Just for the record, a physicist would just call this monte carlo sampling rather than wavefunction collapse. We use similar algorithms all the time for solving lattice models in statistical mechanics for example (see the Ising and six vertex models for examples). There are ways to make this kind of process much faster if you have some kind of scaling behaviour as well. That class of algorithms is referred to as renormalization group methods by physicists, where you start with lower resolution sampling and then gradually increase the resolution. An example of that is that if you want to generate terrain with 100 different height levels for example and heights of adjacent tiles would be within 1 height level of each other, then you can simulate 10x10 blocks first and sample them randomly with the constraint that their average height has to be within 10 height levels of each other. This is a hundred times cheaper to do. Once that is done you fill in the inner squares using that information, but since the average heights per block are already constrained, in each block each square has only 10 possible values to pick from instead of 100, and long range constraints have already been taken care of by the earlier lower res pass. Of course, not every system can be handled in that way, but many can. And you can also decide to have different rules at each resolution scale if you want to model richer behaviour. For example, for procedural terrain generation you can use the lower res for picking biomes, and then do increasing resolution to get transitions between biomes, and then increase resolution to do the actual terrain.

I can really see major utility in combining this with my existing procedurally generated island system to help me navigate designing and placing village prefabs in a natural way. Thanks a lot for sharing

Immediately I notice that the algorithm can be applied to a partially filled grid, which makes me think that one really cool application of this would be to combine it with manual placement. The idea is that a level designer could place tiles of interest to shape the "general feel" of an area, and then run the algorithm on that initial state to fill in the rest. This could also be done in chunks - suppose the designer is able to select a region of cells in the grid to apply WFC to, some of which can be partially filled to start with. This feels promising enough that I might try to make a tool for it in my engine of choice sometime in the next month or so. I could see this being SO useful for speeding up the creation of areas which feel at the same time both hand-crafted and "naturally generated".

4:40 thats actually a part of procedural generation that I love; I really like spending ages finding a way to have my artistic vision generated programatically, rather than setting it in stone myself. I think I enjoy the difficulties of trying to figure out how to define my imagination

Funnily enough, the WFC algorithm for procedural generation actually _is_ based on the wave function collapse in quantum mechanics. Like the whole thing is basically setting each cell to a superposition of all states, then collapsing nearby cells by adjusting their probability distributions by adjusting the possible eigenstates. So it goes from a mixed system to a classical one.

This algorithm feels like a huge advancement in procedurally generated maps! Thank you for sharing it Of course it won’t be right for every situation, but it has so many possibilities. For a first person game, I wonder if it would make sense to combine Wave Function Collapse with the 3D noise underlying marching cubes? Something like at the start, you’d restrict which tiles are placable at each location based on the noise value there. I’ll have to play around with this!

I'm a new programmer and I've always wondered about how world generation was implemented in games. This video is AMAZING and answered every question of mine. The production quality, too- just wonderful. Thank you.

I wrote a caves/maze map generator thing with basically this idea, without ever having heard of it before, so watcing this video kind of surprised me. I think thats pretty cool. My own name for my version was erosion mapping, because it carves out space from a world full of solid blocks, then removes any walls that're only touching other walls, so the new eroded space is perfectly surrounded, ergo maze/cave generation as opposed to world generation.

Dawg, 8:52 your pixel art is fantastic! Your control of contrast and form is unmatched dude. Don’t underplay yourself, you are talented!

Actually, as far as I can tell, the wave function collapse generation method is the same as quantum mechanical wave function collapse, except we randomly generate measurements (determining the boxes type) to reduce the domain (literally the same terminology, use tensor product space if feeling fancy) until we are left with one possible wavefunction (generated mesh). If we want to get fancy, we can describe it as a) wavefunction in the tensor product space of all the tiles' possibilities (so all the number 5's in each tile being almost independent of each other. If one wants more jargon, then 5 in each tile would be the Schmidt rank of the partial trace which eliminates everything except that tile) b) an operator which takes any possibility which is incompatible with our constraints to 0, while letting all other possibilities alone. This is the information propagation stage in the code. So eigenvalues are 0 and 1, and we demand that our wavefunction be a 1-eignevalue of the operator, so we apply it repeatedly in discrete steps c) the random hit generator which pins down the tile is just a simple projective measurement I see no reason why one couldn't possibly generate more complex structures from this, except that any property which is non-local (not entirely dependent on direct neighbours) like connectivity in a maze, the propagation step gets very complicated very quickly.

This is revitalizing my obsession with auto-generation. But it really should be called *domain collapse* but I can see how it is actually similar to the physics concept

i genuinely love seeing videos like this -- really nicely made passion projects. you're not afraid of showing off your pixel art even if you think it needs some more practice, and the explanations are really well done and visual

Awesome video! This is an excellent explanation of wave function collapse. Thoroughly enjoyed watching it and am looking forward to your next video. Keep it up! :)

Even for me who does not use Unity at all, your charming voice is so lovely to hear with the explanations you do. Gorgeous!

This is the first video of this channel I saw. Looooove this kind of in depth content. No matter how many views it gets it deserves more, please know that this is super awesome!

Everything old is new again. This reminds me a lot of Conway's Game of Life and cellular automata in general. Simulations could use this process to great effect, I think. If you had data encoded in each cell regarding say, resources available (food, materials, etc.), you could use that as a basis for things like simulating animal migration, human expansion, stuff like that. Really interesting to learn about practical applications for otherwise dry math concepts.

Really liked how you showed all the progress you made and explained everything, keep it up!

1:14 Wave Function Collapse is actually a really great name for algorithm. I suspect you touch on this in the next video, but this process of the "collapse" of a possibility-space into something concrete is EXACTLY what wave function collapse is in quantum mechanics (albeit much in a much more complex way).

This came at the perfect time for me - I'm about to start working on a terrain generation system for a hobby project and I've been terrified of approaching it. This has really shed some light on a great way to get more control over the generation algorithm, so thank you for sharing! Also the animations were a great help in understanding how it all works, great vid!

Dude came outta nowhere and made a banger video. Super interesting and fun to watch. Please keep at UKposts! I love your content thusfar

I used something similar on a BotW-like that I made with my kid a little while back. One of the things I really liked about this method was the ability to have "landmark" tiles that would generate villages, ruins, and dungeon entrances in reasonably logical locations.

it might be a good idea for a map to generate a few "necessary" tiles in random places before the wave collapse starts to start out with, so that certain needed locations always exist for what ever gameplay takes place in the generated world. then, if any tiles wind up coming into conflict, special case tiles could be inserted as a way to ensure there is a resolution to any strangeness that could cause.

This came across my recommended and after finishing it, I went straight to your channel. Eagerly awaiting more videos!

it's always refreshing to see discussions about world gen stuff... that I can actually understand. Pleasant work, keep it up

An interesting thought that occurred to me while watching this video; when you introduce the grid of cells at 7:11, your rules don't need to be a direct relationship between meshes but could also be their relationships within the grid overall, like you could use Sudoku rules to guarantee that predefined areas contained certain features, or even categorize your meshes further into biomes for more distinct segments. One question, is it possible to generate an "illegal" map where your adjacency rules result in no possible meshes in a space? If that occurs, how do you handle it?

Great googley moogley, this looks like it took you absolutely forever to make. Thank you for the wonderful video. It was extremely interesting with phenomenal visualizations, chill background music and a nice voice. Felt like I was watch the next Sebastian Lague. I eagerly await your next Devlog, but no pressure :)

This is right up my alley! Looking forward to your next release

Fascinating! The explanations were all really well done, the effort you put into those is super appreciated. The simple rules are so elegant, makes me want to do something with procedural generation.

The pixel shader has a great style to it! Would certainly enjoy seeing more about it and other's like it, particularly interested in how it looks in motion.

This is a great WFC collapse explanation! I'd be interesting to hear how you're going to deal with situations where there is no valid tile - or whether your marching cubes rules mean that there will always be a valid tile. Have sub'd and looking forward to the next!

This is one of the best visualizations I’ve ever seen. It really helped me to understand this concept. Very good video.

Incredible video and absolutely beautifully animated. Thank you for sharing your findings and making it much easier for the rest of the community to find alternatives to voxel meshes!

I imagine a hybrid approach of perlin noise layering + wave function collapse could create some pretty nice generation. Overall map structure is important.

You can use some like Binary Space Partition or any of a slew of other techniques in conjunction with this to add weights and biases to what can and can't generate, sometimes altering the generation rules entirely within a given region defined by the algorithm. It's an extra step that would give you tremendous control over how things look and feel at any scale. It's also probably not too much trouble to throw in any kind of geologically-inclined algorithm at this too, since it's all just about the rules for what the tile can 'collapse' into, after all. These things kinda run first, then you apply your wave collapse algorithm afterward to finalize the generation, and it has to adhere to the weights and biases introduced by any prior algorithms that adjusted what tiles can become. Thus, a very fun to tweak and get just right procedural tool is born. I'm probably going to experiment with this in a Godot project at some point, myself.

Hey man I watched both of your videos and I must say, I'm a fan!!! I'd love to see more such content, so PLEASE make more!

you have a realy nice way of explaining stuff and I was left wanting more. keep up the great work!

How do you guarantee that you don't back yourself into a corner where no tile is allowed? You mentioned favoring picking the next tile where the domain is smaller, and then in later clips you have a very directional method (or is that just the animation?). Are there any rules that affect more than just direct neighbors? This reminds me of weighted wang tiles, which matches against each edge type instead of each whole square.

I've been reading about WFC a lot lately. One guy got amazing results by also using NEGATIVE connectivity examples in WFC. This way you don't have to specify all of the possible connectivity rules but can just prevent the rare cases you don't want for some reason. This gives much more artistic control.

This is the best explanation of WFC that I've seen, ever. It explained it simply and made it approachable. I'm really looking forward to your explanation of how you implemented your rule set!

Great description and discussion, really looking forward to your follow up video on this topic!

Well presented. I would not consider that work you put into a solution you are not going to use as wasted. Some lessons just cost more time than others, what matters is that you developed from that point to where you are now. Keep it up!

You're really good at explaining this. Would you perhaps be interested in doing a more detailed walkthrough of working with WFC? It's gonna be a few months till I can get back to programming because of real life stuff, but you've definitely piqued my interest A LOT.

This was quite fun to watch, I'd love to see more of your processes

This is an awesome video!! I love Oskar’s posts. Can’t wait for your next video :)

Consider adding time into the equation (maybe some value between -1 and 1, then you can feed a sine wave in) to the WFC. Eg, water can appear next to sand at -1, but ice at 1. For 2D you could fade between the 2 images for values between -1 and 1, for 3D you’d need a different solution (could use blend shapes if you maintain the same vertex count / order and blend between 2 mesh’s). The sine wave could represent a day / night cycle, a seasonal cycle, or even some sort of “force” (you could simulate earth quakes or waves).

Minesweeper is a game where humans practice collapsing waveforms.

This was surprisingly easy to digest, and very entertaining!

I have no idea how I stumbled across this video but I'm really glad I did. This is a really interesting subject, and something I've wracked my mind against for a long time for similar reasons as you.

The final maps still tend to look too random, like the one at 8:36. Im curious what would the maps look like if there was a weight attached to which tiles should be picked based on neighboring tiles.

if you're still not quite satisfied you can try adding a shell or wrapper for it especially for verticality, either a nested waveform, weighted veroni, poison disc sampling, or my personal favorite reaction diffusion. standard waveform never really had the kind of topology I was looking for but is seriously impressive for what it is.

Randomly found this video on my main page. Absolutely do not regret watching it! Well done

I love the relaxing music in the background

I’d like to see more detail with regards to the tile selection rules, and also the voxel generation if that’s still underlying. It seems that a naive approach would tend to give too random of a distribution at the small scales, leading to no larger overall shape and so no fractal coastlines. Perhaps having a bunch of deeper and deepest ocean tiles, higher mountain tiles, etc. would help, assuming tile selection is what decides the terrain height. Perhaps running multiple iterations of wave-function collapse at progressively narrowing scales would provide better results?

“It would be nice to talk to people with similar interests” I am impressed. Pardon my lengthy ramble. I probably won’t respond further, I often go quite in depth with thoughts I find this content very knowledgeable while also embracing your intrinsic emotional experience. Memes are funny too. I learned something new not just about voxels, marching squares, wave function collapse, tradeoffs during development, but also about the process of being a human working on a game. Interested in kinesis too One thing I found interesting was how the water would often form in illogical ways. It appears you were already experimenting in that direction with southwards flowing water. Talking about the connectivity rules, I suppose you could make dummy layers for things like oceans/lakes/rivers with the same tiles extended in different axis of scales. So, very deep water being 7o, deep water being 6o, water being 5o, and coast being 4o, with lake being maybe 5L, 4L that don’t connect with the o’s and rivers being 4R. Or changing the selection method, or the propagation method. Certainly easy to get distracted by side projects when there are so many interesting ideas and experiments to explore! I do tend to find in my experience working towards a goal with realistic steps tends to curb the side projects. But then I have to set aside the reward of interesting and learning, so the reward of that goal needs to be suitably enticing to make that trade. Normally I don’t really have a lot of interest watching devlogs of people generating hype for their games by regurgitating the same experiences that others have had and expressed. As someone obsessed with learning, it can often feel pretty disappointing and embittering to invest in watching videos and not learning anything from them. But I grit through that expectation for the sake of possibly learning something. This was more than I bargained for haha. The idea was so well explained that I’m now convinced it’s a great method to implement creatively, much preferred to the marching square tedious manual case by case coding.

Really nice and interesting video, looking forward for the next one!

Just came across this vid and it the exact solution to map generation problems I've been having for a while thanks an absolute ton for introducing me to this method

The behavioural research, computational automata, and bee booping are right up my alley. I'm excited to see what other cool stuff you make!

My thought when seeing your first larger wave function collapse example at 8:36 is that the water areas look too small. Iʼm sure there are ways to solve this with the rules you provided, but my first thought was to extend it so each neighbor, in addition to ruling some things out, changed the probabilities of the adjacent tiles, so that choosing a new tile adjacent to water might have a 90% chance of being water and only a 10% chance of being sand. If there are two adjacent water tiles, that would combine to 99% chance of water (9:1 water to sand ratio, squared), up to 99.99% for four adjacent water tiles (which is probably too extreme, but theyʼre illustrative and easy numbers to work with). Is that a thing thatʼs been done before here?

I found the concepts you discussed pretty cool and applicable to data science as well, your teaching style is clear and your voice is great, subbed!

I love it, super excited to see this channels direction

4:12 one reason using a switch is actualy a bad idea, to see what's happening you have to scroll all the way up worse you manually write every case. this is still hard coding even if you're not directly putting the number values there a better way to do this is to make each case a member of a (voxel rule class) , that holds the data about the system then if you serialize that you can save it to a file without having to edit the code directly this way could be much faster because every change you make shows up immediately instead of every {close game, recompile, open game}

I implemented blocks then marching cubes, but I never sucedded to implement dual contouring, I ended up with a big cloud of triangles that I could never fix. This wave function is not about rendering voxels but generating them, it should be compared to simplex noises, geological simulations, etc. I like the freedom it gives, but the huge amount of rules required and the undeterministic results are serious limitation.

Great video. Hope to see more in the near future!

Thank you for this video! So interesting, I had no idea about neither of both algorithms and now I’m looking forward to try by myself!

🤔 how strange. i watched your video and you gained a subscriber

Really nice demos and illustrations, excellent way of explaining the principles you used. 👍

I have enjoyed it. Thank you so much for documenting your WORK!

Awesome! Loved the video! I’m going through something very similar with my own map generation and it’s awesome to see your thought process! Thanks!

This is amaizing. Can't wait for the next video!

This was really neat! Thank you for sharing. I might try it out for some of my games

Looking forward to your next video! You have no idea, this is exactly what I've been looking for, and now I have a name for it. Thank you!

This is an awesome video!! Super informative but not too slow-paced to keep the viewer's attention. I've been wanting to learn how to use Unity so I can experiment with things I find interesting, and this video is quite a nice push to get on that.

Awesome video! Excited to see what’s coming!

this was neat to watch. i had a basic grasp on how this stuff kinda works, but its neat to see it in practice. should do a video showing some of the bugs that can come from stuff out of this too.

This has so much potential it looks really good

This is amazing! cant wait for the breakdown

I found this video while searching around to see what Voxel CAD tools were floating around. I enjoyed seeing the various different approaches you used to do procedural generation. I hope you are able to make more development diary videos of your projects.

Thanks for sharing. Can't wait to see the next video.

Those look awesome, I'm gonna use this. This was an excellent video, sad you've only got two on your channel. Subbed and waiting for more

Appreciate this approachable explanation. The name always reminded me of the Wave Motion Cannon (from Oni), which I understand was itself a reference. On the first watching, it wasn't clear that the number in the cell represented the count of possible solutions to the rules. But I eventually got there! Nice going.

This looks so cool! And your video was so good that it made me subscribe instantly.

Thanks for explaining wave function collapse in a way that's easy to understand!

What the!!! Your channel has TWO videos! One of them I would honestly not give the time of day, but this one is Sebastian Lague level of quality! Thanks for sharing, you earned a sub!

Nice. Im glad i found your channel, great stuff

found your channel just now. I Really like your content, kinda feels like a blend between sebastian lague and primer. really look forward for more videos!

Neato. Just stumbled on "wave function collapse" last night researching a programming language/compiler (Virgil) then your vid appears on my youTube homepage. Intriguing and possibly destiny. Good vid btw. Had thought about random, dynamic texture and object creation with a defined pattern of rules that could be updated on a case basis, but never really got down into the nuts & bolts. Always fascinates me when stumbling into others with similar views. Appears you've got quite a good grasp on the topic - very nice.

Thank you , thank you, thank you. I was struggling with my PG terrain design and you've inspired me to try this approach.

I always told myself I'd never get into game dev, but after just graduating and seeing all the effort and complexity that goes into something like this, I'm rethinking that.

I'm not currently working on any projects that could use this but I'm gonna keep it in mind for my big future dream game. Thanks so much!

Very nice video! I really enjoyed the information, dude, I was thinking oh another video for 10 minutes of blah blah, but I was spell bound! Thanks for sharing, you've got a subscriber here! I'm an aspiring game dev, and I've been considering wave function collapse as a generator for a bunch of things, like terrain, buildings, etc. Thanks again!

ez sub, that shit was badass. I can't wait to hear the technical breakdown, it give me so many idea!

This is truly amazing, your video inspired me to do something like this! I've spent big amount of time working with procedural generation, but never came across this thing. Thanks!

Ok, this was super great. The explanation and visualization was perfect and reminded me of some Sebastian Lague videos (which is meant as a huge compliment). The art style is a mix between nostalgia and new techniques and looks very good. I'd like a video on the pixel shaders as well. Whatever you focus on, UKposts or game development, i'll watch your videos/tutorials. Keep it up!

This was super interesting to watch, loved this. Subscribing!

I love exploring different procedural methods of creating maps and structures. Fractals, and setting rules, limits, and boundaries are neat tricks that with todays technology can work with any kit bashing method approach. Simple and fun, looks good. To truly get realistic or plausible builds these methods must be carefully calibrated and tweaked. Using topographical maps and how water, sand, and even harder stuff flows over time across a landscape can give that plausible yet interesting and even unsuspected look. I like that too, it is a creative process. Thank you for sharing this.

I didn't understand almost the entirety of the video, since it's way above my head. However, it seems like whatever you're doing is really amazing and taking your artistic endeavors to the next level. Keep up the good work and keep moving forward!

Absolutely lovely, I learned a lot and I did not know of any other method than marching cubes! Thank you! Maybe do some more deep dives on the concepts behind the work you do and key methods for procedural work ?

this is a really cool concept that i will be into, cant wait to learn this an be able to implement it into my game

Inspiring. I'll need to learn more congrats on 1 new sub

it's great, in particular, as a CS student, I liked that you put the source in the description or showed it, it's handy.