He's talking about REM cycles, which last 90 minutes. Generally people recommend that you sleep for an amount of time that is a multiple of 90 minutes so that you do not disrupt one of these cycles.

I believe the grandparent was referring to the same phenomenon described in the relevant Wikipedia article at http://en.wikipedia.org/wiki/Sleep#Sleep_stages

I was talking about whatever contemporarily popular science apps and websites like http://sleepyti.me/ are buying into.

Ah, in that case I don't know, but I will ask.

Their latest research indicates that a full sleep cycle lasts approximately 4 hours, which does fly in the face of the older sleep data.

it's entirely possible that their hypothesis will turn out to be incorrect.

That's really interesting, do they have that published anywhere?

I'll ask him for links to his papers, but the bottom of the corporate site has a bunch of links:

http://pulsarinformatics.com/fatigue-facts.html