Continued fractions have been around for hundreds of years and are almost unknown by anyone but mathematicians. They are written in the form [a1; a2, a3, a4, a5, ....] = a1 + 1/(a2 + 1/(a3 + 1/(a4 + 1/(a5 + ... )))). One of the surprising things about continued fractions is the representation of irrational numbers. The golden ratio phi = [1; 1, 1, 1, 1, 1 ... ], the square root of two = [1; 2, 2, 2, 2, 2 ...] and the square root of three = [1; 1, 2, 1, 2, 1, 2 ...].
Fractals are a 20th century development. They are a way to express the “self similarity” often found in nature. The twig looks like the branch, which looks like the tree. The coastline of England seen from space looks like the water’s edge that is seen while standing on the beach, which looks like the edge of the water seen under a microscope. The study of fractals has produced some astounding mathematical objects and led to a deeper understanding of natural phenomena.
Group theory is often the beginning of graduate studies in mathematics and the simplest example of complex mathematical systems that don't deal with numbers. Groups are mathematical objects formed by a set in which one function is defined that obeys three simple rules. It is used to explain how objects can be positioned in space, blood groups and heartbeats. More recently, group theory has been used to find the shortest path solutions to the Rubik’s cube puzzle.
Quaternions are an extension of complex numbers. They were an unusable curiosity – too complex to be useful – until the 20th century invention of Computer Generated Imagery. Spaceships and monsters are described as arrays of quaternions, and rotating the object in space is simply a matter of multiplying the array of quaternions by another array that describes the rotation. The object is only drawn once and in subsequent frames, it's moved mathematically.