Cobblestone farming is the technique of using a cobblestone generator to produce cobblestone without damaging the terrain. Cobblestone generators work on the principle that when a lava stream comes into contact with water, the lava is turned into cobblestone. This fresh cobblestone then prevents the two streams from touching. When this fresh cobblestone is removed, the two fluids will produce another piece of cobblestone. Variants of the generator can also produce smooth stone, but this is generally trickier, because for smooth stone, the lava must enter the water from above.
Many generator designs exist, but the most basic is simply a 10-block long trench, with a water and lava source blocks at opposite ends. This will create cobblestone where the fluids meet: Because of their different ranges, this will not be halfway down the trench, but closer to the lava source block.
When producing cobblestone, one must be careful not to let the flowing water touch the lava source block. Doing so will destroy the lava source, converting it into obsidian. A basic understanding of fluids is helpful to prevent this.
Standard generators have been around for quite a while. Their popularity, however, is limited because cobblestone is so readily available. Standard generators require the player to mine and collect the fresh cobblestone in proximity to the lava. This both presents risks to the player, and reduces efficiency if the dropped item is destroyed by the lava. These drawbacks may be mitigated by design choices, for example by removing the block under the cobblestone, allowing the loot to fall in a safe place.
Notes about the schematics here: Gold blocks indicate "any (suitable) block". For a cobblestone generator, "suitable" probably means "fireproof". Water and lava source blocks may be marked with "s" when there is possible confusion. Cobblestone or smooth stone appear only where they form. An "x" indicates a place to stand while mining the cobblestone.
A lava stream touching a water stream is the simplest type of generator. In a 10 block long trench with sources at either end, the cobble will form next to the lava. With a little more digging, you can manage this more compactly, and even get a current to wash the mined cobblestone away from the lava. This and the next design are easily expandable for multiplayer use.
The "From Below" generator is a small building with the generator proper on the roof. Putting the generator on the roof means very little cobblestone is lost to the lava, but it is a lot more work. This one too uses two lava streams.
Pistons can be used to automate the cobblestone generator and reduce the amount of cobblestone lost. Piston cobblestone generators work on the same principle as standard generators, but, rather than mining, a piston pushes the fresh cobblestone or stone out of the way, allowing the streams to touch once again. Piston cobblestone generators can be used both to create a large supply of cobblestone that the player can mine later, or to supply a self-repairing structure with blocks. The piston can be driven by a clock, or by a circuit to detect when a cobblestone block has appeared. The cobblestone will extend in a long line or pillar; if you don't want it to extend out to the full 13 blocks, you can "cap" it with any unpushable block. Furnaces will do fine, and you have plenty of cobblestone handy to make them.
There are three basic components to consider in a piston cobblestone generator:
The core. This part includes the water and lava that creates the fresh cobblestone in front of the piston. It's generally based on a basic generator plan, with modifications for the piston and redstone.
A clock circuit or block detector, driving a piston. This part generates a signal to drive the piston that pushes fresh cobblestone out. The clock period can be chosen to minimize excessive piston movement.
A block detector is simply a circuit from a power source to the piston, passing through redstone repeaters before and after the spot where the cobblestone will appear. When the block does appear, the repeaters can push current through it to trigger the piston.
A clock generates its signal repeatedly at fixed intervals. Any of the basic repeater clocks will do perfectly well, but you want a total period of at least 7 or 8 (that is, a 4-clock or longer).
The core piston itself is usually non-sticky, but some block-detector CSG designs have a sticky piston with a transparent (non-conducting) block.
Optional secondary pistons. Since pistons can only push a maximum of 12 blocks, the core will only produce at most 13 cobblestone blocks at a time. This can be greatly increased with secondary pistons that guide the row of cobblestone in other directions. Like the core piston, the idea is to get the fresh cobblestone out of the way so that more can be created. A line of secondary pistons may also be used to move the blocks directly into self-repairing structures. Secondary pistons can be triggered by the same clock or detection circuit as the core piston, but this can be noisy if there are many of them. Alternatively, they can get their own clock or detection circuit.
These advanced generator designs consistently produce four cobblestone blocks on every fourth piston cycle. The blocks are pushed upwards, negating any chance of the cobblestone burning from touching lava. Cobblestone Quad-piston "Factory":
Lava flowing into water from above creates smooth stone. Stone can be mined slightly faster than cobblestone, and it can also be collected as stone using a pickaxe with the Silk Touch enchantment. Using smooth stone also gives self-repairing structures a different, more natural look.
Smooth stone generators are rarely designed without pistons, as lava needs to be directly above the stone generated. Lava must flow down into flowing water in front of the piston. As with cobblestone generators, a single-piston design can only make a row of stone up to 13 blocks long.
Examples of a Smooth Stone Generator:
As it is faster to mine, it can be more time efficient to use a smooth stone generator over a cobblestone one.
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