One year I rented an apartment that faced the Piazza del Duomo and stayed 8 months. Not only is the Piazza "home" in my heart, but having studied Italian Art History in Florence I learned about the amazing dome of Brunelleschi, the Giotto bell ower, and about the doors on the baptistery across from the cathedral...there is also a museum of the original works from the Duomo.
I love it all....the people sitting on the steps, the cafes that line the piazza, and especially the celebrations of the church with processions in original Florentine costume and the sound of the drums...Easter, Feast of the 3 Kings to name a couple. For the best time to visit take a walk in the late afternoon, the passagiata or stop by a cafe for an aperativo. I began visiting Florence in 1996. I always look at the dome of the church most of all. I recommend reading about Brunelleschi and his amazing feat in building this dome for the church.
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