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 Ford’s Touching Circles on given points within the Farey Series.

 

 

The first task is to trisect a line to obtain 1/3 of a segment.

Did you get it to work?

 

 

 

 

 

 

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 Ford’s 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 live………………

 

Happily Ever After

 

 

 

 

 

 

 

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