The meaning of physiognomy means the look of your face. When traveling in Italy, you may be struck by the wide eyes and pleasing physiognomy of the Italian people you meet.
The reason physiognomy sounds like it should be something you study in school right after biology, geometry, and astronomy is that people used to think that it was a science by which you could tell someone's character through their facial features. If you've ever read any Nancy Drew stories, you will know how this plays out––anyone with "shifty eyes" is not to be trusted.