alpha=sqrt2-1, lam=3/2-sqrt2 ; alpha+lam=1/2 : alpha=0.414213562373 lam=0.0857864376269 N=11 ; completely periodic (exact)=True
     family: G-period m1=m2=12 (m2=12), v=1, n=6, width=0.056349186, measure=0.67619023 ; billiard period 28 = 12+12+4 (large arc, small arc, segments)
     family: G-period m1=m2=22 (m2=22), v=2, n=11, width=0.014718626, measure=0.32380977 ; billiard period 102 = 44+44+14 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((102, 44, 44, 14), 19), ((28, 12, 12, 4), 41)] -> all in the exact list: True
lam=(sqrt3-1)/40 ; alpha+lam=2/3 : alpha=0.648365396477 lam=0.0183012701892 N=54 ; completely periodic (exact)=True
     family: G-period m1=m2=165 (m2=165), v=3, n=110, width=0.0039104699, measure=0.64522754 ; billiard period 434 = 165+165+104 (large arc, small arc, segments)
     family: G-period m1=m2=54 (m2=54), v=1, n=36, width=0.0065698604, measure=0.35477246 ; billiard period 142 = 54+54+34 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((142, 54, 54, 34), 20), ((434, 165, 165, 104), 40)] -> all in the exact list: True
lam=(sqrt5-2)/50 ; alpha+lam=3/7 : alpha=0.423850069021 lam=0.00472135955 N=211 ; completely periodic (exact)=True
     family: G-period m1=m2=6566 (m2=6566), v=31, n=2814, width=8.0236554e-5, measure=0.52683322 ; billiard period 15884 = 6566+6566+2752 (large arc, small arc, segments)
     family: G-period m1=m2=7413 (m2=7413), v=35, n=3177, width=6.3829325e-5, measure=0.47316678 ; billiard period 35866 = 14826+14826+6214 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
lam=(3-2sqrt2)/100 ; alpha+lam=5/11 : alpha=0.452829725793 lam=0.00171572875254 N=582 ; completely periodic (exact)=True
     family: G-period m1=m2=9497422 (m2=9497422), v=16295, n=4317010, width=6.5336632e-8, measure=0.62052956 ; billiard period 23279264 = 9497422+9497422+4284420 (large arc, small arc, segments)
     family: G-period m1=m2=10761025 (m2=10761025), v=18463, n=4891375, width=3.526341e-8, measure=0.37947044 ; billiard period 52752998 = 21522050+21522050+9708898 (large arc, small arc, segments)
lam=sqrt7/1000 ; alpha+lam=1/3 : alpha=0.330687582022 lam=0.00264575131106 N=377 ; completely periodic (exact)=True
     family: G-period m1=m2=378 (m2=378), v=1, n=126, width=0.0025517557, measure=0.96456367 ; billiard period 880 = 378+378+124 (large arc, small arc, segments)
     family: G-period m1=m2=1131 (m2=1131), v=3, n=377, width=3.1331861e-5, measure=0.035436335 ; billiard period 5266 = 2262+2262+742 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((5266, 2262, 2262, 742), 1), ((880, 378, 378, 124), 59)] -> all in the exact list: True
lam=(sqrt13-3)/7 ; alpha+lam=7/9 : alpha=0.691270452711 lam=0.0865073250663 N=11 ; completely periodic (exact)=True
     family: G-period m1=m2=81 (m2=81), v=7, n=63, width=0.011119671, measure=0.90069337 ; billiard period 422 = 162+162+98 (large arc, small arc, segments)
     family: G-period m1=m2=126 (m2=126), v=11, n=98, width=0.00078814782, measure=0.099306625 ; billiard period 328 = 126+126+76 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((328, 126, 126, 76), 5), ((422, 162, 162, 98), 55)] -> all in the exact list: True
lam=sqrt2/3 (rho1+rho2=1/2) ; alpha+lam=1/1 : alpha=0.528595479209 lam=0.471404520791 N=2 ; completely periodic (exact)=True
     family: G-period m1=m2=3 (m2=3), v=1, n=3, width=0.057190958, measure=0.17157288 ; billiard period 14 = 6+6+2 (large arc, small arc, segments)
     family: G-period m1=m2=2 (m2=2), v=1, n=2, width=0.41421356, measure=0.82842712 ; billiard period 4 = 2+2+0 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((14, 6, 6, 2), 13), ((4, 2, 2, 0), 47)] -> all in the exact list: True
lam=(5sqrt5-6)/10 (N=1) ; alpha+lam=13/10 : alpha=0.78196601125 lam=0.51803398875 N=1 ; completely periodic (exact)=True
     family: G-period m1=m2=60 (m2=60), v=31, n=78, width=0.0098300563, measure=0.58980338 ; billiard period 136 = 60+60+16 (large arc, small arc, segments)
     family: G-period m1=m2=50 (m2=50), v=26, n=65, width=0.0082039325, measure=0.41019662 ; billiard period 226 = 100+100+26 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((136, 60, 60, 16), 39), ((226, 100, 100, 26), 21)] -> all in the exact list: True
lam=sqrt(991)/3148 ; alpha+lam=4/13 : alpha=0.297692259256 lam=0.0100000484363 N=99 ; completely periodic (exact)=True
     family: G-period m1=m2=1300 (m2=1300), v=13, n=400, width=0.00076811673, measure=0.99855176 ; billiard period 2974 = 1300+1300+374 (large arc, small arc, segments)
     family: G-period m1=m2=299 (m2=299), v=3, n=92, width=4.8436275e-6, measure=0.0014482446 ; billiard period 684 = 299+299+86 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((2974, 1300, 1300, 374), 60)] -> all in the exact list: True
lam=pi-3 ; alpha+lam=1/2 : alpha=0.35840734641 lam=0.14159265359 N=7 ; completely periodic (exact)=True
     family: G-period m1=m2=8 (m2=8), v=1, n=4, width=0.0088514249, measure=0.070811399 ; billiard period 18 = 8+8+2 (large arc, small arc, segments)
     family: G-period m1=m2=14 (m2=14), v=2, n=7, width=0.066370614, measure=0.9291886 ; billiard period 62 = 28+28+6 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((18, 8, 8, 2), 4), ((62, 28, 28, 6), 56)] -> all in the exact list: True
lam=(pi-3)/9 ; alpha+lam=5/7 : alpha=0.69855319722 lam=0.0157325170655 N=63 ; completely periodic (exact)=True
     family: G-period m1=m2=448 (m2=448), v=7, n=320, width=0.0012644893, measure=0.56649119 ; billiard period 1202 = 448+448+306 (large arc, small arc, segments)
     family: G-period m1=m2=63 (m2=63), v=1, n=45, width=0.0068810922, measure=0.43350881 ; billiard period 338 = 126+126+86 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((1202, 448, 448, 306), 30), ((338, 126, 126, 86), 30)] -> all in the exact list: True
lam=2^(1/3)-1 ; alpha+lam=3/4 : alpha=0.490078950105 lam=0.259921049895 N=3 ; completely periodic (exact)=True
     family: G-period m1=m2=104 (m2=104), v=27, n=78, width=0.0019737526, measure=0.20527027 ; billiard period 232 = 104+104+24 (large arc, small arc, segments)
     family: G-period m1=m2=100 (m2=100), v=26, n=75, width=0.0079472973, measure=0.79472973 ; billiard period 446 = 200+200+46 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((232, 104, 104, 24), 18), ((446, 200, 200, 46), 42)] -> all in the exact list: True
lam=(2^(1/3)-1)/30 ; alpha+lam=2/5 : alpha=0.391335965004 lam=0.0086640349965 N=115 ; completely periodic (exact)=True
     family: G-period m1=m2=48130 (m2=48130), v=417, n=19252, width=1.8724031e-5, measure=0.9011876 ; billiard period 114678 = 48130+48130+18418 (large arc, small arc, segments)
     family: G-period m1=m2=112765 (m2=112765), v=977, n=45106, width=8.762683e-7, measure=0.098812395 ; billiard period 268682 = 112765+112765+43152 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
lam=(10e-27)/10 ; alpha+lam=1/3 : alpha=0.315051504874 lam=0.018281828459 N=54 ; completely periodic (exact)=True
     family: G-period m1=m2=165 (m2=165), v=3, n=55, width=0.0042604211, measure=0.70296948 ; billiard period 758 = 330+330+98 (large arc, small arc, segments)
     family: G-period m1=m2=54 (m2=54), v=1, n=18, width=0.0055005652, measure=0.29703052 ; billiard period 124 = 54+54+16 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((124, 54, 54, 16), 21), ((758, 330, 330, 98), 39)] -> all in the exact list: True
lam=L (Liouville number sum 10^-k!) ; alpha+lam=1/2 : alpha=0.389999 lam=0.110001 N=9 ; completely periodic (exact)=True
     family: G-period m1=m2=28 (m2=28), v=3, n=14, width=0.009991, measure=0.279748 ; billiard period 64 = 28+28+8 (large arc, small arc, segments)
     family: G-period m1=m2=18 (m2=18), v=2, n=9, width=0.040014, measure=0.720252 ; billiard period 82 = 36+36+10 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((64, 28, 28, 8), 22), ((82, 36, 36, 10), 38)] -> all in the exact list: True
lam=L/7 ; alpha+lam=3/8 : alpha=0.359285571429 lam=0.0157144285714 N=63 ; completely periodic (exact)=True
     family: G-period m1=m2=64 (m2=64), v=1, n=24, width=0.009991, measure=0.639424 ; billiard period 150 = 64+64+22 (large arc, small arc, segments)
     family: G-period m1=m2=504 (m2=504), v=8, n=189, width=0.00071542857, measure=0.360576 ; billiard period 2362 = 1008+1008+346 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((150, 64, 64, 22), 38), ((2362, 1008, 1008, 346), 22)] -> all in the exact list: True
lam=w/3, w=[0;3,10^6,2,10^9,1,10^12,5,1,1,...] ; alpha+lam=2/3 : alpha=0.555555592593 lam=0.111111074074 N=9 ; completely periodic (exact)=True
     family: G-period m1=m2=30 (m2=30), v=3, n=20, width=1.1111102e-7, measure=3.3333306e-6 ; billiard period 74 = 30+30+14 (large arc, small arc, segments)
     family: G-period m1=m2=9 (m2=9), v=1, n=6, width=0.11111074, measure=0.99999667 ; billiard period 22 = 9+9+4 (large arc, small arc, segments)
     direct exact orbit following: complete=True, same families=True
     float billiard, 60 random starts (both senses): [((22, 9, 9, 4), 60)] -> all in the exact list: True
