Learn Polish: Wiem, że żaden wazon się nie stłukł.

This is a perfect example of a spontaneous state change! The base verb stłuc is transitive and means "to shatter [something]". By adding się, it becomes intransitive, meaning the action just happened to the subject without saying who did it. So wazon stłukł would mean the vase shattered something else, but wazon się stłukł means the vase itself shattered.

Since the sentence uses żaden (no/none), why do we also need nie?

Polish requires double negation. Whenever you use a negative pronoun like żaden (no), nikt (nobody), or nic (nothing), you must also put nie in front of the verb. Literally, this sentence says "I know that no vase did not shatter."

I thought negative sentences require the genitive case. Why is żaden wazon in the nominative case?

You are thinking of a very important rule, but it only applies to direct objects! When you negate a verb, the direct object changes from the accusative case to the genitive case. However, żaden wazon is the subject of its clause (it is the thing doing the shattering). Subjects always stay in the nominative case, even when the verb is negated.

What is the infinitive form of stłukł, and why does it look so different?