Hans Walser, [20220111]
Rationale Kreispunkte
Punkte im ersten Quadranten des Einheitskreises mit rationalen Koordinaten.
Mit der üblichen Parametrisierung werden die pythagoreischen Dreiecke generiert und die Hypotenuse auf eins normiert. Die Abbildung 1 zeigt die beiden ersten Beispiele. Es handelt sich um das ägyptische Dreieck mit dem Seitenverhältnis 3:4:5 und das indische Dreieck mit dem Seitenverhältnis 5:12:13.
Abb. 1: Pythagoreische Dreiecke
Die Abbildung 2 zeigt in Folge die ersten 86 Beispiele.
Abb. 2: Die ersten 86 Beispiele
Die Tabelle 1 gibt die rationalen Koordinaten der ersten 86 Punkte.
n |
Koordinaten |
|
n |
Koordinaten |
|
n |
Koordinaten |
1 |
[3/5, 4/5] |
|
31 |
[23/265, 264/265] |
|
61 |
[145/433, 408/433] |
2 |
[5/13, 12/13] |
|
32 |
[165/173, 52/173] |
|
62 |
[93/485, 476/485] |
3 |
[15/17, 8/17] |
|
33 |
[153/185, 104/185] |
|
63 |
[33/545, 544/545] |
4 |
[7/25, 24/25] |
|
34 |
[133/205, 156/205] |
|
64 |
[323/325, 36/325] |
5 |
[21/29, 20/29] |
|
35 |
[105/233, 208/233] |
|
65 |
[299/349, 180/349] |
6 |
[9/41, 40/41] |
|
36 |
[69/269, 260/269] |
|
66 |
[275/373, 252/373] |
7 |
[35/37, 12/37] |
|
37 |
[25/313, 312/313] |
|
67 |
[203/445, 396/445] |
8 |
[11/61, 60/61] |
|
38 |
[195/197, 28/197] |
|
68 |
[155/493, 468/493] |
9 |
[45/53, 28/53] |
|
39 |
[187/205, 84/205] |
|
69 |
[35/613, 612/613] |
10 |
[33/65, 56/65] |
|
40 |
[171/221, 140/221] |
|
70 |
[357/365, 76/365] |
11 |
[13/85, 84/85] |
|
41 |
[115/277, 252/277] |
|
71 |
[345/377, 152/377] |
12 |
[63/65, 16/65] |
|
42 |
[75/317, 308/317] |
|
72 |
[325/397, 228/397] |
13 |
[55/73, 48/73] |
|
43 |
[27/365, 364/365] |
|
73 |
[297/425, 304/425] |
14 |
[39/89, 80/89] |
|
44 |
[221/229, 60/229] |
|
74 |
[261/461, 380/461] |
15 |
[15/113, 112/113] |
|
45 |
[209/241, 120/241] |
|
75 |
[217/505, 456/505] |
16 |
[77/85, 36/85] |
|
46 |
[161/289, 240/289] |
|
76 |
[165/557, 532/557] |
17 |
[65/97, 72/97] |
|
47 |
[29/421, 420/421] |
|
77 |
[105/617, 608/617] |
18 |
[17/145, 144/145] |
|
48 |
[255/257, 32/257] |
|
78 |
[37/685, 684/685] |
19 |
[99/101, 20/101] |
|
49 |
[247/265, 96/265] |
|
79 |
[399/401, 40/401] |
20 |
[91/109, 60/109] |
|
50 |
[231/281, 160/281] |
|
80 |
[391/409, 120/409] |
21 |
[51/149, 140/149] |
|
51 |
[207/305, 224/305] |
|
81 |
[351/449, 280/449] |
22 |
[19/181, 180/181] |
|
52 |
[175/337, 288/337] |
|
82 |
[319/481, 360/481] |
23 |
[117/125, 44/125] |
|
53 |
[135/377, 352/377] |
|
83 |
[279/521, 440/521] |
24 |
[105/137, 88/137] |
|
54 |
[87/425, 416/425] |
|
84 |
[231/569, 520/569] |
25 |
[85/157, 132/157] |
|
55 |
[31/481, 480/481] |
|
85 |
[111/689, 680/689] |
26 |
[57/185, 176/185] |
|
56 |
[285/293, 68/293] |
|
86 |
[39/761, 760/761] |
27 |
[21/221, 220/221] |
|
57 |
[273/305, 136/305] |
|
|
|
28 |
[143/145, 24/145] |
|
58 |
[253/325, 204/325] |
|
|
|
29 |
[119/169, 120/169] |
|
59 |
[225/353, 272/353] |
|
|
|
30 |
[95/193, 168/193] |
|
60 |
[189/389, 340/389] |
|
|
|
Tab. 1: Rationale Koordinaten
Die Punkte fallen wie Sternschnuppen ein (Abb. 3).
Abb. 3: Die ersten 86 Punkte