The history of the Far Lands in Java Edition can be split into three main sections:
Types of Far Lands
A world with far lands can be divided into multiple parts depending on how many axes one given noise generator is overflowing on at a given time at that position. In the diagrams below, normal terrain where nothing is overflowing is marked in gray, one overflow in light blue, two overflows in magenta and three overflows in lime. This can be extended to an arbitrary whole number of dimensions, although there are currently no modifications that allow for 4D or above terrain generation.
Facial / "Edge" / "The Loop"
The most commonly depicted form, this is the result of an overflow on only one axis. On a 2D plane, there are four different overflows possible, corresponding to the four edges of a rectangle. In 3D space, there are six different overflows possible, corresponding to the faces of a cuboid.
Edge / "Corner" / "The Stack"
These are the result of an overflow on two axes. On a 2D plane, four cases of this are possible, corresponding to the vertices of a rectangle. In 3D space, there are twelve such possible intersections, corresponding to the edges of a cuboid.
Vertex / "The Abyss"
A rarely encountered type, this can only be encountered with three axes overflowing at once, which is effectively impossible in any version without mods. There are eight possible regions for this case, corresponding to the vertices of a cuboid.
Noise generator overflows
Java Edition employs a variety of different noise generators for shaping several aspects of terrain. These noise generators experiencing an integer overflow is what results in a set of "Far Lands".
Low noise and high noise overflows
"Low noise" and "high noise" are two extremely similar noise generators used for generating the fundamental shape of terrain itself. As a result, if either of these overflows, the fundamental shape of terrain sees drastic changes. The simultaneous overflowing of these is what results in the conventional Far Lands generating.
Whether low noise or high noise is used for generating terrain at a given position is determined by a third noise generator, referred to as selector noise.
Selector noise overflows
Selector noise is a noise generator used to determine whether low noise or high noise is used to generate terrain at a given point. Once overflowed, where low and high noise are used becomes quite visually obvious, due to its now abrupt changes.
Selector noise overflowing in otherwise normal terrain, with the normal terrain at the bottom. Note how the lower terrain uses low noise and the higher terrain uses high noise. After overflowing, the low terrain stays constant where low noise is selected, and the same can roughly be said for the higher area on the right.
Depth noise overflows
Scale noise overflows
Biome fill noise overflows
The depth of the surface block corresponding to each biome is determined by a noise generator known as biome fill noise. Biome fill noise increments by 0.0625 units each block, or 1 unit every chunk, and thus will overflow at or around X/Z ±34,359,738,368. Overflowing is not visible without either extensive modding or manipulating the noise scale, as the overflow occurs well beyond the block render limit (±2,147,483,647) and coincides almost exactly with the chunk overwrite limit.