My absolute minimum for normal functioning is 4.5 hours a day. The best seems to be 3 hours straight, and a 1.5 hours nap (or two 45 minutes, but that is sub-optimal).
A complete sleep cycle is 90 minutes long. When you wake up in the middle of it, you feel extremely tired and want to go back to sleep (thanks alarm clock). Without being disturbed, we pretty much always wake up at the end of a cycle.
Sleeping in 1.5 hours increments thus makes perfect sense.
A good minimum is 6 hours a day, with a 4.5 hours night and a 1.5 hours nap. With that I can function AND don't feel too tired during the day (except after the mid-day meal, after which I take a nap).
In a 6 hours straight pattern, and no nap, I feel much more tired in the afternoon, but still function ok.
At 7.5 hours straight, I'm fully operational and feel good all day.
At 9 hours straight, I'm over sleeping but this is excellent to "put sleep under my belt". We are actually always more or less in a long term cumulative sleep deficit. Going "marmot" like this thus reduces this deficit and is a great way to improve health (psychological and immune system, especially). After such an over-sleeping period, I can make MANY short nights without any alteration of performance.
Over sleeping is also good to memorize things or sort out difficult and stressful times. Students should sleep a lot. It helps the data to stick in there and be useful later.
Cheers,
David