I’m interested in generating 3D height maps for a 2D game I am working on. I am using this to create land masses like in Minecraft or Dwarf Fortress.
I've created 2D heightmaps before, but I used a very rudimentary algorithm that just interpolated between points of a fully random noise array to create a fixed size map. This doesn't tile however since if I try to add a new map next to it, it doesn't account for the height of the existing map.
I have read about Perlin and Simplex noise, but I’m now confused on how to apply Perlin or Simplex noise to a 2D array of height values.
Any help with this would be greatly appreciated. I have no idea what to do anymore. The term 'octaves' not on sheet music scares me.
Exactly, you have to look for Perlin/Simplex noise. Think of it as a function f(x,y,...) (as many variable as you wish) that will output random-looking noise. The difference with pure noise is that it acts on gradients, so it will look more natural since it "draws" gradients instead of plain noise with high local variability. Simplex noise is pretty much the same as Perlin's but it divides space in simplexes instead of operating in n-dimensional grids like Perlin does. This alleviates computational cost and has some more benefits.
It might seem scary, but it's simple actually. You're scared about octaves, but they're pretty much the same as octaves in music: just higher (or lower) frequency noise mixed with the original output. Talking about sheet music, it's like playing C4 and C5 at the same time. It's still C but it has some flavor added (little spikes in the waveform.) Don't be afraid and keep researching, it's not that hard.
Regarding tiling:
If you mean linear tiling (like Minecraft does) you just have to use the same seed for the noise algorithm. As soon as you approach your new boundaries, just generate the new chunk of data and it will tile perfectly (just like it does if you infinitely fill with noise.)
If you mean torus tiling (repeating tiles, think Pacman for instance) I found the best solution is to generate your noise tile and then interpolating near the borders as if it were tiled. The noise will deform to match sides and it will be completely tileable.