The
FareyFord tessellation and circle packing
The animated circles on the
home page of
my web site represent one of
the most beautiful objects in mathematics, the combination of the
FareyFord tessellation
and circle packing. Here is a quick description.
Consider the number line as the xaxis in the plane, and look at all
the rationals p/q on this number line. (Recall that a rational number
is one
that can be written as the quotient p/q of two integers p and q). For
every such rational number p/q, draw a circle C
_{p/q} of
diameter 1/q
^{2} that touches the xaxis exactly at p/q, and
sits above this xaxis. This looks like a strange idea, but observe the
outcome.


3/1

5/2

2/1

3/2

1/1

1/2

0/1

1/2

1/1

3/2

2/1

5/2

3/1

We see that the circles C_{p/q} touch each other in a funny
pattern. It is here convenient to add the top horizontal line of the
picture, and to consider it as the "circle" C_{1/0} of infinite
radius that touches the xaxis at the infinity point 1/0. This
collection of circles, with disjoint interiors, is the FareyFord circle packing.
We will zoom in to investigate these patterns, but first let us add
some features. Whenever the circles C_{p/q}
and C_{r/s} touch each other, connect the numbers p/q and r/s
by a semicircle centered on the xaxis, as in


3/1

5/2

2/1

3/2

1/1

1/2

0/1

1/2

1/1

3/2

2/1

5/2

3/1

Note that these semicircles pass exactly through the points where the
circles C_{p/q}
and C_{r/s} touch each other. The color coding is a little
difficult to explain here, but we can observe that the
semicircles decompose the halfplane sitting above the xaxis into
curved triangles, each delimited by three semicircles of different
colors. This decomposition of the upper halfplane is known as the FareyFord tessellation.
("Tessellation" is a fancy mathematical word for "tiling", and
mathematicians think of the above triangles as similar to the tiles of
a kitchen floor.)
Let's zoom between 0 and 1/2 in the above picture. We obtain

0/1

1/7

1/6

1/5

2/9

1/4

2/7

1/3

3/8

2/5

3/7

1/2

and then again


1/4

5/19

4/15

3/11

5/18

2/7

5/17

3/10

4/13

5/16

1/3

if we zoom some more between 1/4 and 1/3.
What we observe (and can be proved) is the following.
 The circles C_{p/q} and C_{r/s} touch each other
exactly when psqr=1.
 When three circles C_{p/q} , C_{r/s} and C_{t/u}
with p/q < r/s < t/u, then r = p+t and s = q+u. Namely, the
middle term r/s = (p+t)/(q+u) is the "freshman addition" of the other
two terms p/q
and t/u (I hope you know that this is not the way you are supposed to
add fractions).
The above "addition" of fractions is
known as the
Farey addition,
after John Farey (1766–1826) who experimentally discovered a similar
phenomenon when one lists all rational number p/q with q less that a
given number (and knew nothing about circle packings or tessellations).
The real discoverer (in 1928) of the FareyFord tessellation and circle
packing is Lester Randolph Ford (1886–1967). The FareyFord
tessellation and circle packing have many more interesting properties
and appear in many branches
of mathematics, from hyperbolic noneuclidean geometry to number
theory. One reason is that they are closely related to the group SL
_{2}(
Z) of 2by2 matrices with integer
coefficients and determinant 1. Want to know more? An excellent
reference is of course Chapter 8 of my book
Lowdimensional geometry: from euclidean
surfaces to hyperbolic knots.
Now, you are going to say that this is not exactly the original
picture of circles moving within a circle. For this, we need to deform
the halfplane delimited by the xaxis into the interior of the circle
of radius 1 centered at the origin. The map f from the plane to itself
defined by
f(x,y) = ( (x^{2}+y^{2}1)/(x^{2}+y^{2}+1)
,
2x/(x^{2}+y^{2}+1) )
does just that. In fact, f sends every circle in the plane to a circle
(interpreting a straight line as a circle of infinite radius), and in
particular sends the upper halfplane to the interior of the unit
circle. If we draw the images under f of the circles and semicircles
of the FareyFord tessellation and circle packing, we obtain
which looks more like what we wanted.
Incidentally, the map f is better expressed in terms of the complex
coordinate z = x+iy, where it becomes
f(z) = (zi)/(z+i).
In hyperbolic geometry, it provides an isometry between the upper
halfplane model and the disk model for the hyperbolic plane.
Wait, wait! These circles are not moving. Well, here we jump from
mathematics to art. The FareyFord tessellation and circle packing are
invariant under horizontal translation of 1, and our coloring scheme
under horizontal translation of 3. If we perform these translations in
small increments, and combine them in an animated file, we get
We now just need to apply the above map f to go from the upper
halfplane to the interior of the circle,
et voilà. The mathematical
content is the same as before, but it's clearly prettier.