# (period, big-arc hits, small-arc hits, segment hits): number of starts;  None = not closed within the cap
Source Example 3: (1/4, 1/sqrt(30)): rho1=0.250000000000 rho2=0.182574185835
   exact reduced map (cap 60000 chords): [((6, 4, 0, 2), 31), (None, 89)]
   float billiard   (cap 60000 reflections): [((6, 4, 0, 2), 31), (None, 89)]
   start-by-start mismatches: 0
Companion paper 4.3: ((5-sqrt5)/10, sqrt5/10): rho1=0.276393202250 rho2=0.223606797750
   exact reduced map (cap 60000 chords): [((14, 6, 6, 2), 37), ((4, 2, 2, 0), 83)]
   float billiard   (cap 60000 reflections): [((14, 6, 6, 2), 37), ((4, 2, 2, 0), 83)]
   start-by-start mismatches: 0
my choice: rho1+rho2 = 1/4, alpha = sqrt2-1, lam = 3/2-sqrt2 : rho1=0.207106781187 rho2=0.042893218813
   exact reduced map (cap 60000 chords): [((102, 44, 44, 14), 38), ((28, 12, 12, 4), 82)]
   float billiard   (cap 60000 reflections): [((102, 44, 44, 14), 38), ((28, 12, 12, 4), 82)]
   start-by-start mismatches: 0
Companion paper 8.2: r1=5/11+1/(22pi), r2=5/11-1/(220pi) (r1+10 r2=5), float billiard, cap 100000: [(None, 60)]
