Looking up...
Das Buch liegt auf dem Tisch.
The book is on the table.
Indicates a position on a surface or location
Sie setzt sich auf den Stuhl.
She sits on the chair.
Commonly used to describe physical positioning.
Indicates a state of being active or in operation
Das Licht ist an.
The light is on.
Often used with verbs like 'sein' (to be) to describe states.
Indicates a direction or movement upward
Er steigt auf den Berg.
He climbs up the mountain.
Used with verbs of movement to indicate upward direction.
Remember that 'auf' can take either the dative (location) or accusative (movement) case depending on the context.
Use 'auf' with accusative when indicating movement upward or onto something, and with dative when indicating a stationary position.
From Middle High German 'ûf', from Old High German 'ûf', from Proto-Germanic 'up' (up).
The preposition 'auf' is often used with the dative case to indicate location and with the accusative case to indicate movement.