What is a region of mostly calm weather at the center of strong tropical cyclones called?

I live in Puerto Rico, USA, and I recall vividly when the eye of Hurricane Georges passed over our house. All was calm, the sky was blue and surrounding us were thick gray storm clouds spinning sideways. It lasted all of 15 minutes...then that churning, spinning eye wall slammed into our house taking with it parts of my roof.

this picture would also make a great wall feature