The Farey-Ford tessellation
This animation by Francis Bonahon shows the Farey-Ford tessellation of the unit disc. It can be constructed using basic properties of fractions.
Each of the internal dark blue circles touches the big outside circle at precisely one point on the boundary. These boundary points are joined to each other by curves that are coloured red, yellow or cyan. The red, yellow and cyan curves divide up the disc into hyperbolic triangles, which differ from ordinary Euclidean triangles in that their angles add up to less than 180 degrees. These hyperbolic triangles tessellate (i.e., tile) the interior of the big circle, and the red, yellow and cyan colours are assigned in such a way that each hyperbolic triangle has one side of each colour.
The construction of the tessellation is based on the properties of rational numbers (fractions). Recall that each rational number can be written as a fraction in lowest terms, p/q, where p and q are integers, q is greater than zero, and p and q have no common factors bigger than 1. For example, the fractions 3/7 and 6/14 both represent the same number, but the representation 3/7 is the one that satisfies the above conditions. The rational number 0 will be represented as 0/1.
If p/q is a fraction (in lowest terms), Â we can think of it as a point on the real line. We can then draw a dark blue circle of radius 1/(q^2) that sits above the real line and touches it only at the point p/q. This circle will be denoted by C(p/q). Remarkably, if we do this for every fraction, the circles fit together neatly. This is called the Farey-Ford circle packing, and there is a picture of it here (http://goo.gl/9utvPB). The horizontal line y=1 in the picture is considered to be a circle of infinite radius centred at the point at infinity, 1/0.
There is a simple rule to tell whether the circles C(p/q) and C(r/s) touch each other: they touch each other if and only if the absolute value of the number ps-qr is equal to 1. For example, the circles C(2/7) and C(3/11), which touch the real axis at the points 2/7 and 3/11 respectively, also touch each other, because (2x11)-(7x3) is equal to +1. If two circles touch each other, then the boundary points at which the two circles touch the real axis are then connected to each other by a semicircular arc, coloured red, yellow or cyan. There is a picture of this here (http://goo.gl/1AX48r). The big dark blue circles on the left and the right of the picture are C(1/4) and C(1/3), and the third biggest dark blue circle in the middle is C(2/7).
It is also possible for three circles to be mutually touching, which is what gives rise to the hyperbolic triangles in the tessellation. It can be shown that if p/q < r/s < t/u and the circles C(p/q), C(r/s) and C(t/u) all touch each other, then we must have (p+t)/(q+u) = r/s. Notice that this is how we teach students not to add fractions: we have simply added the numerators together and then added the denominators together! For example, the circles C(1/4), C(3/11) and C(2/7) all touch each other, because 1/4 < 3/11 < 2/7, and 3/11 = (1+2)/(4+7). This âwrongâ kind of addition is called Farey addition, after John Farey (1766-1826). The tessellation itself was discovered by Lester Randolph Ford (1886-1967).
If we now identify the point (x,y) in the plane with the complex number x+iy (where i is a square root of -1) then we can transform the upper half plane to a disc of radius 1 centred at the origin by the transformation f(z)=(i-z)/(z+i). This transformation has the property that it sends circlines to circlines, which means that any circle or straight line will be sent under the transformation to some other circle or straight line. The main picture shows the aforementioned disc of radius 1. The static point on the left of the circle is the point at infinity, and the big static dark blue circle on the left of the picture is the line y=1 mentioned above.
But what about the animation? Well, if q is the denominator of a fraction p/q in lowest terms, then adding an integer to p/q will produce another fraction with the same denominator (in lowest terms). Since the size of each circle is only determined by the denominator of the corresponding fraction, it follows that the Farey-Ford circle packing is periodic under shifting to the left or to the right by an integer distance. It is therefore possible to animate the circle packing on the real line by gradually moving it to the right. You can see an animation of that here (http://goo.gl/l79WBQ). Applying the transformation f(z) above then yields the animation in the main picture.
Picture credit: All the pictures mentioned above come from Francis Bonahon's web page at the University of Southern California (http://www-bcf.usc.edu/~fbonahon/STML49/FareyFord.html). His web page also contains a more detailed discussion of the topics in this post.
