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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Пікірлер: 155

@lanog40 Жыл бұрын

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 Жыл бұрын

Oh fascinating!

@jordan4192 Жыл бұрын

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 Жыл бұрын

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

@Cappurniggas Жыл бұрын

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 ай бұрын

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

@petecopeland9906 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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

@ModusTollendoTollens Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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

@DrTrefor Жыл бұрын

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

@MushookieMan Жыл бұрын

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

@nonamehere9658 Жыл бұрын

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.

@leafy_5 8 ай бұрын

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

@egycg3569 Жыл бұрын

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

@DrTrefor Жыл бұрын

Glad you like them!

@lolmomz Жыл бұрын

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

@DrTrefor Жыл бұрын

Oh very cool!

@OchiiDinUmbraa Жыл бұрын

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 Жыл бұрын

Great suggestion!

@williammanley5859 Жыл бұрын

Would also be very interested in seeing this!

@christopherlocke Жыл бұрын

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 Жыл бұрын

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.

@YassFuentes 11 ай бұрын

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 ай бұрын

Glad you enjoyed it!

@rafaelalmada723 Жыл бұрын

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

@DrTrefor Жыл бұрын

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

@boscoyu_sci Жыл бұрын

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?

@perappelgren948 Жыл бұрын

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

@alexandrubobaru 11 ай бұрын

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.

@airsquid8532 Жыл бұрын

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

@DrTrefor Жыл бұрын

I appreciate that!

@tantzer6113 Жыл бұрын

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 Жыл бұрын

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.

@TheGuruNetOn Жыл бұрын

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

@Illogical. Жыл бұрын

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

@ATOM-vv3xu Жыл бұрын

one of your best videos yet, very interesting!

@DrTrefor Жыл бұрын

Glad you enjoyed!

@yash1152 Жыл бұрын

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 Жыл бұрын

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.

@cs127 Жыл бұрын

amazing video as always!

@Gandhi_Physique Жыл бұрын

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

@svetlanapodkolzina1081 Жыл бұрын

Thank you for another interesting lecture!

@DrTrefor Жыл бұрын

Glad you enjoyed it!

@GeoffryGifari Жыл бұрын

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

@andrewsemenenko8826 Жыл бұрын

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!

@drabart6121 Жыл бұрын

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 Жыл бұрын

Great suggestion!

@Unidentifying Жыл бұрын

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

@geckoo9190 Жыл бұрын

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.

@GeoffryGifari Жыл бұрын

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?

@flmbray Жыл бұрын

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.

@romanemul1 Жыл бұрын

Just bought a brilliant year membership through your link ;)

@DrTrefor Жыл бұрын

That's awesome, hope you enjoy!

@Illogical. Жыл бұрын

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.

@erikasolnc Жыл бұрын

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

@techni6018 4 ай бұрын

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

@flamencoprof Жыл бұрын

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

11 ай бұрын

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

@djwilliams8 10 ай бұрын

Reminds me of my dissertation into Rayleigh Benard convection cells.

@astk5214 Жыл бұрын

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

@General12th Жыл бұрын

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

@DrTrefor Жыл бұрын

I think it should work if you reload?

@coolParadigmes 5 ай бұрын

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 ?

@jsalsman Жыл бұрын

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

@user-ds1zm3ty6b 10 ай бұрын

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.

@kappascopezz5122 Жыл бұрын

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

@blim8777 11 ай бұрын

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?

@laurapavone3513 Жыл бұрын

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

@3moirai Жыл бұрын

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 Жыл бұрын

Tonnes more applications, might do more:)

@3snoW_ Жыл бұрын

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

@DrTrefor Жыл бұрын

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!

@geraldsnodd Жыл бұрын

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 " :) ]

@SteinGauslaaStrindhaug Жыл бұрын

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 Жыл бұрын

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.

@wjrasmussen666 Жыл бұрын

Any repository for your code?

@rwarazor Жыл бұрын

Voronoi diagram is also closely related to Delaunay triangulation!

@zilog1 Жыл бұрын

I was wondering where that setting came from in after effects

@kruksog Жыл бұрын

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 Жыл бұрын

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

@rwarazor Жыл бұрын

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 Жыл бұрын

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

@monkyyy0 Жыл бұрын

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 Жыл бұрын

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

@NashBrooklyn Жыл бұрын

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 ай бұрын

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

@choppergamer Жыл бұрын

nice cup topology shirt

@RealCraft_MC Жыл бұрын

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

@phobosmoon4643 Жыл бұрын

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.

@ramelo07 10 ай бұрын

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

@Petch85 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

@@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.

@tallskinnygeek Жыл бұрын

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

@LineOfThy Жыл бұрын

Nobody missed that

@yannickpullens112 Жыл бұрын

So much better when put on 2x speed 👌

@brandongammon6978 Жыл бұрын

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 Жыл бұрын

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 Жыл бұрын

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.

@-homerow- Жыл бұрын

Immersion Exposure Therapy for Trypophobia

@neshirst-ashuach1881 Жыл бұрын

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

@DrTrefor Жыл бұрын

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 Жыл бұрын

@@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 Жыл бұрын

@@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.)

@celtc7875 11 ай бұрын

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

@GroovingPict 11 ай бұрын

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.

@Pasakoye Жыл бұрын

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

@cantthinkofnameyeah7249 Жыл бұрын

This pattern is in the Mars Victoria crater.

@josephbilling3886 Жыл бұрын

ofc the fucking exponential shows up. Every time.

@onecalledchuck1664 Жыл бұрын

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

@realdragon Жыл бұрын

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

@lorenbooker9486 26 күн бұрын

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

@Shiva4D Жыл бұрын

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 )))

@OhMyGodMuffins Жыл бұрын

Hydrology has entered the chat.

@tamlynburleigh9267 Жыл бұрын

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!

@trucid2 5 ай бұрын

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

@jamessconiers1968 Жыл бұрын

they probably grew those polygonal structures.

@claymitchell752 2 ай бұрын

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

@RM-yw6xe Жыл бұрын

John Snow knew nothing. ;)

@potato_power9829 Жыл бұрын

its because polygon is the bestagon

@StevenSiew2 Жыл бұрын

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

@scottwasik79 Жыл бұрын

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

@maxp3141 10 ай бұрын

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

@maxp3141 10 ай бұрын

At 11:50 - there it is! :)

@OneWhoWalksAlone Жыл бұрын

🍿

@floppy8568 Жыл бұрын

the way he pronounces Voronoi

@floppy8568 Жыл бұрын

voronoi mispronounciation counter: 13

@aurabozzi228 Жыл бұрын

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 :)

@specygamer30 11 ай бұрын

the trees look cursed

@addymant Жыл бұрын

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.

@FoxDog1080 Жыл бұрын

I thought this was normal and intuitive

@seanewing204 Жыл бұрын

It's because hexagons are the bestagons.

@FranciscoCastelluccio 11 ай бұрын

Because hexagon is the bestagon

@partofyoutube1297 10 ай бұрын

Cell lab:

@trentHV 11 ай бұрын

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@romanscerbak5167 Жыл бұрын

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).

@asherasher9249 Жыл бұрын

why did you censor russian

@pom791 Жыл бұрын

@@asherasher9249 same thought lol

@htomerif 11 ай бұрын

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 ай бұрын

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 ай бұрын

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

@CheeseLordAlmightytheOneGod 11 ай бұрын

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

@htomerif 11 ай бұрын

@@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 ай бұрын

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