So infinity is an important part of being a fractal. This means that parts of the fractal continue to get smaller into infinity or that the length of the fractal grows to infinity. On and on and on... infinitely.
The Koch Curve is one of the earliest fractal curves and is another example of infinity.
The curve begins with a straight line that is divided into three equal segments. An equilateral triangle is created over the center segment and then the bottom of that triangle is removed. This is the first generation of a Koch Curve. Being the first generation means that this is the shape, the form, that will be used to create the rest of the curve. This segment will replace every line in the original generation.
Eventually the length of the Koch Curve will become infinite because one needs to measure each segment and add it to the total length of the curve.