..and my 1.6" thick 14" mirror rides on a mere 6-pointer, with the flotation points spaced at around 58% of radius.
The thing to do is download David Lewis's program PLOP and have a go at modelling the various options. I think you'll find that anything more than 6 points for a 12" mirror is overkill:
http://www.davidlewistoronto.com/plop/