Origami had a two growth spurts in the 20th century. The first centered around a man named Akira Yoshizawa. First, Yoshizawa created origami models that were more anatomically complete than most previously made. While previous models may have represented, for instance, a horse, by only three legs and a head (using the four corners of the square sheet of paper), for Yoshizawa a horse must have four legs, a head, and a tail. By making legs/points from the center and edges of the square (still without cutting), Yoshizawa was able to create crabs, octopus, spiders, etc. with the correct number of limbs.

Related to this striving for realism in depiction of natural subjects within the constraints of the uncut square, Yoshizawa imbued his models with the curves and organic shapes of the natural objects they were meant to represent. Whereas traditional models were geometric, abstract approximations of animals using creases and the polygonal planes they formed, Yoshizawa used softer folds and curves in carefully sculpted proportions, which enabled not only realistic, three-dimensional depiction, but also expressive potential that had not been a part of traditional origami. He did this in part through wet-folding, which is exactly what it sounds like -- you dampen the paper, fold and curve it, then when dry it holds the shapes you've given it like hardened clay. Hoàng Tiến Quyết is a good example of a contemporary origami artist advancing this aesthetic of expressive folding.

Yoshizawa also helped to codify a system of notation that allowed the communication of instructions (we call them diagrams) for these complex models so we could fold them too! It's hard to overstate how grateful we who love origami are to Yoshizawa. Through his models he showed what was possible -- a level of realism and expression that raised origami from craft to art. Through his published diagrams he allowed the rest of us to attempt to reproduce some of his works, and to develop an understanding of their structure.

Part 2 of the growth spurt is just the result of this increasing understanding of origami structure. Mathematicians began to formalize the rules governing, for instance, a pattern of creases that can be folded flat, and the constraints on the number and length of flaps (e.g. legs for origami animals) that can be folded out of an uncut square. Armed with this increasing mathematical understanding, in the 1990s the members of the Origami Tanteidan ("Origami Detectives") undertook the "Bug Wars," a friendly competition among expert origamists striving to best each other in producing origami insects and arthropods of increasing complexity and realism, with ever more and longer legs, antennae, wings, etc. Bugs have continued to be a favorite subject of modern origami because of the enjoyable challenge they pose to the designer and the folder. Satoshi Kamiya is one of the foremost practitioners of origami complexity, and Ryujin is his most famous creation. I recommend Robert Lang's Origami Design Secrets to those interested in the intersection of mathematics and origami design.