The Farey Experience
Recall that the Farey Series FN is the set of all
fractions in lowest terms between 0 and 1 whose denominators do
not exceed N, arranged in order of magnitude.
For example,
F1: 
, 
F2: 
, 
, 
You complete the following:
F3: 
, 
, 
, 
, 
F4: 
, 
, 
, 
, 
, 
, 
F5: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
F6: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
Look how these previous problems
progressed. Can you come up with a
general rule to move from FN-1 to FN? Is your rule is truth or tale?
Complete the following:
F7: 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
Can you establish a
general rule for determining the number of fractions added when moving from FN-1
to FN when N is prime? Are
you on the right track?
Do not be weary this journey is not over. The following excursions will lead you
to discover how to construct FordÕs Touching Circles on given points within the
Farey Series.
The first task is to
trisect a line to obtain 1/3 of a segment.
The second
task is to construct the corresponding circle at the point of rational number 1/3, with a
diameter of 1/3^2.
Recall FordÕs Touching Circles are such that for any rational number p/q, draw
a circle of diameter 1/q^2. See a possible construction.
Using this method it is possible
to construct a circle corresponding to any rational number in the Farey
Series. Consecutive
circles will be tangent.
You have encountered far more than
just a tale and because of this mathematical journey you will now liveÉÉÉÉÉÉ
Happily Ever After
