(1) the hexagons of Section 3
H  : directions [30, 240, 90, 300, 150, 180] degrees, lengths [1, 1, (-1 + 2*sqrt3), 3, 1, 1]
   closes exactly; vertices as printed: (0.0000,0.0000), (0.8660,0.5000), (0.3660,-0.3660), (0.3660,2.0981), (1.8660,-0.5000), (1.0000,0.0000)
   signed turning angles at v_0..v_5 (degrees): [-150, -150, -150, -150, -150, 30], total -720
   intersecting pairs of non-adjacent edges (all proper crossings): e_0 x e_2 at (0.3660,0.2113), e_1 x e_5 at (0.5774,0.0000), e_2 x e_5 at (0.3660,0.0000)
H_1: directions [30, 240, 90, 300, 150, 180] degrees, lengths [2, 1, (-2 + 3*sqrt3), 5, 2, 2]
   closes exactly; vertices as printed: (0.0000,0.0000), (1.7321,1.0000), (1.2321,0.1340), (1.2321,3.3301), (3.7321,-1.0000), (2.0000,0.0000)
   signed turning angles at v_0..v_5 (degrees): [-150, -150, -150, -150, -150, 30], total -720
   intersecting pairs of non-adjacent edges (all proper crossings): e_0 x e_2 at (1.2321,0.7113)
   e_S(A*) = (pi/6)(20 - 4a + 4c) for all 32 odd sets; 0 for S = {0,..,4}, at least 4 pi/3 otherwise
(2) the forced directions of Section 3.3
   u_0..u_5 at [0, 210, 60, 270, 120, 150] degrees; signed turns [-150, -150, -150, -150, -150, 30]; u_0, -u_1, u_2, -u_3, u_4 at [0, 30, 60, 90, 120] degrees
(3) the family A_n (exact, units of pi)
   n = 6..12: the only tight odd set is {0,..,4}; all e_S >= 0; sum alpha = 13 pi/3;
   sum (pi - alpha_i) = (n-5) pi + 2 pi/3; forced directions 0, 210, 60, 270, 120, 120 + j*30/(n-5);
   signed turns (-150 x 5, +30/(n-5) x (n-5)) degrees, total -720
   n = 7..12: the polygon with the forced directions closes with positive lengths (mu = 0.05)
(4) the example after Lemma 4.3 (exact, units of pi)
   A = (9/10, 9/10, 1/10, 1/10, 1/10) pi: sum = 21/10 pi, min e_S = 3/10 pi, so A is in G_5;
   s = 0 (the two large angles): (L, U) = (17/10, 1/5) pi, empty;  s = 2: (L, U) = (1/10, 1/5) pi
(5) the pseudo-triangle of Figure 4 (degrees; floating point)
   alpha = [155, 20, 20, 160, 30, 155, 25, 15]  S = [0, 3, 5]  rho = {(0, 3): 40, (3, 5): 30, (5, 0): 40}  Theta = {0: 65.0, 3: 55.0, 5: 60.0}
   chain 0 -> 3: directions ['20.0', '0.0', '-20.0'], lengths ['2.3412', '1.6000', '2.3412'], apex of D: (3.0000,1.0919)
   chain 3 -> 5: directions ['140.0', '110.0'], lengths ['3.2503', '3.2503'], apex of D: (3.5101,2.0892)
   chain 5 -> 0: directions ['-95.0', '-120.0', '-135.0'], lengths ['2.4560', '1.4000', '2.0993'], apex of D: (2.1353,2.1353)
   vertices v_0..v_7: (0.0000,0.0000) (2.2000,0.8007) (3.8000,0.8007) (6.0000,0.0000) (3.5101,2.0892) (2.3985,5.1435) (2.1844,2.6968) (1.4844,1.4844)
   signed turning angles: +alpha_i for i in S, -alpha_i otherwise; total 360; simple
ALL CHECKS PASSED
