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There are two sort of classes of alchemists in Fullmetal alchemist.

Standard alchemists, like Armstrong and Mustang, tend to use alchemy as a kind of machine that lets them physically manipulate matter. Armstrong carves rocks into pointed spearheads and Mustang can control the oxygen density in air, and they use that to produce the effects they want. That is not really related to alchemy. Alchemy is about change on a more fundamental level than physical form.

The other type of alchemist are people like Ed who have seen the truth and can do circleless alchemy. Edward does a lot of transmutation, which is more consistent with what alchemists wanted to accomplish. It is still different though. Edward performed alchemy in a chemistry mindset of identifying and categorizing the composition of a substance and invoking chemical reactions to change it into a different substance. This is explained in the show as a process where the alchemist draws energy from the tectonic activity underground, and applies that to produce an effect.

The closest analogue to this is the magnum opus of alchemy, Transmutation of base metals into gold. The way that supposedly works is that physical matter is dependent on purity to determine its properties. Basically saying the difference between lead and gold is that lead is unbalanced in its intrinsic fundamental components, and if you balance them then it would turn into gold.

Alchemists also pursued this rebalancing through standard physical reactions and didn’t necessarily involve siphoning power from the world.

Beyond this, there are other interpretations or implementations of alchemy that are distinct from the material path. There is plant alchemy and spiritual alchemy as well, which aren’t represented or alluded to as much in the show.