60の倍数の角度を持つ三角形の整列 ～ ナンバリング
60の倍数の角度を持つ三角形を通し番号付きで列挙します。
番号は元々(0,1)区間の有理数に与えられたものですが、これまで書いてきたように、(0,1)区間の有理数と60の倍数を持つ三角形を無理矢理1対1対応させ、流用したものです。
各行最後に４つの数字が載っていて、そのうち一つが括弧で囲まれています。これを除いた三つの数字で三角形を作ると、60度または120度または180度の三角形が作れます。行中の
「鋭」は60度の角度を持つ鋭角三角形
「純」は60度の角度を持つ鈍角三角形
「重」は120度の角度を持つ（鈍角）三角形
「退」はつぶれた（縮退した）三角形
を表しています。
1 : t = 1/2 , x = -1/4 = -0.250 , 鈍 { 3 ,( 5), 7 , 8 }
2 : t = 1/3 , x = -1/1 = -1.000 , 重 { 0 , 1 , 1 ,( 1)}
3 : t = 2/3 , x = 1/4 = 0.250 , 鋭 {( 3), 5 , 7 , 8 }
4 : t = 1/4 , x = -13/8 = -1.625 , 重 { 5 , 16 , 19 ,( 21)}
5 : t = 2/5 , x = -13/20 = -0.650 , 鈍 { 7 ,( 33), 37 , 40 }
6 : t = 3/5 , x = 1/15 = 0.067 , 鋭 {( 7), 8 , 13 , 15 }
7 : t = 3/4 , x = 11/24 = 0.458 , 鋭 {( 13), 35 , 43 , 48 }
8 : t = 1/5 , x = -11/5 = -2.200 , 重 { 3 , 5 , 7 ,( 8)}
9 : t = 2/7 , x = -37/28 = -1.321 , 重 { 9 , 56 , 61 ,( 65)}
10 : t = 3/8 , x = -37/48 = -0.771 , 鈍 { 11 ,( 85), 91 , 96 }
11 : t = 3/7 , x = -11/21 = -0.524 , 鈍 { 5 ,( 16), 19 , 21 }
12 : t = 4/7 , x = -1/56 = -0.018 , 鈍 { 55 ,( 57), 97 , 112 }
13 : t = 5/8 , x = 11/80 = 0.138 , 鋭 {( 69), 91 , 139 , 160 }
14 : t = 5/7 , x = 13/35 = 0.371 , 鋭 {( 11), 24 , 31 , 35 }
15 : t = 4/5 , x = 23/40 = 0.575 , 鋭 {( 17), 63 , 73 , 80 }
16 : t = 1/6 , x = -11/4 = -2.750 , 重 { 7 , 8 , 13 ,( 15)}
17 : t = 2/9 , x = -23/12 = -1.917 , 重 { 11 , 24 , 31 ,( 35)}
18 : t = 3/11 , x = -47/33 = -1.424 , 重 { 7 , 33 , 37 ,( 40)}
19 : t = 3/10 , x = -73/60 = -1.217 , 重 { 13 , 120 , 127 ,( 133)}
20 : t = 4/11 , x = -73/88 = -0.830 , 鈍 { 15 ,( 161), 169 , 176 }
21 : t = 5/13 , x = -47/65 = -0.723 , 鈍 { 9 ,( 56), 61 , 65 }
22 : t = 5/12 , x = -23/40 = -0.575 , 鈍 { 17 ,( 63), 73 , 80 }
23 : t = 4/9 , x = -11/24 = -0.458 , 鈍 { 13 ,( 35), 43 , 48 }
24 : t = 5/9 , x = -1/15 = -0.067 , 鈍 { 7 ,( 8), 13 , 15 }
25 : t = 7/12 , x = 1/56 = 0.018 , 鋭 {( 55), 57 , 97 , 112 }
26 : t = 8/13 , x = 23/208 = 0.111 , 鋭 {( 185), 231 , 361 , 416 }
27 : t = 7/11 , x = 13/77 = 0.169 , 鋭 {( 32), 45 , 67 , 77 }
28 : t = 7/10 , x = 47/140 = 0.336 , 鋭 {( 93), 187 , 247 , 280 }
29 : t = 8/11 , x = 71/176 = 0.403 , 鋭 {( 105), 247 , 313 , 352 }
30 : t = 7/9 , x = 11/21 = 0.524 , 鋭 {( 5), 16 , 19 , 21 }
31 : t = 5/6 , x = 13/20 = 0.650 , 鋭 {( 7), 33 , 37 , 40 }
32 : t = 1/7 , x = -23/7 = -3.286 , 退 { 7 , 8 ,( 13), 15 }
33 : t = 2/11 , x = -109/44 = -2.477 , 重 { 65 , 88 , 133 ,( 153)}
34 : t = 3/14 , x = -169/84 = -2.012 , 重 { 85 , 168 , 223 ,( 253)}
35 : t = 3/13 , x = -71/39 = -1.821 , 重 { 16 , 39 , 49 ,( 55)}
36 : t = 4/15 , x = -59/40 = -1.475 , 重 { 19 , 80 , 91 ,( 99)}
37 : t = 5/18 , x = -83/60 = -1.383 , 重 { 23 , 120 , 133 ,( 143)}
38 : t = 5/17 , x = -107/85 = -1.259 , 重 { 11 , 85 , 91 ,( 96)}
39 : t = 4/13 , x = -121/104 = -1.163 , 重 { 17 , 208 , 217 ,( 225)}
40 : t = 5/14 , x = -121/140 = -0.864 , 鈍 { 19 ,( 261), 271 , 280 }
41 : t = 7/19 , x = -107/133 = -0.805 , 鈍 { 13 ,( 120), 127 , 133 }
42 : t = 8/21 , x = -83/112 = -0.741 , 鈍 { 29 ,( 195), 211 , 224 }
43 : t = 7/18 , x = -59/84 = -0.702 , 鈍 { 25 ,( 143), 157 , 168 }
44 : t = 7/17 , x = -71/119 = -0.597 , 鈍 { 24 ,( 95), 109 , 119 }
45 : t = 8/19 , x = -169/304 = -0.556 , 鈍 { 135 ,( 473), 553 , 608 }
46 : t = 7/16 , x = -109/224 = -0.487 , 鈍 { 115 ,( 333), 403 , 448 }
47 : t = 5/11 , x = -23/55 = -0.418 , 鈍 { 16 ,( 39), 49 , 55 }
48 : t = 6/11 , x = -13/132 = -0.098 , 鈍 { 119 ,( 145), 229 , 264 }
49 : t = 9/16 , x = -13/288 = -0.045 , 鈍 { 275 ,( 301), 499 , 576 }
50 : t = 11/19 , x = 1/209 = 0.005 , 鋭 {( 104), 105 , 181 , 209 }
51 : t = 10/17 , x = 11/340 = 0.032 , 鋭 {( 329), 351 , 589 , 680 }
52 : t = 11/18 , x = 13/132 = 0.098 , 鋭 {( 119), 145 , 229 , 264 }
53 : t = 13/21 , x = 11/91 = 0.121 , 鋭 {( 40), 51 , 79 , 91 }
54 : t = 12/19 , x = 71/456 = 0.156 , 鋭 {( 385), 527 , 793 , 912 }
55 : t = 9/14 , x = 47/252 = 0.187 , 鋭 {( 205), 299 , 439 , 504 }
56 : t = 9/13 , x = 37/117 = 0.316 , 鋭 {( 40), 77 , 103 , 117 }
57 : t = 12/17 , x = 143/408 = 0.350 , 鋭 {( 265), 551 , 721 , 816 }
58 : t = 13/18 , x = 61/156 = 0.391 , 鋭 {( 95), 217 , 277 , 312 }
59 : t = 11/15 , x = 23/55 = 0.418 , 鋭 {( 16), 39 , 49 , 55 }
60 : t = 10/13 , x = 131/260 = 0.504 , 鋭 {( 129), 391 , 469 , 520 }
61 : t = 11/14 , x = 167/308 = 0.542 , 鋭 {( 141), 475 , 559 , 616 }
62 : t = 9/11 , x = 61/99 = 0.616 , 鋭 {( 19), 80 , 91 , 99 }
63 : t = 6/7 , x = 59/84 = 0.702 , 鋭 {( 25), 143 , 157 , 168 }
64 : t = 1/8 , x = -61/16 = -3.812 , 退 { 32 , 45 ,( 67), 77 }
65 : t = 2/13 , x = -157/52 = -3.019 , 退 { 104 , 105 ,( 181), 209 }
66 : t = 3/17 , x = -131/51 = -2.569 , 重 { 40 , 51 , 79 ,( 91)}
67 : t = 3/16 , x = -229/96 = -2.385 , 重 { 133 , 192 , 283 ,( 325)}
68 : t = 4/19 , x = -313/152 = -2.059 , 重 { 161 , 304 , 409 ,( 465)}
69 : t = 5/23 , x = -227/115 = -1.974 , 重 { 56 , 115 , 151 ,( 171)}
70 : t = 5/22 , x = -409/220 = -1.859 , 重 { 189 , 440 , 559 ,( 629)}
71 : t = 4/17 , x = -241/136 = -1.772 , 重 { 105 , 272 , 337 ,( 377)}
72 : t = 5/19 , x = -143/95 = -1.505 , 重 { 24 , 95 , 109 ,( 119)}
73 : t = 7/26 , x = -529/364 = -1.453 , 重 { 165 , 728 , 823 ,( 893)}
74 : t = 8/29 , x = -649/464 = -1.399 , 重 { 185 , 928 , 1033 ,( 1113)}
75 : t = 7/25 , x = -239/175 = -1.366 , 重 { 32 , 175 , 193 ,( 207)}
76 : t = 7/24 , x = -143/112 = -1.277 , 重 { 31 , 224 , 241 ,( 255)}
77 : t = 8/27 , x = -179/144 = -1.243 , 重 { 35 , 288 , 307 ,( 323)}
78 : t = 7/23 , x = -191/161 = -1.186 , 重 { 15 , 161 , 169 ,( 176)}
79 : t = 5/16 , x = -181/160 = -1.131 , 重 { 21 , 320 , 331 ,( 341)}
80 : t = 6/17 , x = -181/204 = -0.887 , 鈍 { 23 ,( 385), 397 , 408 }
81 : t = 9/25 , x = -191/225 = -0.849 , 鈍 { 17 ,( 208), 217 , 225 }
82 : t = 11/30 , x = -179/220 = -0.814 , 鈍 { 41 ,( 399), 421 , 440 }
83 : t = 10/27 , x = -143/180 = -0.794 , 鈍 { 37 ,( 323), 343 , 360 }
84 : t = 11/29 , x = -239/319 = -0.749 , 鈍 { 40 ,( 279), 301 , 319 }
85 : t = 13/34 , x = -649/884 = -0.734 , 鈍 { 235 ,( 1533), 1663 , 1768 }
86 : t = 12/31 , x = -529/744 = -0.711 , 鈍 { 215 ,( 1273), 1393 , 1488 }
87 : t = 9/23 , x = -143/207 = -0.691 , 鈍 { 32 ,( 175), 193 , 207 }
88 : t = 9/22 , x = -241/396 = -0.609 , 鈍 { 155 ,( 637), 727 , 792 }
89 : t = 12/29 , x = -409/696 = -0.588 , 鈍 { 287 ,( 1105), 1273 , 1392 }
90 : t = 13/31 , x = -227/403 = -0.563 , 鈍 { 88 ,( 315), 367 , 403 }
91 : t = 11/26 , x = -313/572 = -0.547 , 鈍 { 259 ,( 885), 1039 , 1144 }
92 : t = 10/23 , x = -229/460 = -0.498 , 鈍 { 231 ,( 689), 829 , 920 }
93 : t = 11/25 , x = -131/275 = -0.476 , 鈍 { 72 ,( 203), 247 , 275 }
94 : t = 9/20 , x = -157/360 = -0.436 , 鈍 { 203 ,( 517), 643 , 720 }
95 : t = 6/13 , x = -61/156 = -0.391 , 鈍 { 95 ,( 217), 277 , 312 }
96 : t = 7/13 , x = -11/91 = -0.121 , 鈍 { 40 ,( 51), 79 , 91 }
97 : t = 11/20 , x = -37/440 = -0.084 , 鈍 { 403 ,( 477), 763 , 880 }
98 : t = 14/25 , x = -37/700 = -0.053 , 鈍 { 663 ,( 737), 1213 , 1400 }
99 : t = 13/23 , x = -11/299 = -0.037 , 鈍 { 144 ,( 155), 259 , 299 }
100 : t = 15/26 , x = -1/780 = -0.001 , 鈍 { 779 ,( 781), 1351 , 1560 }
101 : t = 18/31 , x = 11/1116 = 0.010 , 鋭 {( 1105), 1127 , 1933 , 2232 }
102 : t = 17/29 , x = 13/493 = 0.026 , 鋭 {( 240), 253 , 427 , 493 }
103 : t = 13/22 , x = 23/572 = 0.040 , 鋭 {( 549), 595 , 991 , 1144 }
104 : t = 14/23 , x = 59/644 = 0.092 , 鋭 {( 585), 703 , 1117 , 1288 }
105 : t = 19/31 , x = 61/589 = 0.104 , 鋭 {( 264), 325 , 511 , 589 }
106 : t = 21/34 , x = 167/1428 = 0.117 , 鋭 {( 1261), 1595 , 2479 , 2856 }
107 : t = 18/29 , x = 131/1044 = 0.125 , 鋭 {( 913), 1175 , 1813 , 2088 }
108 : t = 17/27 , x = 23/153 = 0.150 , 鋭 {( 65), 88 , 133 , 153 }
109 : t = 19/30 , x = 61/380 = 0.161 , 鋭 {( 319), 441 , 661 , 760 }
110 : t = 16/25 , x = 143/800 = 0.179 , 鋭 {( 657), 943 , 1393 , 1600 }
111 : t = 11/17 , x = 37/187 = 0.198 , 鋭 {( 75), 112 , 163 , 187 }
112 : t = 11/16 , x = 107/352 = 0.304 , 鋭 {( 245), 459 , 619 , 704 }
113 : t = 16/23 , x = 239/736 = 0.325 , 鋭 {( 497), 975 , 1297 , 1472 }
114 : t = 19/27 , x = 59/171 = 0.345 , 鋭 {( 56), 115 , 151 , 171 }
115 : t = 17/24 , x = 97/272 = 0.357 , 鋭 {( 175), 369 , 481 , 544 }
116 : t = 18/25 , x = 347/900 = 0.386 , 鋭 {( 553), 1247 , 1597 , 1800 }
117 : t = 21/29 , x = 241/609 = 0.396 , 鋭 {( 184), 425 , 541 , 609 }
118 : t = 19/26 , x = 407/988 = 0.412 , 鋭 {( 581), 1395 , 1759 , 1976 }
119 : t = 14/19 , x = 227/532 = 0.427 , 鋭 {( 305), 759 , 949 , 1064 }
120 : t = 13/17 , x = 109/221 = 0.493 , 鋭 {( 56), 165 , 199 , 221 }
121 : t = 17/22 , x = 383/748 = 0.512 , 鋭 {( 365), 1131 , 1351 , 1496 }
122 : t = 18/23 , x = 443/828 = 0.535 , 鋭 {( 385), 1271 , 1501 , 1656 }
123 : t = 15/19 , x = 157/285 = 0.551 , 鋭 {( 64), 221 , 259 , 285 }
124 : t = 13/16 , x = 251/416 = 0.603 , 鋭 {( 165), 667 , 763 , 832 }
125 : t = 14/17 , x = 299/476 = 0.628 , 鋭 {( 177), 775 , 877 , 952 }
126 : t = 11/13 , x = 97/143 = 0.678 , 鋭 {( 23), 120 , 133 , 143 }
127 : t = 7/8 , x = 83/112 = 0.741 , 鋭 {( 29), 195 , 211 , 224 }
128 : t = 1/9 , x = -13/3 = -4.333 , 退 { 3 , 5 ,( 7), 8 }
129 : t = 2/15 , x = -71/20 = -3.550 , 退 { 40 , 51 ,( 79), 91 }
130 : t = 3/20 , x = -373/120 = -3.108 , 退 { 240 , 253 ,( 427), 493 }
131 : t = 3/19 , x = -167/57 = -2.930 , 重 { 55 , 57 , 97 ,( 112)}
132 : t = 4/23 , x = -481/184 = -2.614 , 重 { 297 , 368 , 577 ,( 665)}
133 : t = 5/28 , x = -709/280 = -2.532 , 重 { 429 , 560 , 859 ,( 989)}
134 : t = 5/27 , x = -109/45 = -2.422 , 重 { 32 , 45 , 67 ,( 77)}
135 : t = 4/21 , x = -131/56 = -2.339 , 重 { 75 , 112 , 163 ,( 187)}
136 : t = 5/24 , x = -167/80 = -2.087 , 重 { 87 , 160 , 217 ,( 247)}
137 : t = 7/33 , x = -157/77 = -2.039 , 重 { 40 , 77 , 103 ,( 117)}
138 : t = 8/37 , x = -1177/592 = -1.988 , 重 { 585 , 1184 , 1561 ,( 1769)}
139 : t = 7/32 , x = -877/448 = -1.958 , 重 { 429 , 896 , 1171 ,( 1325)}
140 : t = 7/31 , x = -407/217 = -1.876 , 重 { 95 , 217 , 277 ,( 312)}
141 : t = 8/35 , x = -1033/560 = -1.845 , 重 { 473 , 1120 , 1417 ,( 1593)}
142 : t = 7/30 , x = -251/140 = -1.793 , 重 { 111 , 280 , 349 ,( 391)}
143 : t = 5/21 , x = -61/35 = -1.743 , 重 { 13 , 35 , 43 ,( 48)}
144 : t = 6/23 , x = -421/276 = -1.525 , 重 { 145 , 552 , 637 ,( 697)}
145 : t = 9/34 , x = -913/612 = -1.492 , 重 { 301 , 1224 , 1399 ,( 1525)}
146 : t = 11/41 , x = -659/451 = -1.461 , 重 { 104 , 451 , 511 ,( 555)}
147 : t = 10/37 , x = -1069/740 = -1.445 , 重 { 329 , 1480 , 1669 ,( 1809)}
148 : t = 11/40 , x = -1237/880 = -1.406 , 重 { 357 , 1760 , 1963 ,( 2117)}
149 : t = 13/47 , x = -851/611 = -1.393 , 重 { 120 , 611 , 679 ,( 731)}
150 : t = 12/43 , x = -1417/1032 = -1.373 , 重 { 385 , 2064 , 2281 ,( 2449)}
...... （中略） .....
2848 : t = 47/114 , x = -2123/3572 = -0.594 , 鈍 { 1449 ,( 5695), 6541 , 7144 }
2849 : t = 73/177 , x = -2557/4307 = -0.594 , 鈍 { 875 ,( 3432), 3943 , 4307 }
2850 : t = 92/223 , x = -24337/41032 = -0.593 , 鈍 { 16695 ,( 65369), 75121 , 82064 }
2851 : t = 85/206 , x = -20761/35020 = -0.593 , 鈍 { 14259 ,( 55781), 64111 , 70040 }
2852 : t = 97/235 , x = -13499/22795 = -0.592 , 鈍 { 4648 ,( 18147), 20863 , 22795 }
2853 : t = 116/281 , x = -38593/65192 = -0.592 , 鈍 { 26599 ,(103785), 119329 , 130384 }
2854 : t = 109/264 , x = -11351/19184 = -0.592 , 鈍 { 7833 ,( 30535), 35113 , 38368 }
2855 : t = 83/201 , x = -3289/5561 = -0.591 , 鈍 { 1136 ,( 4425), 5089 , 5561 }
2856 : t = 88/213 , x = -7379/12496 = -0.591 , 鈍 { 5117 ,( 19875), 22867 , 24992 }
2857 : t = 119/288 , x = -13487/22848 = -0.590 , 鈍 { 9361 ,( 36335), 41809 , 45696 }
2858 : t = 131/317 , x = -24503/41527 = -0.590 , 鈍 { 8512 ,( 33015), 37993 , 41527 }
2859 : t = 112/271 , x = -35809/60704 = -0.590 , 鈍 { 24895 ,( 96513), 111073 , 121408 }
2860 : t = 105/254 , x = -31441/53340 = -0.589 , 鈍 { 21899 ,( 84781), 97591 , 106680 }
2861 : t = 117/283 , x = -19511/33111 = -0.589 , 鈍 { 6800 ,( 26311), 30289 , 33111 }
2862 : t = 98/237 , x = -9119/15484 = -0.589 , 鈍 { 6365 ,( 24603), 28327 , 30968 }
2863 : t = 67/162 , x = -4259/7236 = -0.589 , 鈍 { 2977 ,( 11495), 13237 , 14472 }
2864 : t = 65/157 , x = -5987/10205 = -0.587 , 鈍 { 2109 ,( 8096), 9331 , 10205 }
2865 : t = 94/227 , x = -25021/42676 = -0.586 , 鈍 { 17655 ,( 67697), 78037 , 85352 }
2866 : t = 111/268 , x = -34861/59496 = -0.586 , 鈍 { 24635 ,( 94357), 108787 , 118992 }
2867 : t = 99/239 , x = -13859/23661 = -0.586 , 鈍 { 4901 ,( 18760), 21631 , 23661 }
2868 : t = 104/251 , x = -30553/52208 = -0.585 , 鈍 { 21655 ,( 82761), 95449 , 104416 }
2869 : t = 121/292 , x = -41341/70664 = -0.585 , 鈍 { 29323 ,(112005), 129187 , 141328 }
2870 : t = 109/263 , x = -16763/28667 = -0.585 , 鈍 { 5952 ,( 22715), 26203 , 28667 }
2871 : t = 80/193 , x = -18049/30880 = -0.584 , 鈍 { 12831 ,( 48929), 56449 , 61760 }
2872 : t = 73/176 , x = -14989/25696 = -0.583 , 鈍 { 10707 ,( 40685), 46963 , 51392 }
2873 : t = 95/229 , x = -12683/21755 = -0.583 , 鈍 { 4536 ,( 17219), 19879 , 21755 }
2874 : t = 100/241 , x = -28081/48200 = -0.583 , 鈍 { 20119 ,( 76281), 88081 , 96400 }
2875 : t = 83/200 , x = -19333/33200 = -0.582 , 鈍 { 13867 ,( 52533), 60667 , 66400 }
2876 : t = 71/171 , x = -2353/4047 = -0.581 , 鈍 { 847 ,( 3200), 3697 , 4047 }
2877 : t = 76/183 , x = -5387/9272 = -0.581 , 鈍 { 3885 ,( 14659), 16939 , 18544 }
2878 : t = 59/142 , x = -9721/16756 = -0.580 , 鈍 { 7035 ,( 26477), 30607 , 33512 }
2879 : t = 37/89 , x = -1907/3293 = -0.579 , 鈍 { 693 ,( 2600), 3007 , 3293 }
2880 : t = 38/91 , x = -3949/6916 = -0.571 , 鈍 { 2967 ,( 10865), 12613 , 13832 }
2881 : t = 61/146 , x = -10153/17812 = -0.570 , 鈍 { 7659 ,( 27965), 32479 , 35624 }
2882 : t = 79/189 , x = -2833/4977 = -0.569 , 鈍 { 1072 ,( 3905), 4537 , 4977 }
2883 : t = 74/177 , x = -4967/8732 = -0.569 , 鈍 { 3765 ,( 13699), 15919 , 17464 }
2884 : t = 87/208 , x = -20557/36192 = -0.568 , 鈍 { 15635 ,( 56749), 65971 , 72384 }
2885 : t = 105/251 , x = -14963/26355 = -0.568 , 鈍 { 5696 ,( 20659), 24019 , 26355 }
2886 : t = 100/239 , x = -27121/47800 = -0.567 , 鈍 { 20679 ,( 74921), 87121 , 95600 }
2887 : t = 77/184 , x = -16069/28336 = -0.567 , 鈍 { 12267 ,( 44405), 51643 , 56672 }
2888 : t = 85/203 , x = -9767/17255 = -0.566 , 鈍 { 3744 ,( 13511), 15721 , 17255 }
2889 : t = 116/277 , x = -36361/64264 = -0.566 , 鈍 { 27903 ,(100625), 117097 , 128528 }
2890 : t = 129/308 , x = -44941/79464 = -0.566 , 鈍 { 34523 ,(124405), 144787 , 158928 }
2891 : t = 111/265 , x = -16631/29415 = -0.565 , 鈍 { 6392 ,( 23023), 26797 , 29415 }
2892 : t = 106/253 , x = -30301/53636 = -0.565 , 鈍 { 23335 ,( 83937), 97717 , 107272 }
2893 : t = 119/284 , x = -38173/67592 = -0.565 , 鈍 { 29419 ,(105765), 123139 , 135184 }
2894 : t = 101/241 , x = -13739/24341 = -0.564 , 鈍 { 5301 ,( 19040), 22171 , 24341 }
2895 : t = 70/167 , x = -13189/23380 = -0.564 , 鈍 { 10191 ,( 36569), 42589 , 46760 }
2896 : t = 73/174 , x = -4763/8468 = -0.562 , 鈍 { 3705 ,( 13231), 15421 , 16936 }
2897 : t = 107/255 , x = -5113/9095 = -0.562 , 鈍 { 1991 ,( 7104), 8281 , 9095 }
2898 : t = 128/305 , x = -43873/78080 = -0.562 , 鈍 { 34207 ,(121953), 142177 , 156160 }
2899 : t = 115/274 , x = -35401/63020 = -0.562 , 鈍 { 27619 ,( 98421), 114751 , 126040 }
2900 : t = 123/293 , x = -20231/36039 = -0.561 , 鈍 { 7904 ,( 28135), 32809 , 36039 }
2901 : t = 144/343 , x = -55441/98784 = -0.561 , 鈍 { 43343 ,(154225), 179857 , 197568 }
2902 : t = 131/312 , x = -15287/27248 = -0.561 , 鈍 { 11961 ,( 42535), 49609 , 54496 }
2903 : t = 97/231 , x = -4189/7469 = -0.561 , 鈍 { 1640 ,( 5829), 6799 , 7469 }
2904 : t = 92/219 , x = -7523/13432 = -0.560 , 鈍 { 5909 ,( 20955), 24451 , 26864 }
2905 : t = 121/288 , x = -13007/23232 = -0.560 , 鈍 { 10225 ,( 36239), 42289 , 46464 }
2906 : t = 129/307 , x = -22163/39603 = -0.560 , 鈍 { 8720 ,( 30883), 36043 , 39603 }
2907 : t = 108/257 , x = -31057/55512 = -0.559 , 鈍 { 24455 ,( 86569), 101041 , 111024 }
2908 : t = 95/226 , x = -24001/42940 = -0.559 , 鈍 { 18939 ,( 66941), 78151 , 85880 }
2909 : t = 103/245 , x = -14099/25235 = -0.559 , 鈍 { 5568 ,( 19667), 22963 , 25235 }
2910 : t = 82/195 , x = -5951/10660 = -0.558 , 鈍 { 4709 ,( 16611), 19399 , 21320 }
2911 : t = 53/126 , x = -2483/4452 = -0.558 , 鈍 { 1969 ,( 6935), 8101 , 8904 }
2912 : t = 51/121 , x = -3419/6171 = -0.554 , 鈍 { 1376 ,( 4795), 5611 , 6171 }
2913 : t = 78/185 , x = -15973/28860 = -0.553 , 鈍 { 12887 ,( 44833), 52477 , 57720 }
2914 : t = 97/230 , x = -24673/44620 = -0.553 , 鈍 { 19947 ,( 69293), 81127 , 89240 }
2915 : t = 89/211 , x = -10379/18779 = -0.553 , 鈍 { 4200 ,( 14579), 17071 , 18779 }
2916 : t = 100/237 , x = -8723/15800 = -0.552 , 鈍 { 7077 ,( 24523), 28723 , 31600 }
2917 : t = 119/282 , x = -12347/22372 = -0.552 , 鈍 { 10025 ,( 34719), 40669 , 44744 }
2918 : t = 111/263 , x = -16103/29193 = -0.552 , 鈍 { 6545 ,( 22648), 26533 , 29193 }
2919 : t = 84/199 , x = -18433/33432 = -0.551 , 鈍 { 14999 ,( 51865), 60769 , 66864 }
2920 : t = 87/206 , x = -19729/35844 = -0.550 , 鈍 { 16115 ,( 55573), 65143 , 71688 }
2921 : t = 117/277 , x = -17831/32409 = -0.550 , 鈍 { 7289 ,( 25120), 29449 , 32409 }
2922 : t = 128/303 , x = -14219/25856 = -0.550 , 鈍 { 11637 ,( 40075), 46987 , 51712 }
2923 : t = 109/258 , x = -10307/18748 = -0.550 , 鈍 { 8441 ,( 29055), 34069 , 37496 }
2924 : t = 101/239 , x = -13259/24139 = -0.549 , 鈍 { 5440 ,( 18699), 21931 , 24139 }
2925 : t = 112/265 , x = -32593/59360 = -0.549 , 鈍 { 26767 ,( 91953), 107857 , 118720 }
2926 : t = 93/220 , x = -22453/40920 = -0.549 , 鈍 { 18467 ,( 63373), 74347 , 81840 }
2927 : t = 63/149 , x = -5147/9387 = -0.548 , 鈍 { 2120 ,( 7267), 8527 , 9387 }
2928 : t = 58/137 , x = -8677/15892 = -0.546 , 鈍 { 7215 ,( 24569), 28861 , 31784 }
2929 : t = 83/196 , x = -17749/32536 = -0.546 , 鈍 { 14787 ,( 50285), 59083 , 65072 }
2930 : t = 97/229 , x = -12107/22213 = -0.545 , 鈍 { 5053 ,( 17160), 20167 , 22213 }
2931 : t = 86/203 , x = -19021/34916 = -0.545 , 鈍 { 15895 ,( 53937), 63397 , 69832 }
2932 : t = 89/210 , x = -6779/12460 = -0.544 , 鈍 { 5681 ,( 19239), 22621 , 24920 }
2933 : t = 103/243 , x = -4537/8343 = -0.544 , 鈍 { 1903 ,( 6440), 7573 , 8343 }
2934 : t = 92/217 , x = -21697/39928 = -0.543 , 鈍 { 18231 ,( 61625), 72481 , 79856 }
2935 : t = 67/158 , x = -11497/21172 = -0.543 , 鈍 { 9675 ,( 32669), 38431 , 42344 }
2936 : t = 59/139 , x = -4439/8201 = -0.541 , 鈍 { 1881 ,( 6320), 7441 , 8201 }
2937 : t = 76/179 , x = -14713/27208 = -0.541 , 鈍 { 12495 ,( 41921), 49369 , 54416 }
2938 : t = 79/186 , x = -5291/9796 = -0.540 , 鈍 { 4505 ,( 15087), 17773 , 19592 }
2939 : t = 65/153 , x = -1789/3315 = -0.540 , 鈍 { 763 ,( 2552), 3007 , 3315 }
2940 : t = 54/127 , x = -7381/13716 = -0.538 , 鈍 { 6335 ,( 21097), 24877 , 27432 }
2941 : t = 57/134 , x = -8209/15276 = -0.537 , 鈍 { 7067 ,( 23485), 27703 , 30552 }
2942 : t = 43/101 , x = -2327/4343 = -0.536 , 鈍 { 1008 ,( 3335), 3937 , 4343 }
2943 : t = 26/61 , x = -1693/3172 = -0.534 , 鈍 { 1479 ,( 4865), 5749 , 6344 }
2944 : t = 25/58 , x = -1489/2900 = -0.513 , 鈍 { 1411 ,( 4389), 5239 , 5800 }
2945 : t = 41/95 , x = -1991/3895 = -0.511 , 鈍 { 952 ,( 2943), 3517 , 3895 }
2946 : t = 54/125 , x = -6877/13500 = -0.509 , 鈍 { 6623 ,( 20377), 24373 , 27000 }
2947 : t = 51/118 , x = -6121/12036 = -0.509 , 鈍 { 5915 ,( 18157), 21727 , 24072 }
2948 : t = 61/141 , x = -1453/2867 = -0.507 , 鈍 { 707 ,( 2160), 2587 , 2867 }
2949 : t = 74/171 , x = -4271/8436 = -0.506 , 鈍 { 4165 ,( 12707), 15223 , 16872 }
2950 : t = 71/164 , x = -11773/23288 = -0.506 , 鈍 { 11515 ,( 35061), 42019 , 46576 }
2951 : t = 55/127 , x = -3527/6985 = -0.505 , 鈍 { 1729 ,( 5256), 6301 , 6985 }
2952 : t = 62/143 , x = -8917/17732 = -0.503 , 鈍 { 8815 ,( 26649), 31981 , 35464 }
2953 : t = 85/196 , x = -16741/33320 = -0.502 , 鈍 { 16579 ,( 50061), 60091 , 66640 }
2954 : t = 95/219 , x = -3481/6935 = -0.502 , 鈍 { 1727 ,( 5208), 6253 , 6935 }
2955 : t = 82/189 , x = -5183/10332 = -0.502 , 鈍 { 5149 ,( 15515), 18631 , 20664 }
2956 : t = 79/182 , x = -14401/28756 = -0.501 , 鈍 { 14355 ,( 43157), 51847 , 57512 }
2957 : t = 89/205 , x = -9131/18245 = -0.500 , 鈍 { 4557 ,( 13688), 16447 , 18245 }
2958 : t = 76/175 , x = -13297/26600 = -0.500 , 鈍 { 13303 ,( 39897), 47953 , 53200 }
2959 : t = 53/122 , x = -6457/12932 = -0.499 , 鈍 { 6475 ,( 19389), 23311 , 25864 }
2960 : t = 57/131 , x = -3707/7467 = -0.496 , 鈍 { 1880 ,( 5587), 6727 , 7467 }
2961 : t = 84/193 , x = -16081/32424 = -0.496 , 鈍 { 16343 ,( 48505), 58417 , 64848 }
2962 : t = 101/232 , x = -23221/46864 = -0.495 , 鈍 { 23643 ,( 70085), 84427 , 93728 }
2963 : t = 91/209 , x = -9419/19019 = -0.495 , 鈍 { 4800 ,( 14219), 17131 , 19019 }
2964 : t = 98/225 , x = -7271/14700 = -0.495 , 鈍 { 7429 ,( 21971), 26479 , 29400 }
2965 : t = 115/264 , x = -10007/20240 = -0.494 , 鈍 { 10233 ,( 30247), 36457 , 40480 }
2966 : t = 105/241 , x = -12503/25305 = -0.494 , 鈍 { 6401 ,( 18904), 22789 , 25305 }
2967 : t = 78/179 , x = -13789/27924 = -0.494 , 鈍 { 14135 ,( 41713), 50293 , 55848 }
2968 : t = 75/172 , x = -12709/25800 = -0.493 , 鈍 { 13091 ,( 38509), 46459 , 51600 }
2969 : t = 99/227 , x = -11063/22473 = -0.492 , 鈍 { 5705 ,( 16768), 20233 , 22473 }
2970 : t = 106/243 , x = -8447/17172 = -0.492 , 鈍 { 8725 ,( 25619), 30919 , 34344 }
2971 : t = 89/204 , x = -5951/12104 = -0.492 , 鈍 { 6153 ,( 18055), 21793 , 24208 }
2972 : t = 79/181 , x = -7019/14299 = -0.491 , 鈍 { 3640 ,( 10659), 12871 , 14299 }
2973 : t = 86/197 , x = -16621/33884 = -0.491 , 鈍 { 17263 ,( 50505), 60997 , 67768 }
2974 : t = 69/158 , x = -10681/21804 = -0.490 , 鈍 { 11123 ,( 32485), 39247 , 43608 }
2975 : t = 45/103 , x = -2267/4635 = -0.489 , 鈍 { 1184 ,( 3451), 4171 , 4635 }
2976 : t = 46/105 , x = -1559/3220 = -0.484 , 鈍 { 1661 ,( 4779), 5791 , 6440 }
2977 : t = 71/162 , x = -3707/7668 = -0.483 , 鈍 { 3961 ,( 11375), 13789 , 15336 }
2978 : t = 89/203 , x = -8723/18067 = -0.483 , 鈍 { 4672 ,( 13395), 16243 , 18067 }
2979 : t = 82/187 , x = -14797/30668 = -0.482 , 鈍 { 15871 ,( 45465), 55141 , 61336 }
2980 : t = 93/212 , x = -18997/39432 = -0.482 , 鈍 { 20435 ,( 58429), 70891 , 78864 }
2981 : t = 111/253 , x = -13523/28083 = -0.482 , 鈍 { 7280 ,( 20803), 25243 , 28083 }
2982 : t = 104/237 , x = -7907/16432 = -0.481 , 鈍 { 8525 ,( 24339), 29539 , 32864 }
2983 : t = 79/180 , x = -4559/9480 = -0.481 , 鈍 { 4921 ,( 14039), 17041 , 18960 }
2984 : t = 83/189 , x = -2509/5229 = -0.480 , 鈍 { 1360 ,( 3869), 4699 , 5229 }
2985 : t = 112/255 , x = -9131/19040 = -0.480 , 鈍 { 9909 ,( 28171), 34219 , 38080 }
2986 : t = 123/280 , x = -33013/68880 = -0.479 , 鈍 { 35867 ,(101893), 123787 , 137760 }
2987 : t = 105/239 , x = -12023/25095 = -0.479 , 鈍 { 6536 ,( 18559), 22549 , 25095 }
2988 : t = 98/223 , x = -20917/43708 = -0.479 , 鈍 { 22791 ,( 64625), 78541 , 87416 }
2989 : t = 109/248 , x = -25861/54064 = -0.478 , 鈍 { 28203 ,( 79925), 97147 , 108128 }
2990 : t = 91/207 , x = -3001/6279 = -0.478 , 鈍 { 1639 ,( 4640), 5641 , 6279 }
2991 : t = 62/141 , x = -2783/5828 = -0.478 , 鈍 { 3045 ,( 8611), 10471 , 11656 }
2992 : t = 59/134 , x = -7513/15812 = -0.475 , 鈍 { 8299 ,( 23325), 28399 , 31624 }
2993 : t = 85/193 , x = -7787/16405 = -0.475 , 鈍 { 4309 ,( 12096), 14731 , 16405 }
2994 : t = 100/227 , x = -21529/45400 = -0.474 , 鈍 { 23871 ,( 66929), 81529 , 90800 }
2995 : t = 89/202 , x = -17041/35956 = -0.474 , 鈍 { 18915 ,( 52997), 64567 , 71912 }
2996 : t = 93/211 , x = -9287/19623 = -0.473 , 鈍 { 5168 ,( 14455), 17617 , 19623 }
2997 : t = 108/245 , x = -25033/52920 = -0.473 , 鈍 { 27887 ,( 77953), 95017 , 105840 }
2998 : t = 97/220 , x = -20173/42680 = -0.473 , 鈍 { 22507 ,( 62853), 76627 , 85360 }
2999 : t = 71/161 , x = -5399/11431 = -0.472 , 鈍 { 3016 ,( 8415), 10261 , 11431 }
3000 : t = 64/145 , x = -8737/18560 = -0.471 , 鈍 { 9823 ,( 27297), 33313 , 37120 }
end
ソースは次です。
#include<stdio.h>
#include<stdlib.h>
int _gcm(int a,int b){return (b==0)?a:_gcm(b,a%b);}
int gcm(int a,int b){
if(a<0)a=-a;if(b<0)b=-b;if(a==b) return a==0?1:a;
return (a>b)?_gcm(a,b):_gcm(b,a);
}
int main(int argc,char *argv[]){
int f[3001][4],i,j,m,n,*p,*q,xa,xb,mn,mm,nn,la,lb,lc,ld,g;
double x;
p=f[1],p[0]=0,p[1]=1,p[2]=1,p[3]=1;
for(i=2;i<3001;i++){
j=i&1;p=f[i];q=f[i>>1];
if(j==0){p[0]=q[0];p[1]=q[1];p[2]=q[0]+q[2];p[3]=q[1]+q[3];}
else {p[0]=q[0]+q[2];p[1]=q[1]+q[3];p[2]=q[2];p[3]=q[3];}
}
for(i=1;i<3001;i++){
p=f[i];m=p[0]+p[2];n=p[1]+p[3];mm=3*m*m;nn=n*n;xa=mm-nn;xb=2*m*n;
la=mm+nn;lb=xb+xa;lc=xb-xa;ld=2*xb;
g=gcm(gcm(la,lb),gcm(lc,ld));la=la/g;lb=lb/g;lc=lc/g;ld=ld/g;
x=(double)xa/(double)xb;g=gcm(xa,xb);
// if(lc<20 || ld<20 || (lb<20 && lb>-20) ){
printf("%4d : t =%4d/%-4d , x = %6d/%-5d = %6.3f ,",i,m,n,xa/g,xb/g,x);
if(x>-1){
if(x>0) printf(" %s {(%6d),%7d ,%7d ,%7d }\n","鋭",lc,lb,la,ld);
else printf(" %s {%7d ,(%6d),%7d ,%7d }\n","鈍",lb,lc,la,ld);
}else{
if(x>-3)printf(" %s {%7d ,%7d ,%7d ,(%6d)}\n","重",-lb,ld,la,lc);
else printf(" %s {%7d ,%7d ,(%6d),%7d }\n","退",ld,-lb,la,lc);
}
// }
}
printf("end\n");
return 0;
}
生成数を増やし、最小辺が20未満のものだけを出力させて、順番を入れ替えて見やすくしてみました。
2 : t = 1/3 , x = -1/1 = -1.000 , 重 { 0 , 1 , 1 ,( 1)}
1 : t = 1/2 , x = -1/4 = -0.250 , 鈍 { 3 ,( 5), 7 , 8 }
3 : t = 2/3 , x = 1/4 = 0.250 , 鋭 {( 3), 5 , 7 , 8 }
8 : t = 1/5 , x = -11/5 = -2.200 , 重 { 3 , 5 , 7 ,( 8)}
128 : t = 1/9 , x = -13/3 = -4.333 , 退 { 3 , 5 ,( 7), 8 }
4 : t = 1/4 , x = -13/8 = -1.625 , 重 { 5 , 16 , 19 ,( 21)}
11 : t = 3/7 , x = -11/21 = -0.524 , 鈍 { 5 ,( 16), 19 , 21 }
30 : t = 7/9 , x = 11/21 = 0.524 , 鋭 {( 5), 16 , 19 , 21 }
8192 : t = 1/15 , x = -37/5 = -7.400 , 退 { 5 , 16 ,( 19), 21 }
6 : t = 3/5 , x = 1/15 = 0.067 , 鋭 {( 7), 8 , 13 , 15 }
16 : t = 1/6 , x = -11/4 = -2.750 , 重 { 7 , 8 , 13 ,( 15)}
24 : t = 5/9 , x = -1/15 = -0.067 , 鈍 { 7 ,( 8), 13 , 15 }
32 : t = 1/7 , x = -23/7 = -3.286 , 退 { 7 , 8 ,( 13), 15 }
5 : t = 2/5 , x = -13/20 = -0.650 , 鈍 { 7 ,( 33), 37 , 40 }
18 : t = 3/11 , x = -47/33 = -1.424 , 重 { 7 , 33 , 37 ,( 40)}
31 : t = 5/6 , x = 13/20 = 0.650 , 鋭 {( 7), 33 , 37 , 40 }
9 : t = 2/7 , x = -37/28 = -1.321 , 重 { 9 , 56 , 61 ,( 65)}
21 : t = 5/13 , x = -47/65 = -0.723 , 鈍 { 9 ,( 56), 61 , 65 }
254 : t = 13/15 , x = 47/65 = 0.723 , 鋭 {( 9), 56 , 61 , 65 }
14 : t = 5/7 , x = 13/35 = 0.371 , 鋭 {( 11), 24 , 31 , 35 }
17 : t = 2/9 , x = -23/12 = -1.917 , 重 { 11 , 24 , 31 ,( 35)}
191 : t = 7/15 , x = -13/35 = -0.371 , 鈍 { 11 ,( 24), 31 , 35 }
512 : t = 1/11 , x = -59/11 = -5.364 , 退 { 11 , 24 ,( 31), 35 }
10 : t = 3/8 , x = -37/48 = -0.771 , 鈍 { 11 ,( 85), 91 , 96 }
38 : t = 5/17 , x = -107/85 = -1.259 , 重 { 11 , 85 , 91 ,( 96)}
255 : t = 8/9 , x = 37/48 = 0.771 , 鋭 {( 11), 85 , 91 , 96 }
7 : t = 3/4 , x = 11/24 = 0.458 , 鋭 {( 13), 35 , 43 , 48 }
23 : t = 4/9 , x = -11/24 = -0.458 , 鈍 { 13 ,( 35), 43 , 48 }
143 : t = 5/21 , x = -61/35 = -1.743 , 重 { 13 , 35 , 43 ,( 48)}
2048 : t = 1/13 , x = -83/13 = -6.385 , 退 { 13 , 35 ,( 43), 48 }
19 : t = 3/10 , x = -73/60 = -1.217 , 重 { 13 , 120 , 127 ,( 133)}
41 : t = 7/19 , x = -107/133 = -0.805 , 鈍 { 13 ,( 120), 127 , 133 }
2046 : t = 19/21 , x = 107/133 = 0.805 , 鋭 {( 13), 120 , 127 , 133 }
20 : t = 4/11 , x = -73/88 = -0.830 , 鈍 { 15 ,( 161), 169 , 176 }
78 : t = 7/23 , x = -191/161 = -1.186 , 重 { 15 , 161 , 169 ,( 176)}
2047 : t = 11/12 , x = 73/88 = 0.830 , 鋭 {( 15), 161 , 169 , 176 }
35 : t = 3/13 , x = -71/39 = -1.821 , 重 { 16 , 39 , 49 ,( 55)}
47 : t = 5/11 , x = -23/55 = -0.418 , 鈍 { 16 ,( 39), 49 , 55 }
59 : t = 11/15 , x = 23/55 = 0.418 , 鋭 {( 16), 39 , 49 , 55 }
1024 : t = 1/12 , x = -47/8 = -5.875 , 退 { 16 , 39 ,( 49), 55 }
15 : t = 4/5 , x = 23/40 = 0.575 , 鋭 {( 17), 63 , 73 , 80 }
22 : t = 5/12 , x = -23/40 = -0.575 , 鈍 { 17 ,( 63), 73 , 80 }
288 : t = 7/27 , x = -97/63 = -1.540 , 重 { 17 , 63 , 73 ,( 80)}
32768 : t = 1/17 , x = -143/17 = -8.412 , 退 { 17 , 63 ,( 73), 80 }
39 : t = 4/13 , x = -121/104 = -1.163 , 重 { 17 , 208 , 217 ,( 225)}
81 : t = 9/25 , x = -191/225 = -0.849 , 鈍 { 17 ,( 208), 217 , 225 }
16382 : t = 25/27 , x = 191/225 = 0.849 , 鋭 {( 17), 208 , 217 , 225 }
36 : t = 4/15 , x = -59/40 = -1.475 , 重 { 19 , 80 , 91 ,( 99)}
62 : t = 9/11 , x = 61/99 = 0.616 , 鋭 {( 19), 80 , 91 , 99 }
176 : t = 11/27 , x = -61/99 = -0.616 , 鈍 { 19 ,( 80), 91 , 99 }
40 : t = 5/14 , x = -121/140 = -0.864 , 鈍 { 19 ,( 261), 271 , 280 }
158 : t = 9/29 , x = -299/261 = -1.146 , 重 { 19 , 261 , 271 ,( 280)}
16383 : t = 14/15 , x = 121/140 = 0.864 , 鋭 {( 19), 261 , 271 , 80 }
セット毎にまとめました。「退」の三角形には大きな番号がつく傾向があるのがよく分かると思います。有理数の生成方法や、変換式を調整することで改善できるかもしれませんが、この辺で妥協しておきます。
明日は、プログラムについて、特に、区間(0,1)の有理数の生成方法について説明を加えたいと思います。
#60度 #120度 #180度 #三角形 #ナンバリング #整列 #有理数生成