Very well done and interesting essay. I most enjoyed the section on the mathematics of groups. This part of mathematics, in my opinion, plays the central role in our understanding of fundamental particles. With electrons/leptons being organized as SU(2) and quarks/hadrons showing properties that make them an SU(3) symmetry, the importance of these concepts in understanding fundamental particles cannot be overstated.
I hope you get a chance to have a look at my essay here. I model particles of the standard model, starting with objects that fall into S(3) and SO(3) groupings and continue with objects that are capable of being described with SU(2) and SU(3) symmetries.
Good luck in the essay and thank you for the great read.
Regards,
Ed Unverricht