How do SPARQL triple patterns get joined?

Through variables that appear in more than one pattern.

Only when wrapped in `OPTIONAL`.

Joining patterns through shared variables — Semantic Web Academy

Two patterns sharing a variable form an implicit join:

SELECT ?ninja ?master WHERE {
  ?ninja a :Ninja .
  ?master :teaches ?ninja .
}

The engine returns every (ninja, master) pair where the master teaches a known ninja.

Why it works: a row is kept only if every triple pattern matches with the same binding for any variable that appears more than once. In the example, ?ninja must be the same node in both patterns, which is exactly an inner join on ?ninja. Same idea in tabular form:

Pattern	Binds
`?ninja a :Ninja .`	`?ninja` → every Ninja
`?master :teaches ?ninja .`	+ `?master` per pair

Theory