Building a Procedural Hex Map with Wave Function Collapse

Furthermore, various heuristics can speed up the backtracking a lot compared to a brute force approach. As anyone who has implemented a Sudoku solver can attest, a brute force backtracking is easy to implement but will immediately get bogged down with slowness.

[1] https://en.wikipedia.org/wiki/Knuth%27s_Algorithm_X

(Dwarf Fortress is the one procedural game I know - besides maybe its more faithful clones - which isn't static. But the procedural generation there is also mostly not incremental, it generates the whole world beforehand and only some of the fill-in details are incrementally generated).

The holy grail has to be a graph-based refinement/embellishment system, where it generates just the nodes, temporally and spatially, which matter for the situation the player is in.

The goal of the original algorithm ( https://github.com/mxgmn/WaveFunctionCollapse ) is to infer the constraints from a sample, and then run a constraint solver.

Hard-coding the constraints skips the whole point of the algorithm (which is also badly named by the way).

The maps are pretty, but the per-tile build constraints of the WFC build approach means that pretty unnatural generations end up happening because non-local influence is difficult to take into account. I think this may be OK for games where you discover tiles one at a time, but for a full map generator it's not great, and better solutions exist. Red Blob Games did a writeup of a noise-based method which looks superior imo. You can use moisture-tracking approaches for rivers, lay roads, bridges and other artificial elements in a separate pass, and it will likely end up faster and more robust. I think WFC is an interesting programming problem, though, so it was likely fun to implement.

Nonetheless, this was an excellent write-up and impressive demo.

As an aside, if the author reads this, did you consider using bitfields for the superposition state (ie, what options are available for a tile)? I did a wfc implementation a while back and moved to bitfields after a while.. the speedup was incredible. It became faster to just recompute a chunk from scratch than backtrack because the inner loop was nearly completely branchless. I think my chunks were 100 tiles cubed or something.

[0] https://catlikecoding.com/unity/tutorials/hex-map/

There is, a hexagonal grid is isomorphic to a [skewed] rectangular grid, i.e. it can also be indexed with a coordinate pair (u, v). The neighbours are at offsets (+1, 0), (-1, 0), (0, +1), and (0, -1) - just as in a rectangular grid - and the two additional neighbours are at (+1, -1) and (-1, +1). The coordinates in the plane are (x, y) = (√3(u + v/2), 3v/2) or some variation of this, depending on how exactly one lays out the hexagons and picks axes.

It is surprisingly hard to find a good illustration for this, or maybe I am using the wrong search terms, but here [1] is the best one I could quickly find.

[1] https://www.researchgate.net/figure/Rhomboidal-and-hexagonal...

For guiding the search, you might want to consider search steps that select only one feature, for example that a pair of adjacent tiles should be connected by a road, and just propagate that information. That could be used as a way to guide the search on high-level features first, and then later realize the plans by doing the normal search.

[0] https://store.steampowered.com/app/1455840/Dorfromantik/

In the level data, the start/goal tiles (approx 10x10 blocks of solid ground under the player and goal) and slopes aren't represented by their tile offset in the level data -- whilst all other "ground" tiles are. Instead, for slopes you're only told the start and end coordinates, and they often overlap.

So to render the slopes correctly I had to work out all the rules for which tiles were allowed next to each other, and solve some ambiguity rules -- I figured out that shallow slopes take precedence over steep ones. Eventually I cracked it, but it took quite a week or so of iteration to figure it out.

https://en.wikipedia.org/wiki/Dorfromantik

In this rabbit example I made 8 years ago, the WFC solver ensures that the animation must loop, which means you will always end up with an equal number of births and deaths.

https://xcancel.com/MattRix/status/979020989181890560

I say this having had a couple of fun "hex-based strategy game hobby projects" over the years (sidenote -- trying to cover a sphere in hexes is actually a non-trivial matter). Invariably I ended up with "to make a map from scratch, first you create the universe" where I'd go through all of the ages, compute waterflows and precipitation, and on and on. Maybe I made the requirements too unreasonable and that's precisely why I never yielded a working game from it.

I was also wishing I could zoom in to human size and run around HAHAHA

They are essentially making the entire game based on similar concepts and then using them to develop their core content. Simon is an inspiration and has said they won't be taking investor money so they can stay true to the users and creators.

> Model synthesis (also wave function collapse or 'wfc') is a family of constraint-solving algorithms commonly used in procedural generation, especially in the video game industry.

> [...] One of the differences between Merrell & Gumin's implementation and 'wave function collapse' lies in the decision of which cell to 'collapse' next. Merrell's implementation uses a scanline approach, whereas Gumin's always selects as next cell the one with the lowest number of possible outcomes

And then `## Developments` mentions:

"Hierarchical semantic wave function collapse" (2023) Alaska, Bidarra: .. citations of: https://scholar.google.com/scholar?cites=1671019743611687613...