Continued fractions are a way to represent numbers as . They appear in various areas of mathematics, from number theory to dynamical systems to geometry. I will describe functions which generate these fractions. The continued fraction expansions provide a nice description of paths on the geometric surfaces. The geometric properties of these pictures help us to describe patterns in the continued fraction expansions, and the continued fraction expansions provide a compact description of the geometry. Various types of continued fractions correspond to different rules for the numerators and denominators. The first example is the “regular” or “simple” continued fractions, where all of the numerators, \(e_i\) , are 1. I then show how the dynamics and geometry change when all of the numerators are \(\pm 1\) and the denominators, \(a_i\) , are all even. Mathematical Background: Since I am using functions to generate continued fractions, you should be comfortable with function composition, as well as preimages of points. I will refer to functions that are onto (every point has a preimage) and one-to-one (different points have different images) as invertible. There is also a lot of adding and multiplying fractions and use of the fact that . For the geometry, you should be comfortable with rotating objects in space. Complex numbers will also show up throughout. If you are unfamiliar with complex variables, you can think of the point \(z=x+iy\) as \((x,y)\) in the plane. Circles of radius r centered at \(x+iy\) in the complex plane can be written as \(x+iy+ r e^{it}\) where \(t\in (0,2\pi )\) . Here, all of our circles are centered on the x-axis, so \(y=0\) .