Reddit mentions of Project Origami

oh shoot I've waited to answer something like this, I used to do stuff in here

1. The Huzita-Hatori axioms. Pretty fundamental in orimath (haven't found a better name for it yet) and you can do some cool things in it such as trisecting an angle that you can't do using a compass. My main man Robert Lang, the Dr. Manhattan of origami, proved the 7th axiom.
   
   
2. I'm not sure if I have the right link here but there should be a course online about combinatorial geometry that heavily involves origami
   
   
3. I'm aware that Thomas Hull wrote a book on a related topic here
   
   
4. Kawasaki's theorem is interesting.
   
   I'd say you're looking mostly at things in combinatorial geometry when it comes to understanding this area, although this paper involves some abstract algebra as well.
   
   As the LovepeaceandStarTrek mentioned, Erik Demaine's a wizard at this stuff. He's a guru at MIT who I think taught a course or two on designing the kinds of things he comes up with - WAIT! I found it, here
   
   Aside from all of that, there's this and this

Check out Thomas Hull, Robert Lang, or Joseph O'Rourke. Ex:

• http://www.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1466567910
  
• http://www.amazon.com/How-Fold-It-Mathematics-Polyhedra/dp/0521145473
  
• http://www.langorigami.com/

I'm a huge origami fan. Great books for teachers:

https://smile.amazon.com/Origamics-Mathematical-Explorations-Through-Folding/dp/9812834907/ref=mp_s_a_1_2?ie=UTF8&qid=1538344551&sr=8-2&pi=AC_SX236_SY340_FMwebp_QL65&keywords=origami+explorations+math&dpPl=1&dpID=41HUKVEXbXL&ref=plSrch

https://smile.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1466567910/ref=mp_s_a_1_9?ie=UTF8&qid=1538344609&sr=8-9&pi=AC_SX236_SY340_FMwebp_QL65&keywords=math+origami&dpPl=1&dpID=515Ywuc4QYL&ref=plSrch