Bu konu matematik dergilerinde
Langley's Adventitious Angles olarak irdelenmiş.
Derece cinsinden açıların rasyonel olduğu durumlar akademik olarak
The number of intersection points made by diagonals of a regular polygon adlı makalede kapsamlı ele alınmış. Bu durumun oluşması için düzgün çokgenlerde 3 köşegenin kesişmesi gerektiği sonucu üzerine düzgün çokgen üzerinde bu modellemeler yapılmış. Düzgün çokgenlerde kaç köşegenenin tek bir noktada kesişebildiği, düzgün çokgende köşegenlerin kesişiminden oluşan nokta sayıları gibi farklı noktalar da tespit edilmiş. Daha detaylı okumalar için ilgili makalelerdeki referanslara bakabilirsiniz.
Bir aileye ait olmayan tüm soru tipleri aşağıdaki tabloda hangi çokgene ait olduklarına göre sıralanmış bir şekilde verilmiş. Bağıntı Türü sütunu (bu makalede anlatılan şekliyle) birbirinden türetilebilen modelleri göstermekte. (Açılar $180^\circ$ ile çarpılmalıdır.)
$$
\begin{array}{c||c|c|c|c||c|c|c||c}
\text{Çokgen} & \text {Model #} & a_1 & a_2 & a_3 & b_1 & b_2 & b_3
& \text{Bağıntı Türü} \\ \hline \hline
30 & 6 & 1/10 & 2/15 & 3/10 & 2/15 & 1/6 & 1/6
& 2(R_5:R_3) \\
& 7 & 1/15 & 1/15 & 7/15 & 1/15 & 1/10 & 7/30 & \\
& 8 & 1/30 & 7/30 & 4/15 & 1/15 & 1/10 & 3/10 & \\
& 9 & 1/30 & 1/10 & 7/15 & 1/15 & 1/15 & 4/15 & \\
& 10 & 1/30 & 1/15 & 19/30 & 1/15 & 1/10 & 1/10 & \\ \hline
& 11 & 1/15 & 1/6 & 4/15 & 1/10 & 1/10 & 3/10
& (R_5:R_3)+2R_3 \\
& 12 & 1/15 & 2/15 & 11/30 & 1/10 & 1/6 & 1/6 & \\
& 13 & 1/30 & 1/6 & 13/30 & 1/10 & 2/15 & 2/15 & \\
& 14 & 1/30 & 1/30 & 7/10 & 1/30 & 1/15 & 2/15 & \\ \hline
& 15 & 1/30 & 7/30 & 3/10 & 1/15 & 2/15 & 7/30
& R_5+R_3+2R_2 \\
& 16 & 1/30 & 1/6 & 11/30 & 1/15 & 1/10 & 4/15 & \\
& 17 & 1/30 & 1/10 & 13/30 & 1/30 & 2/15 & 4/15 & \\
& 18 & 1/30 & 1/15 & 8/15 & 1/30 & 1/10 & 7/30 & \\ \hline
42 & 19 & 1/14 & 5/42 & 5/14 & 2/21 & 5/42 & 5/21
& (R_7:5R_3) \\
& 20 & 1/21 & 4/21 & 13/42 & 1/14 & 1/6 & 3/14 & \\
& 21 & 1/42 & 3/14 & 5/14 & 1/21 & 1/6 & 4/21 & \\
& 22 & 1/42 & 1/6 & 19/42 & 1/14 & 2/21 & 4/21 & \\
& 23 & 1/42 & 1/6 & 13/42 & 1/21 & 1/14 & 8/21 & \\
& 24 & 1/42 & 1/21 & 13/21 & 1/42 & 1/14 & 3/14 & \\ \hline
60 & 25 & 1/20 & 1/12 & 29/60 & 1/15 & 1/10 & 13/60
& 2(R_5:R_3) \\
& 26 & 1/20 & 1/12 & 9/20 & 1/15 & 1/12 & 4/15 & \\
& 27 & 1/20 & 1/12 & 5/12 & 1/20 & 1/10 & 3/10 & \\
& 28 & 1/60 & 4/15 & 3/10 & 1/20 & 1/12 & 17/60 & \\
& 29 & 1/60 & 13/60 & 9/20 & 1/12 & 1/10 & 2/15 & \\
& 30 & 1/60 & 13/60 & 5/12 & 1/20 & 2/15 & 1/6 & \\ \hline
& 31 & 1/12 & 1/6 & 17/60 & 2/15 & 3/20 & 11/60
& (R_5:3R_3)+2R_2 \\
& 32 & 1/12 & 2/15 & 19/60 & 1/10 & 3/20 & 13/60 & \\
& 33 & 1/15 & 11/60 & 13/60 & 1/12 & 1/10 & 7/20 & \\
& 34 & 1/20 & 11/60 & 3/10 & 1/12 & 7/60 & 4/15 & \\
& 35 & 1/20 & 1/10 & 23/60 & 1/15 & 1/12 & 19/60 & \\
& 36 & 1/30 & 7/60 & 19/60 & 1/20 & 1/15 & 5/12 & \\
& 37 & 1/30 & 1/12 & 7/12 & 1/15 & 1/10 & 2/15 & \\
& 38 & 1/30 & 1/20 & 11/20 & 1/30 & 1/15 & 4/15 & \\
& 39 & 1/60 & 3/10 & 7/20 & 1/12 & 7/60 & 2/15 & \\
& 40 & 1/60 & 4/15 & 23/60 & 1/12 & 1/10 & 3/20 & \\
& 41 & 1/60 & 7/30 & 5/12 & 1/15 & 7/60 & 3/20 & \\
& 42 & 1/60 & 13/60 & 11/30 & 1/20 & 1/12 & 4/15 & \\
& 43 & 1/60 & 1/6 & 31/60 & 1/15 & 1/10 & 2/15 & \\
& 44 & 1/60 & 1/6 & 5/12 & 1/20 & 1/15 & 17/60 & \\
& 45 & 1/60 & 2/15 & 9/20 & 1/30 & 1/12 & 17/60 & \\
& 46 & 1/60 & 1/10 & 31/60 & 1/30 & 1/15 & 4/15 & \\ \hline
84 & 47 & 1/12 & 3/14 & 19/84 & 11/84 & 13/84 & 4/21
& (R_7:R_3)+2R_2 \\
& 48 & 1/14 & 11/84 & 23/84 & 1/12 & 2/21 & 29/84 & \\
& 49 & 1/21 & 13/84 & 23/84 & 1/14 & 1/12 & 31/84 & \\
& 50 & 1/42 & 1/12 & 7/12 & 1/21 & 1/14 & 4/21 & \\
& 51 & 1/84 & 25/84 & 5/14 & 5/84 & 1/12 & 4/21 & \\
& 52 & 1/84 & 5/21 & 5/12 & 5/84 & 1/14 & 17/84 & \\
& 53 & 1/84 & 3/14 & 37/84 & 1/21 & 1/12 & 17/84 & \\
& 54 & 1/84 & 1/6 & 43/84 & 1/21 & 1/14 & 4/21 & \\ \hline
90 & 55 & 1/18 & 13/90 & 7/18 & 11/90 & 2/15 & 7/45
& (R_5:R_3)+2R_3 \\
& 56 & 1/45 & 19/90 & 16/45 & 1/18 & 1/10 & 23/90 & \\
& 57 & 1/90 & 23/90 & 31/90 & 2/45 & 1/15 & 5/18 & \\
& 58 & 1/90 & 17/90 & 47/90 & 1/18 & 4/45 & 2/15 & \\ \hline
120 & 59 & 13/120 & 3/20 & 31/120 & 2/15 & 19/120 & 23/120 &
(R_5:R_3)+3R_2 \\
& 60 & 1/12 & 19/120 & 29/120 & 1/10 & 13/120 & 37/120 & \\
& 61 & 1/20 & 23/120 & 29/120 & 1/15 & 13/120 & 41/120 & \\
& 62 & 1/60 & 13/120 & 73/120 & 1/20 & 1/12 & 2/15 & \\
& 63 & 1/120 & 7/20 & 43/120 & 7/120 & 11/120 & 2/15 & \\
& 64 & 1/120 & 3/10 & 49/120 & 7/120 & 1/12 & 17/120 & \\
& 65 & 1/120 & 4/15 & 53/120 & 1/20 & 11/120 & 17/120 & \\
& 66 & 1/120 & 13/60 & 61/120 & 1/20 & 1/12 & 2/15 & \\ \hline
210 & 67 & 1/15 & 41/210 & 8/35 & 1/14 & 31/210 & 61/210
& (R_7:(R_5:2R_3)) \\
& 68 & 13/210 & 1/10 & 83/210 & 1/14 & 4/35 & 9/35 & \\
& 69 & 1/35 & 2/15 & 97/210 & 1/14 & 17/210 & 47/210 & \\
& 70 & 1/210 & 3/14 & 121/210 & 11/210 & 1/15 & 3/35 & \\ \hline
\end{array}
$$
Aşağıdaki tabloda ise açı ölçüsüne göre sıralanmış bir biçimde modeller listelenmekte.
$$
\begin{array}{ccc| c}
&& [\![ A, B ]\!] & \text {Model #} \\ \hline
(6/7, 270/7, 726/7) & : & (66/7, 12, 108/7) & 70 \\ \hline
(3/2, 39, 183/2) & : & (9, 15, 24) & 66 \\ \hline
(3/2, 48, 159/2) & : & (9, 33/2, 51/2) & 65 \\ \hline
(3/2, 54, 147/2) & : & (21/2, 15, 51/2) & 64 \\ \hline
(3/2, 63, 129/2) & : & (21/2, 33/2, 24) & 63 \\ \hline
(2, 34, 94) & : & (10, 16, 24) & 58 \\ \hline
(2, 46, 62) & : & (8, 12, 50) & 57 \\ \hline
(15/7, 30, 645/7) & : & (60/7, 90/7, 240/7) & 54 \\ \hline
(15/7, 270/7, 555/7) & : & (60/7, 15, 255/7) & 53 \\ \hline
(15/7, 300/7, 75) & : & (75/7, 90/7, 255/7) & 52 \\ \hline
(15/7, 375/7, 450/7) & : & (75/7, 15, 240/7) & 51 \\ \hline
(3, 18, 93) & : & (6, 12, 48) & 46 \\ \hline
(3, 39/2, 219/2) & : & (9, 15, 24) & 62 \\ \hline
(3, 24, 81) & : & (6, 15, 51) & 45 \\ \hline
(3, 30, 75) & : & (9, 12, 51) & 44 \\ \hline
(3, 30, 93) & : & (12, 18, 24) & 43 \\ \hline
(3, 39, 66) & : & (9, 15, 48) & 42 \\ \hline
(3, 39, 75) & : & (9, 24, 30) & 30 \\ \hline
(3, 39, 81) & : & (15, 18, 24) & 29 \\ \hline
(3, 42, 75) & : & (12, 21, 27) & 41 \\ \hline
(3, 48, 54) & : & (9, 15, 51) & 28 \\ \hline
(3, 48, 69) & : & (15, 18, 27) & 40 \\ \hline
(3, 54, 63) & : & (15, 21, 24) & 39 \\ \hline
(4, 38, 64) & : & (10, 18, 46) & 56 \\ \hline
(30/7, 60/7, 780/7) & : & (30/7, 90/7, 270/7) & 24 \\ \hline
(30/7, 15, 105) & : & (60/7, 90/7, 240/7) & 50 \\ \hline
(30/7, 30, 390/7) & : & (60/7, 90/7, 480/7) & 23 \\ \hline
(30/7, 30, 570/7) & : & (90/7, 120/7, 240/7) & 22 \\ \hline
(30/7, 270/7, 450/7) & : & (60/7, 30, 240/7) & 21 \\ \hline
(36/7, 24, 582/7) & : & (90/7, 102/7, 282/7) & 69 \\ \hline
(6, 6, 126) & : & (6, 12, 24) & 14 \\ \hline
(6, 9, 99) & : & (6, 12, 48) & 38 \\ \hline
(6, 12, 96) & : & (6, 18, 42) & 18 \\ \hline
(6, 12, 114) & : & (12, 18, 18) & 10 \\ \hline
(6, 15, 105) & : & (12, 18, 24) & 37 \\ \hline
(6, 18, 78) & : & (6, 24, 48) & 17 \\ \hline
(6, 18, 84) & : & (12, 12, 48) & 9 \\ \hline
(6, 21, 57) & : & (9, 12, 75) & 36 \\ \hline
(6, 30, 66) & : & (12, 18, 48) & 16 \\ \hline
(6, 30, 78) & : & (18, 24, 24) & 13 \\ \hline
(6, 42, 48) & : & (12, 18, 54) & 8 \\ \hline
(6, 42, 54) & : & (12, 24, 42) & 15 \\ \hline
(60/7, 195/7, 345/7) & : & (90/7, 15, 465/7) & 49 \\ \hline
(60/7, 240/7, 390/7) & : & (90/7, 30, 270/7) & 20 \\ \hline
(9, 15, 75) & : & (9, 18, 54) & 27 \\ \hline
(9, 15, 81) & : & (12, 15, 48) & 26 \\ \hline
(9, 15, 87) & : & (12, 18, 39) & 25 \\ \hline
(9, 18, 69) & : & (12, 15, 57) & 35 \\ \hline
(9, 33, 54) & : & (15, 21, 48) & 34 \\ \hline
(9, 69/2, 87/2) & : & (12, 39/2, 123/2) & 61 \\ \hline
(10, 26, 70) & : & (22, 24, 28) & 55 \\ \hline
(78/7, 18, 498/7) & : & (90/7, 144/7, 324/7) & 68 \\ \hline
(12, 12, 84) & : & (12, 18, 42) & 7 \\ \hline
(12, 24, 66) & : & (18, 30, 30) & 12 \\ \hline
(12, 30, 48) & : & (18, 18, 54) & 11 \\ \hline
(12, 33, 39) & : & (15, 18, 63) & 33 \\ \hline
(12, 246/7, 288/7) & : & (90/7, 186/7, 366/7) & 67 \\ \hline
(90/7, 150/7, 450/7) & : & (120/7, 150/7, 300/7) & 19 \\ \hline
(90/7, 165/7, 345/7) & : & (15, 120/7, 435/7) & 48 \\ \hline
(15, 24, 57) & : & (18, 27, 39) & 32 \\ \hline
(15, 57/2, 87/2) & : & (18, 39/2, 111/2) & 60 \\ \hline
(15, 30, 51) & : & (24, 27, 33) & 31 \\ \hline
(15, 270/7, 285/7) & : & (165/7, 195/7, 240/7) & 47 \\ \hline
(18, 24, 54) & : & (24, 30, 30) & 6 \\ \hline
(39/2, 27, 93/2) & : & (24, 57/2, 69/2) & 59 \\ \hline
\end{array}
$$
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