Du kletterst auf die Kiste und springst zu mir herüber.

Questions & Answers about Du kletterst auf die Kiste und springst zu mir herüber.

Why does this sentence use herüber instead of hinüber?
The prefix her- indicates movement towards the speaker, while hin- indicates movement away from the speaker. Because the sentence says you are jumping zu mir (to me), the movement is coming towards the speaker, making herüber (over to here) the correct choice.
Is herüber just an adverb here, or is it part of a verb?
It is acting as the prefix of the separable verb herüberspringen (to jump over to here). In a standard present-tense sentence like this, the prefix breaks off from the verb (springst) and goes to the very end of the clause.
Why is it auf die Kiste and not auf der Kiste?
auf is a two-way preposition. Because climbing onto the box involves a change of location towards a destination, it takes the accusative case (die Kiste). If you were already on the box and just climbing around on top of it, it would take the dative case (auf der Kiste).
Since there is movement towards me, why is it zu mir instead of the accusative zu mich?
While it is true that movement towards a goal often triggers the accusative case with two-way prepositions (like auf in this exact sentence), the preposition zu is strictly a dative preposition. It always forces the following noun or pronoun into the dative case (mir), regardless of movement.

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