Alpha Gamma Delta's badge was orginally the opposite of what it is now. What it is now is the Delta at the back, with a Gamma on top of that, and at the front is the Alpha.
Everyone has to buy their badge, and the most basic is just gold. You can also get a diamond or pearl in the center, or you can get it with all pearls (14 of them) on it. There is one that is all diamonds, but the International president is the only person allowed to wear it.
For a pic, you can go here:
http://www.burrpatt.com/alphagammadelta/AGDBadges.asp