I’ve written a lot about Wave Function Collapse. Developed in 2016 by Maxim Gumin, it’s an algorithm for generating tilemaps and pixel textures based on the constraint solving with extra randomization . But did you know most of the key ideas come from a paper written a full decade earlier? Today, we’ll be looking into **Model Synthesis**, the 2007 PhD dissertation of **Paul Merrell**, and some of the elaborations he’s designed, particularly **Modifying in Blocks**.

# Beginner

This category contains articles suitable for most readers just starting out.

See Intermediate and Advanced for more articles.

# Arc Consistency Explained

I’ve been doing a lot of experiments with WaveFunctionCollapse, which as we’ve covered before, is essentially a constraint solver turned into a procedural generator.

The underlying solver WaveFunctionCollapse came with is called **Arc Consistency 3**, or AC-3 for short. But AC-3 is actually one of a whole family of Arc Consistency algorithms. Today, my solver and most others uses AC-4, a more advanced algorithm. Let’s talk a bit about how those both work.

# Recursive Subdivision Variants

You are probably familiar with Recursive Subdivision – also known as Binary Space Partitioning – as a procedural generation technique. Like all the best proc gen, it’s a simple idea, that produces complex output. I’m here to discuss some variants that others have used to produce interesting results.

Continue reading# Triangle Grids

Ah, the triangle grid. Square grids are virtually ubiquitous, laying out out everything from the pixels in an image to houses in a city block. The hex grid has a decent showing too, particularly in board games. But triangle grids – regular tilings of the 2d plane with equilateral triangles – just don’t seem popular. I’ve seen claims they are useless, or that the maths is hard. But I’m here to prove both of these are wrong: the maths is actually **easier** than working with hexes, and triangles have all sorts of neat advantages.

I’ve worked out all the maths in my reference code on github, but it’s worth explaining why and how to use these grids.

Continue reading# Lock and Key Dungeons

Lock and key dungeons are, well, video game levels with locks preventing progress, and collectable keys that let you proceed.

The concept is a lot broader than it sounds. Locks/keys aren’t necessary physical objects, but anything that works in a similar way, which can often be quite abstract.

Once you are familiar with the pattern, **you begin spotting it everywhere**. It’s most prominent in puzzle games and metroidvanias, but it’s applicable to any game which has an authored progression path.

In this article, we’ll look at lock and key dungeons, then how to analyse and design them.

Continue reading# Dungeon Generation in Binding of Isaac

The Binding of Isaac, and its remake, Binding Of Isaac: Rebirth are one of my favourite games of all time. It’s a roguelite twin stick shooter, much like Enter the Gungeon.

The dungeons it generates are particularly iconic. I’ve seen countless tutorials online offering how to do Isaac-like generation, but I was interested in how the original did it. To my suprise, most tutorials get it wrong. In this article I go over how the generation works, including a Javascript demo.

Continue reading# Wave Function Collapse Explained

*A simple guide to constraint solving*

Since developing DeBroglie and Tessera, I’ve had a lot of requests to explain what it is, how it works. The generation can often seem quite magical, but actually the rules underlying it are quite simple.

So, what is the Wave Function Collapse algorithm (WFC)? Well, it’s an algorithm developed by Maxim Gumin based on work by Paul Merrell for generating tile based images based off simple configuration or sample images. If you’ve come here hoping to learn about quantum physics, you are going to be disappointed.

WFC is capable of a lot of stuff – just browse Maxim’s examples, or check out #wavefunctioncollapse on twitter, or see my youtube video.

WFC is explained briefly in Maxim’s README, but I felt it needed a fuller explanation from first principals. It is a slight twist on a much more broad concept – constraint programming. So much of this article is going to **explain constraint programming, and we’ll get back to WFC at the end**.

# Dual Contouring Tutorial

# Marching Cubes 3d Tutorial

In the first article I showed how the Marching Cubes algorithm works in 2d.

In this tutorial, I cover how it can be extended to 3d.

Continue reading

# Marching Cubes Tutorial

In Minecraft, you can dig in any direction – removing a block at a time with well defined edges. But other games manage to destruct terrain smoothly, without all the blockiness of Minecraft.

The following tutorial in Marching Cubes, a technique for achieving destructible terrain, and more generally, creating a smooth boundary mesh to something solid. In this series, we’ll cover 2d in this first article, follwed by 3d in the next , and Dual Contouring in the third. This last is a more advanced technique for achieving the same effect.