104 Lattice polygons
1.3 The Farey sequence ..................
Letnbe a positive integer. The Farey sequence of ordernis the sequence
of rational numbers in[0,1]of denominators≤narranged in increasing
order. We write0=^01 and1=^11.
F 1 :^01 ,^11.
F 2 :^01 ,^12 ,^11.
F 3 :^01 ,^13 ,^12 ,^23 ,^11.
F 4 :^01 ,^14 ,^13 ,^12 ,^23 ,^34 ,^11.
F 5 :^01 ,^15 ,^14 ,^13 ,^25 ,^12 ,^35 ,^23 ,^34 ,^45 ,^11.
F 6 :^01 ,^16 ,^15 ,^14 ,^13 ,^25 ,^12 ,^35 ,^23 ,^34 ,^45 ,^56 ,^11.
F 7 :^01 ,^17 ,^16 ,^15 ,^14 ,^27 ,^13 ,^25 ,^37 ,^12 ,^47 ,^35 ,^23 ,^57 ,^34 ,^45 ,^56 ,^67 ,^11.
Theorem 1.3. 1.Ifhkandh
′
k′are successive terms ofFn, then
kh′−hk′=1 and k+k′>n.
2.Ifhk,h′
k′, andh′′
k′′are three successive terms ofFn, then
h′
k′=
h+h′′
k+k′′.
The rational numbers in[0,1]can be represented by lattice points in
the first quadrant (below the liney = x). Restricting to the left side
of the vertical linex= n, we can record the successive terms of the
Farey sequenceFnby rotating a ruler about 0 counterclockwise from the
positivex-axis. The sequence of “visible” lattice points swept through
corresponds toFn.IfPandQare two lattice points such that triangle
OP Qcontains no other lattice points in its interior or boundary, then the
rational numbers corresponding toPandQare successive terms in a
Farey sequence (of order≤their denominators).
Corollary 1.4.A primitive lattice triangle has area^12.