I will cut a piece of brazillian amethyst in a trillion faceted shape in this video.   I use an ultra tec faceting machine  fitted with a digital angle readout.
This machine is an older v2 model and not being made  since around 2008.
For info on an ultratec  machine look here:
www.ultratec-facet.com

For faceting diagrams for different designs , look here:
www.pinterest.com
www.gemologyproject.com
wwwfacetingdiagrams.com

I use and reccomend gearloose lapidary products.
www.gearloose.com