Petals Around the Rose: Fraternity Register

I suppose it's the engineers and analysis-types who get confused on this one. My guess is that most algebraists can figure it out rather quickly. I guessed the "technique" before a single roll. I really don't get how one can't make the connections when being told that the name is significant. It's all flowers and symmetry and using the "force". Then again, I think my boundless laziness is what allowed me to "see" the answer.