год назад

Cool! Are the vornoi cells composed of groups of atoms with identical magnetic spin direction?

@Cappurniggas Год назад

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...

@graemecook8131 11 месяцев назад

I wondered if this occurred immediately, when he said “crystallization” I assumed I was right

@petecopeland9906 Год назад

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.

@DrTrefor Год назад

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.

@pseudolullus Год назад

@@DrTrefor Grains boundaries in metallic materials do show some Voronoi-like tesselation tho, at least in several cases

@ModusTollendoTollens Год назад

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.

@Kyoz Год назад

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.

@morgan0 Год назад

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.

@kevalan1042 Год назад

interesting that a initially quadratic growth (when none of the circles overlap yet) is well approximated by an exponential

@DrTrefor Год назад

For sure, you might not initially expect that at all!

@MushookieMan Год назад

If you can approximate an exponential with a quadratic, why not the other way around? heh

@nonamehere9658 Год назад

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.

@egycg3569 Год назад

Incredible, i came across these textures alot while working with graphics, and wondered what they really are about. Thanks a million times.

@DrTrefor Год назад

Glad you like them!

@YassFuentes 11 месяцев назад

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!

@DrTrefor 11 месяцев назад

Glad you enjoyed it!

@lolmomz Год назад

I used Voronoi segmentation in my master thesis to quantify certain proteins in the plasma membrane!

@DrTrefor Год назад

Oh very cool!

@leafy_5 7 месяцев назад

As a math degree turned graphic designer I friggin love Voronoi textures. Really great explanation!

@OchiiDinUmbraa Год назад

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.

@DrTrefor Год назад

Great suggestion!

@williammanley5859 Год назад

Would also be very interested in seeing this!

@christopherlocke Год назад

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.

@DrTrefor Год назад

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.

@rafaelalmada723 Год назад

This is awesome. My (ongoing) PhD deals heavily with Voronoi tesselation

@DrTrefor Год назад

Oh cool! Feel free to share any particularly cool resources here:)

@perappelgren948 Год назад

Good. Really good. Your enthusiasm and deep knowledge makes it simple. Great!

@boscoyu_sci Год назад

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?

@cs127 Год назад

amazing video as always!

@alexandrubobaru 11 месяцев назад

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.

@ATOM-vv3xu Год назад

one of your best videos yet, very interesting!

@DrTrefor Год назад

Glad you enjoyed!

@svetlanapodkolzina1081 Год назад

Thank you for another interesting lecture!

@DrTrefor Год назад

Glad you enjoyed it!

@Illogical. Год назад

This concept was one of my first programming projects in python!

@TheGuruNetOn Год назад

1:50 many of these diagrams remind me of tissue cell diagrams in Biology.

@tantzer6113 Год назад

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.

@tantzer6113 Год назад

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.

@airsquid8532 Год назад

Nothing to say this time, just wanted to leave a comment encouraging you to keep doing what you're doing :)

@DrTrefor Год назад

I appreciate that!

@yash1152 Год назад

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

@DrTrefor Год назад

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.

@romanemul1 Год назад

Just bought a brilliant year membership through your link ;)

@DrTrefor Год назад

That's awesome, hope you enjoy!

@Gandhi_Physique Год назад

When I did polyhedral crystal simulation for rare earth magnets, I did Voronoi tessellation. Interesting to see this elsewhere.

@Unidentifying Год назад

I've noticed these structures too occasionally during a salt crystallization project

@drabart6121 Год назад

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.

@DrTrefor Год назад

Great suggestion!

@GeoffryGifari Год назад

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

@djwilliams8 10 месяцев назад

Reminds me of my dissertation into Rayleigh Benard convection cells.

@Illogical. Год назад

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.

@geckoo9190 Год назад

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.

@andrewsemenenko8826 Год назад

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!

@flamencoprof Год назад

Also reminds me of the street layouts in old European once-walled city centres.

11 месяцев назад

7:15 I think one more condition you are assuming in this model is that all seeds start growing at the same time.

@jsalsman Год назад

Learned this stuff in the 1980s, but clicked on the video to finally get how to pronounce it!

@erikasolnc Год назад

all this time I thought this was just a Blender node thing

@kappascopezz5122 Год назад

Cool, I only knew voronoi patterns from procedural generation that is meant to look natural (like procedural textures or world maps)

@techni6018 4 месяца назад

Nice video, Where can i find this simulation of the growing circle code?

@3snoW_ Год назад

At 9:38 some editing is missing from the video. But it didn't distract from the explanation, it remained clear.

@DrTrefor Год назад

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!

@laurapavone3513 Год назад

I lost you at the P point®️ but i got that it has been a very interesting explanation 🤓

@GeoffryGifari Год назад

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?

@geraldsnodd Год назад

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 " :) ]

@flmbray Год назад

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.

@General12th Год назад

Hi Dr. Bazett! Does the video and audio start to desync around 9:40?

@DrTrefor Год назад

I think it should work if you reload?

@yannickpullens112 Год назад

So much better when put on 2x speed 👌

@user-ds1zm3ty6b 10 месяцев назад

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.

@zilog1 Год назад

I was wondering where that setting came from in after effects

@coolParadigmes 5 месяцев назад

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 ?

@SteinGauslaaStrindhaug Год назад

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?

@DrTrefor Год назад

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.

@3moirai Год назад

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.

@DrTrefor Год назад

Tonnes more applications, might do more:)

@choppergamer Год назад

nice cup topology shirt

@wjrasmussen666 Год назад

Any repository for your code?

@blim8777 11 месяцев назад

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?

@rwarazor Год назад

Voronoi diagram is also closely related to Delaunay triangulation!

@NashBrooklyn Год назад

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 -

@user-sj1gj8wv9y 11 месяцев назад

wow, i managed to predict that there will be e to the power of pi*(something with t) in formula

@astk5214 Год назад

Everyone wants to be convex nobody wants to be concave, in the end we all straight

@kruksog Год назад

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.

@DrTrefor Год назад

Not by hand exactly, but yes there a few algorithms to generate the diagram and I might do that in a future video.

@rwarazor Год назад

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.

@rwarazor Год назад

and I wouldn't even bother with more than 2 dimensions, if you value your sanity

@monkyyy0 Год назад

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

@Gandhi_Physique Год назад

@@rwarazor You can generate 3d Voronoi models using a tool called Neper. I used that for a research internship.

@phobosmoon4643 Год назад

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.

@Petch85 Год назад

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?

@DrTrefor Год назад

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.

@Petch85 Год назад

@@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.

@Pasakoye Год назад

Oh its just growing circles. That makes it a bit easier to implement.

@RealCraft_MC Год назад

Is this why Voronoi Textures could texture anything in the digital world??

@ramelo07 10 месяцев назад

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

@cantthinkofnameyeah7249 Год назад

This pattern is in the Mars Victoria crater.

@tallskinnygeek Год назад

1:10 - You can't trick me, I saw that you moved the points around.

@LineOfThy Год назад

Nobody missed that

@neshirst-ashuach1881 Год назад

Great video, but I'm confused about your explanation for equidistribution - surely thats the same as independance?

@DrTrefor Год назад

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.

@neshirst-ashuach1881 Год назад

@@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?

@strikeemblem2886 Год назад

@@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.)

@josephbilling3886 Год назад

ofc the fucking exponential shows up. Every time.

@celtc7875 11 месяцев назад

I think it’s even in universal boundaries in a multiverse

@brandongammon6978 Год назад

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.

@brandongammon6978 Год назад

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 ??

@brandongammon6978 Год назад

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.

@onecalledchuck1664 Год назад

It’s just an approximation to lessen the computational load on the simulation.

@GroovingPict 11 месяцев назад

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.

@-homerow- Год назад

Immersion Exposure Therapy for Trypophobia

@realdragon Год назад

I winder how it would look like if if the speed of growth in different directions would be uneven

@lorenbooker9486 25 дней назад

Is this the way cosmologists modeled the early universe and ran into issues associated with the reionization epoch?

@potato_power9829 Год назад

its because polygon is the bestagon

@OhMyGodMuffins Год назад

Hydrology has entered the chat.

@tamlynburleigh9267 Год назад

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!

@jamessconiers1968 Год назад

they probably grew those polygonal structures.

@claymitchell752 2 месяца назад

Why would you not just say 'equidistant' instead of equal distance. Such a great word

@RM-yw6xe Год назад

John Snow knew nothing. ;)

@Shiva4D Год назад

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 )))

@trucid2 5 месяцев назад

There's a giant hexagon on Saturn. Is that another example, or is it fundamentally different

@scottwasik79 11 месяцев назад

Just like a lower frequency that's all it's coming to

@FranciscoCastelluccio 11 месяцев назад

Because hexagon is the bestagon

@OneWhoWalksAlone Год назад

🍿

@FoxDog1080 Год назад

I thought this was normal and intuitive

@floppy8568 Год назад

the way he pronounces Voronoi

@floppy8568 Год назад

voronoi mispronounciation counter: 13

@maxp3141 10 месяцев назад

At 9.15: I hear exp-music sounding from a distance. Let’s see..

@maxp3141 10 месяцев назад

At 11:50 - there it is! :)

@seanewing204 Год назад

It's because hexagons are the bestagons.

@StevenSiew2 Год назад

You assumed that each crystal seed is EQUALLY strong. What if some seeds are STRONGER than others? Then it would not be equal distance.

@specygamer30 11 месяцев назад

the trees look cursed

@aurabozzi228 Год назад

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 :)

@addymant Год назад

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.

@partofyoutube1297 10 месяцев назад

Cell lab:

@trentHV 11 месяцев назад

ܗܣ

@htomerif 11 месяцев назад

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.

@CheeseLordAlmightytheOneGod 11 месяцев назад

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.

@htomerif 11 месяцев назад

@@CheeseLordAlmightytheOneGod which has nothing to do with voronoi diagrams.

@CheeseLordAlmightytheOneGod 11 месяцев назад

@@htomerif it fills up the closet point to a site of nuclration aka the water well

@htomerif 11 месяцев назад

@@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.

@CheeseLordAlmightytheOneGod 11 месяцев назад

@htomerif your haven't every been invited to a party ever, now have you‽

@romanscerbak5167 Год назад

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).