Pnyx and the Filopappou hill are located right next to the Acropolis with then main entrance being near the entrance to the Acropolis. Both of these hills have some ancient ruins on them, Pnyx has an ancient theatre and Filopappou has a monument to the muses. While these sites are fine, the real draw is the views. Pnyx is the easier hill to climb, and it has a great view of the Acropolis. Filopappou also has a great view of the Acropolis on the way up, however at the top it has the best panoramic view in all of Athens.
One of the great hidden gems of these two hills can be found at the base of Filopappou hill about 30 meters from the entrance. There you'll find a couple of small signs leading you to Socrates prison. These small caves are thought to be the place where Socrates was imprisoned and eventually poisoned. It was later used as a bunker during the Ottoman period. Unlike the hills it's not that difficult to get to, and there are rarely any other tourist around.
For the most part nothing particular happens on these hills with one exception. On clean Monday every year, flocks of locals climb Pnyx hill to picnic and fly kites. This picnic represents the first day of lent and the last day of carnival. The meal follows the Greek Orthodox rules of lent. The kites are said to help you wash away your sins from carnival. They say that the higher your kite goes the more sins are washed away.
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