115
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Berikut pemetaan (mapping) angka Seratus Limabelas (115) ke piramida data dari diagram berupa konsep, detil bagan dan modul² yang dipakai sebagai dasar pemrograman.
Sebelum masuk ke detail, berikut ini daftar keistimewaan angka 115 menurut wikipedia:
- 115 memiliki jumlah kuadrat pembagi: sigma (115) = 1 + 5 + 23 + 115 = 144 = 12 ^ {2}.
- Ada 115 pohon berakar yang berbeda dengan tepat delapan simpul,
- 115 cara yang tidak sama untuk menempatkan enam benteng pada papan catur 6 × 6 sedemikian rupa sehingga tidak ada dua benteng yang saling menyerang, dan 115 solusi untuk masalah melipat perangko untuk strip tujuh prangko,
- 115 juga merupakan bilangan piramidal heptagonal . [5] Nomor Woodall ke-115 ,
- 2 ^ {115} -1 = 4 \; 776 \; 913 \; 109 \; 852 \; 041 \; 418 \; 248 \; 056 \; 622 \; 882 \; 488 \; 319, adalah bilangan prima .
- Simak untuk keistimewaan² lainnya.
True Prime Pairs:
(5,7), (11,13), (17,19)
|------------ 6¤ -------------|------------- 6¤ ------------|
|---- {23} ----|-------- 66 -------|-- {23} -|----- 80 -----|
+----+----+----+----+----+----+----+----+----+----+----+----+
| 5 | 7 | 11 | 13 | 17 | 19 | 17 | 12 |{11}| 19 | 18 | 43 |
+----+----+----+----+----+----+----+----+----+----+----+----+
|----- 23 -----|----- 49 -----|--------- {77} ---------| 43 |- Φ(1,2,3) = Φ(6,12,18) = Φ(13,37,61)
True Prime Pairs:
(5,7), (11,13), (17,19)
{-25} {-6} 11|12 23 33|34 53 71 114
Δ Δ Δ Δ Δ Δ Δ Δ
|---------36'-------|---36'---|-- {29}--|- {30} --|- {61} --|
+----+----+----+----+----+----+----+----+----+----+----+----+
| 5'| 7'| 11'| 13'| 17'| 19'| 17 | 12*| 11*| 19 |{18}|{43}|
+----+----+----+----+----+----+----+----+----+----+----+----+
|----- 48 -----| 11 |- {37} --| 43 |
Δ Δ Δ Δ Δ Δ Δ
| | 48 59 77|78 96 139
| | |
| | 71 {89} 96 114
| | -- Δ Δ Δ Δ
| | | +----+----+----+
----------------------> Δ25 |{18}| 7 | 18 | 43
| | +----+----+----+
| -- Δ Δ Δ Δ
| 96 114 121 139
| Δ Δ Δ Δ
-------------> 96/6 = {-16} 2 9 {27}layer| 1st | 2nd | 3rd |∑(2,3)
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | 7 | 19 | 38 | 62 | 63 | 64 | 93 | 94 | 95 | 139 |
i + 1 +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 5¨
| | 8 | 20 | 39 | 65 | 66 | 68 | 96 | 97 | 98 | |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | {9} | 21 | 40 | 43 | 67 | 69 | 99 | 100 | 101 | 286 |
+ {2} +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 7¨
| |{10} | 22 | 41 | 44 | 45 | 70 | 102 | 103 | 104 | |
q +-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | 11 |{23} | 42 | 46 | 47 | 71 | 105 | 106 | 107 | 114 |
+ 3 +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 11¨
| | 12 | 24 |{25} | 48 | 49 | 72 | 108 | 109 | 110 | |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | 13 | 26 | 27 |{50} | 51 | 73 | 74 | 111 | 112 | 247 |
+ 4 +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 13¨
| | 14 | 28 | 29 | 52 |{53} | 75 | 76 | 113 | 114 | |
r +-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | 15 | 30 | 31 | 54 | 55 |{77} | 78 | 79 |{80} | 157 |
+ 5 +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 17¨
| | 16 | 32 | 33 | 56 | 57 | 81 |{82} |{83} | 84 | |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
| | 17 | 34 | 35 | 58 | 59 | 85 |{86} |{87} | 88 | 786 |
o + 6 +-----+-----+-----+-----+-----+-----+-----+-----+-----+------ 19¨
| | 18 | 36 | 37 | 60 | 61 |{89} | 90 | 91 |{92} | |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+------ ---
∑ | 21 | 150 | | | | | | | | | 1729
|--------------------------------------------------- 16¨ ---|
|--------------------------------------- 15¨ ---|
|--------------------------- 14¨ ---|
|--------------- 13¨ ---|
|--- 12¨ ---|
- 139 = 168 - 29 = (102 + 66) - 29
The sum of the first 29 semiprimes is divisible by 29 (1247/29=43 which is also a prime).
-----+-----+-----+-----+-----+
19¨ | 1 | 2 | 3 | 4 | 4¤
-----+-----+-----+-----+-----+
17¨ | 5 | 6 | 7 | 8 | 4¤
+-----+-----+-----+-----+
12¨ | 9 | 10 | {2¤} (M dan F)
+-----+-----+-----+
{11¨}|{11} | 12 | 13 | {3¤} <----------- d(11) = d(17+12)= d(29)
-----+-----+-----+-----+-----+
19¨ | 14 | 15 | 16 | 17 | {4¤}
+-----+-----+-----+-----+
18¨ | 18 | 19 | 20 | 3¤
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
43¨ | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 |{29} | 9¤
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+1 + 7 = 8 = 2 x 2 x 2 = 2³ » 23
-----+-----+-----+-----+-----+ ---
19¨ | 3¨ | 4¨ | 6¨ | 6¨ | 4¤ |
-----+-----+-----+-----+-----+ |
17¨ | 5¨ | 3¨ | 2¨ | 7¨ | 4¤ |
+-----+-----+-----+-----+ |
12¨ | 6¨ | 6¨ | 2¤ (M dan F) |
+-----+-----+-----+ 17¤
11¨ | 3¨ | 3¨ | 5¨ | 3¤ |
-----+-----+-----+-----+-----+ |
19¨ | 4¨ | 4¨ | 5¨ | 6¨ | 4¤ |
+-----+-----+-----+-----+ ---
{18¨}| 5¨ | 5¨ | 8¨ | 3¤ |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 12¤
43¨ | 3¨ | 5¨ | 5¨ | 5¨ | 3¨ | 7¨ | 5¨ | 3¨ | 7¨ | 9¤ (C1 dan C2) |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+ ---
|----- 13¨ -----|-------------- {30¨} --------------|
|----- {13¨} -----|------ 15¨ ------|------ 15¨ ------|
| 1 2 3 | 4 5 6 | 7 8 9 |
Δ Δ Δ True Prime Pairs:
(5,7), (11,13), (17,19)
----+-----+-----+-----+-----+ -----------------------------------------------
786| 1,2 | 2 | 2,3 | 3,4 | {19} |
----+-----+-----+-----+-----+ |
{86}| 4 | 4,5 | 5,6 |{6,7}| 17 Base Zone
+-----+-----+-----+-----+ |
{78}|{7,8}| 8,9 | 12 (M dan F) ----> Δ |
+-----+-----+-----+ -----------
{67}| 9,11|11,12|12,14| 11 <----------- Mid Zone |
----+-----+-----+-----+-----+ |
{6}|15,16|17,18|18,20|21,22| 19 Mirror Zone
+-----+-----+-----+-----+ |
{8}|23,25|25,27|27,29| 18 |
+-----+-----+-----+-----+-----+-----+-----+-----+-------+ -----------
{7}|29,33|33,36|36,39|39,41|41,45|46,51|51,57|58,66|{67,77}| 43 (C1 dan C2)<---Δ
----+-----+-----+-----+-----+-----+-----+-----+-----+-------+ -----------
| 1 2 3 | 4 5 6 | 7 8 9 |
|------ 29' ------|--------------- 139' ----------------|
|------ 102¨ -----|--------------- 66¨ ----------------|Maka selanjutnya topik bahasan untuk angka tigabelas (13) kita bagi dalam dua (2) grup yaitu 13 ke 68 untuk Skema-23 (mulai dari Pratinjau) dan 13 ke 139 untuk Skema-34.
i | Φ | # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ∑° | ∑
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1 | 3 | 1:1:0 | 1 | 2 | {3} | - | - | - | - | - | 102 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
2 | 4 | 1:2:1 | 4 | 5 | 6 | 7 | - | - | - | - | 66 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ {786}
3 | 6 |*1:2:2 | 8 | 9 | 10 | 11 | 12 | {13}| - | - | 329 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
4 | 6 |*1:3:3 | 14 | 15 | 16 | 17 | 18 | {19}| - | - | 289 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
5 | 5 | 1:3:4 | 20 | 21 | 22 | 23 | 24 | - | - | - | 83 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
6 | 3 |*1:3:5 | 25 | 26 | 27 | - | - | - | - | - | 65 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 581
7 | 2 |*1:4:6 | 28 | 29 | - | - | - | - | - | - | 202 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
8 | 7 | 1:4:7 | 30 | 31 | 32 | 33 | 34 | 35 | {36}| - | 231 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
9 | 6 |*1:4:8 | 37 | 38 | 39 | 40 | 41 | {42}| - | - | 329 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ {618}
10 | {6} |*1:4:9 | 43 | 44 | 45 | 46 | 47 | {48}| - | - | 289 |
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========- 61 = 43 + 18 = 18th prime
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========
11 | 3 | 2:1:0 | 49 | 50 | 51 | - | - | - | - | - | 90 | 3Φ
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
12 | 3 | 2:2:1 | 52 | 53 | {54}| - | - | - | - | - | 56 | 241
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
13 | 5 |*2:2:2 | 55 | 56 | 57 | 58 | {59}| - | - | - | 95 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
14 | 4 |*2:3:3 | 60 | 61 | 62 | 63 | - | - | - | - | 32 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
15 | 4 | 2:3:4 | 64 | 65 | 66 | 67 | - | - | - | - | 126 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 836
16 | 5 |*2:3:5 | 68 | 69 | 70 | 71 | 72 | - | - | - | 38 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
17 | 6 |*2:4:6 | 73 | 74 | 75 | 76 | 77 | {78}| - | - | 640 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
{18}| 5 | 2:4:7 | {79}| 80 | 81 | 82 | {83}| - | - | - | {61}|
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
19 | 5 |*2:4:8 | 84 | 85 | 86 | 87 | {88}| - | - | - | 330 | 1072
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
20 | {8} |*2:4:9 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | {96}|{681}| |
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========- 115 = 23 x (2 + 3) = 23 x (1*1*5)
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========
21 | 3 | 3:1:0 | 97 | 98 | {99}| - | - | - | - | - |{299}| 3Φ
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
22 | 5 | 3:2:1 |{100}| 101 | 102 | 103 | 104 | - | - | - | 791 | |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
23 | 5 |*3:2:2 | 105 | 106 | 107 | 108 | 109 | - | - | - | 561 | 6ΦΦ
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
24 | 5 |*3:3:3 | 110 | 111 | 112 | 113 |{114}| - | - | - | 155 | |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
{25}| 3 | 3:3:4 |{115}| 116 |{117}| - | - | - | - | - | 1210|{6ΦΦ9}<-
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
26 | 7 |*3:3:5 | 118 | 119 | 120 | 121 | 122 | 123 |{124}| - | 1879|
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
27 | 5 |*3:4:6 |{125}| 126 | 127 |{128}|{129}| - | - | - | 155 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
28 | 3 | 3:4:7 | 130 | 131 |{132}| - | - | - | - | - | 37 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
29 | {7} |*3:4:8 | 133 | 134 | 135 | 136 | 137 | 138 |{139}| - |{922}|
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+======
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========
21 | 3 | 3:1:0 | 97 | 98 | {99}| - | - | - | - | - |{299}| 3Φ
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
22 | 5 | 3:2:1 |{100}| 101 | 102 | 103 | 104 | - | - | - | 791 | |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
23 | 5 |*3:2:2 | 105 | 106 | 107 | 108 | 109 | - | - | - | 561 | 6ΦΦ
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
24 | 5 |*3:3:3 | 110 | 111 | 112 | 113 |{114}| - | - | - | 155 | |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
{25}| 3 | 3:3:4 | {3}| 4 | {5}| - | - | - | - | - | 1210|{6ΦΦ9}<-
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
26 | 7 |*3:3:5 | 6 | 7 | 8 | 9 | 10 | {11}| {12}| - | 1879|
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
27 | 5 |*3:4:6 | {13}| 14 | 15 | {16}| {17}| - | - | - | 155 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
28 | 3 | 3:4:7 | 18 | 19 | {20}| - | - | - | - | - | 37 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
29 | {7} |*3:4:8 | 21 | 22 | 23 | 24 | 25 | 26 | {27}| - |{922}|
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+======Perhatikan bahwa (24,25,26) adalah formasi sentral 114 ke 139 dimana baris ke 26 yang dimulai dari 118 ke 124 terjadi skema angka silang 18 dan 79 yaitu pada angka 1879.
Urutan ini akan diawali konfigurasi baris ke-4: tiga (3) blok, kemudian mengisi blok sejumlah (2, 4, 4, 3, 4, 9) dengan urutan loncat baris seperti ini:
- 53 + 139 = 192
id: 115 - 139
# | id | Φ | Δ |
-----+-----+-----+-----+
1 | {53}|{139}| {27}| Δ112
-----+-----+-----+-----+
2 | 54 | 138 | 26 |
-----+-----+-----+-----+
3 | 55 |{137}| 25 |
-----+-----+-----+-----+
4 | 56 | 136 | 24 |
-----+-----+-----+-----+
5 | 57 | 135 | 23 |
-----+-----+-----+-----+
6 | 58 | 134 | 22 |
-----+-----+-----+-----+
7 | 59 | 133 | 21 |
-----+-----+-----+-----+
8 | 60 | 132 | 20 |
-----+-----+-----+-----+
9 | 61 | 131 | 19 |
-----+-----+-----+-----+
{10}| 62 | 130 | {18}|
-----+-----+-----+-----+
{11}| 63 | 129 | {17}|
-----+-----+-----+-----+
12 | 64 | 128 | 16 |
-----+-----+-----+-----+
13 | {65}|{127}| 15 |
-----+-----+-----+-----+
14 | {66}|{126}| 14 |
-----+-----+-----+-----+
15 | 67 | 125 | 13 |
-----+-----+-----+-----+
16 | {68}|{124}| 12 |
-----+-----+-----+-----+
17 | {69}|{123}| 11 |
-----+-----+-----+-----+
18 | 70 | 122 | 10 |
-----+-----+-----+-----+
{19}| 71 | 121 | {9}|
-----+-----+-----+-----+
20 | 72 | 120 | 8 |
-----+-----+-----+-----+
21 | 73 | 119 | 7 |
-----+-----+-----+-----+
22 | 74 | 118 | 6 |
-----+-----+-----+-----+
23 | 75 | 117 | 5 |
-----+-----+-----+-----+
24 | 76 | 116 | 4 |
-----+-----+-----+-----+
25 | {77}|{115}| 3 |
=====+=====+=====+=====+
26 | {78}|{114}| {18}| Δ60
-----+-----+-----+-----+
27 | 79 | 113 | 17 |
-----+-----+-----+-----+
28 | 80 | 112 | 16 |
-----+-----+-----+-----+
{29}| 81 | 111 | 15 |
-----+-----+-----+-----+
{30}| 82 | 110 | 14 |
-----+-----+-----+-----+
31 | 83 | 109 | {13}|
-----+-----+-----+-----+
32 | 84 | 108 | 12 |
-----+-----+-----+-----+
{33}| 85 | 107 | 11 |
-----+-----+-----+-----+
34 | 86 | 106 | 10 |
-----+-----+-----+-----+
35 | 87 | 105 | 9 |
-----+-----+-----+-----+
36 | 88 | 104 | {8}|
-----+-----+-----+-----+
37 | 89 | 103 | 7 |
-----+-----+-----+-----+
38 | 90 | 102 | 6 |
-----+-----+-----+-----+
{39}| 91 | 101 | 5 |
-----+-----+-----+-----+
40 | 92 | 100 | 4 |
-----+-----+-----+-----+
41 | {93}| {99}| {3}|
-----+-----+-----+-----+
42 | 94 | 98 | 2 |
-----+-----+-----+-----+
43 | 95 | 97 | 1 |
-----+-----+-----+-----+
- | {Φ96} | - |
-----+-----+-----+-----+
Δ Δ | Δ |
#77 139 | 114 |
i | Φ | # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ∑° | ∑
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
1 | 3 | 1:1:0 | 1 | 2 | 3 | - | - | - | - | - | 102 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
2 | 4 | 1:2:1 | 4 | 5 | 6 | 7 | - | - | - | - | 66 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 786
3 | 6 |*1:2:2 | 8 | 9 | 10 | 11 | 12 | 13 | - | - | 329 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
4 | 6 |*1:3:3 | 14 | 15 | 16 | 17 | 18 | 19 | - | - | 289 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
5 | 5 | 1:3:4 | 20 | 21 | 22 | 23 | 24 | - | - | - | 83 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
6 | 3 |*1:3:5 | 25 | 26 | 27 | - | - | - | - | - | 65 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 581
{7} | 2 |*1:4:6 | {28}| {29}| - | - | - | - | - | - | 202 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
8 | 7 | 1:4:7 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | - | 231 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
9 | 6 |*1:4:8 | 37 | 38 | 39 | 40 | 41 | 42 | - | - | 329 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ {618}
10 | 6 |*1:4:9 | 43 | 44 | 45 | 46 | 47 | 48 | - | - | 289 |
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+========
Disini kita akan lakukan Proses Enkapsulasi dari Metoda OOP untuk mengisi blok² yang terbentuk dengan urutan: 3 (13'), 5 (19), 2 (17'), 6 (18'), 1 (19'), dan 9 (43).
--------
Case A
--------
Case B
----------------+
| ⅓
+--- } ⅔
Case A | ⅓
+---------
| ⅓ |
-----------------+ Φ = ⅔
| ⅓ |
+---------
Case B | ⅓
+--- } ⅔
| ⅓
---------Berikut ini akan diuraikan teknik yang dipakai untuk mendapatkan korelasi dari dua buah kasus yaitu dengan metoda segitiga dalam lingkaran
Deskripsi dan bentuknya terlihat pada artikel berjudul Computational Geometric Topology in Arrangement Theory seperti berikut ini:
The Simpson lines of a triangle trace out a deltoid. If you divide the circle into 3n number of intervals, draw the corresponding Simpson lines, and alternate three colors then the result is a 3-net (or web). The existence of nets are a crux in understanding various topological properties in arrangement theory.
Deltoid adalah hypocycloid dengan tiga cusps, juga dikenal sebagai tricuspoid atau Steiner hypocycloid setelah ahli matematika Swiss Jakob Steiner yang menyelidiki kurva pada 1856.
Di antara yang pertama mempelajari sifat-sifatnya adalah Leonhard Euler saat mengerjakan masalah dalam optik pada 1745.
Deltoid, dinamakan demikian karena terlihat seperti delta Yunani huruf besar (Δ), dibentuk oleh titik pada keliling lingkaran yang berputar di dalam lingkaran lain dengan radius tiga kali lebih besar.
Persamaan parametrik cycloid dengan lingkaran dalam jari-jari r adalah:
x ( t ) = 2 r cos t + r cos 2 t
y ( t ) = 2 r sin t - r sin 2 t
Panjang jalur deltoid adalah 16 r / 3, dan area di dalam deltoid adalah 2π r 2 .
Jika garis singgung ditarik ke deltoid di beberapa titik, P , dan titik-titik di mana garis singgung melintasi deltoid dua cabang lainnya disebut titik A dan B , maka panjang AB sama dengan 4 r.
Jika garis singgung deltoid ditarik pada titik A dan B , mereka akan tegak lurus, dan mereka akan berpotongan pada titik di dalam deltoid yaitu rotasi 180 ° dari titik P tentang pusat lingkaran tetap
114 is the biggest difference of two consecutive six-digit primes (492113 and 492227).
- 3³ = 27
- Φ(13:9) = Φ(29th prime) = Φ(109) = (2+69) + 68 = 71 + 68 = 139
True Prime Vektors ζ(s):
(2,3), (29,89), (36,68), (72,42), (100,50), (2,3), (29,89), ...infinity
----------------------+-----+-----+-----+ ---
7 --------- 1,2:1| 1 | 30 | 40 | {71} (2,3) ‹----------------- |
| +-----+-----+-----+-----+ | |
| 8 ‹------ 3:2| 1 | 30 | 40 | 90 | 161 (7) ‹--- | 5¨
| | +-----+-----+-----+-----+ | | |
| | 6 ‹-- 4,6:3| 1 | 30 | 200 | 231 (10,11,12) ‹--|--- | |
| | | +-----+-----+-----+-----+ | | | ---
--|--|-----» 7:4| 1 | 30 | 40 | 200 | 271 (13) --› | 5® | |
| | +-----+-----+-----+-----+ | | |
--|---› 8,9:5| 1 | 30 | 200 | 231 (14,15) ---------› | 7¨
289 | +-----+-----+-----+-----+-----+ | |
| --› 10:6 | 20 | 5 | 10 | 70 | 90 | 195 (19) --› Φ |{6®} |
--------------------+-----+-----+-----+-----+-----+ | ---
67 --------› 11:7| 5 | 9 | 14 (20) --------› ¤ | |
| +-----+-----+-----+ | |
| 78 ‹----- 12:8| 9 | 60 | 40 | 109 (26) «------------ | 11¨
| | +-----+-----+-----+ | | |
| | 86‹- {13:9}| 9 | 60 | 69 ({27}) «-- Δ (Rep Fork) | 2® | |
| | | +-----+-----+-----+ | | ---
| | ---› 14:10| 9 | 60 | 40 | 109 (28) ------------- | |
| | +-----+-----+-----+ | |
| ---› 15,18:11| 1 | 30 | 40 | {71} (29,30,31,32) ---------- 13¨
329 | +-----+-----+-----+ |
| ‹--------- 19:12| 10 | 60 | 70 (36) ‹--------------------- Φ |
-------------------+-----+-----+ ---
786 ‹------- 20:13| 90 | 90 (38) ‹-------------- ¤ |
| +-----+-----+ |
| 618 ‹- 21,22:14| 8 | 40 | 48 (40,41) ‹---------------------- 17¨
| | +-----+-----+-----+-----+-----+ | |
| | 594 ‹- 23:15| 8 | 40 | 70 | 60 | 100 | 278 (42) «-- |{6'®} |
| | | +-----+-----+-----+-----+-----+ | | ---
--|--|-»24,27:16| 8 | 40 | 48 (43,44,45,46) ------------|---- |
| | +-----+-----+ | |
--|---› 28:17| 100 | 100 (50) --------------------------» 19¨
168 | +-----+ |
| 102-›{29}:18| 50 | 50({68}) ---------> Δ |
----------------------+-----+- 1 x 3 x 9 = 27
- 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 = 6³
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ |
{25}| 3 | 3:3:4 | {3}| {4}| {5}| - | - | - | - | - | 1210|{6ΦΦ9}<-
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
26 | 7 |*3:3:5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | - | 1879|
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
27 | 5 |*3:4:6 | 13 | 14 | 15 | 16 | 17 | - | - | - | 155 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
28 | 3 | 3:4:7 | 18 | 19 | {20}| - | - | - | - | - | 37 |
----+-----+-------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
29 | {7} |*3:4:8 | 21 | 22 | 23 | 24 | 25 | 26 | {27}| - |{922}|
====+=====+=======+=====+=====+=====+=====+=====+=====+=====+=====+=====+======Berikut ini daftar keistimewaan angka 216 menurut wikipedia:
- 216 = 3³ + 4³ + 5³ = 6³, it is the smallest cube that is also the sum of three cubes (Plato was among the first to notice this, and mentioned it in Book VIII of Republic).
- It is also the sum of a twin prime (107 + 109). But since there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number.
- This multiplicative magic square
- has magic constant 216.
- It has been conjectured that each natural number not equal to 216 can be written in the form p + Tx, where p is 0 or a prime, and Tx = x(x+1)/2 is a triangular number.[1]
- In base 10, it is a Harshad number.
- There are 216 fixed hexominoes, the polyominoes made from 6 squares.
- 216 is a Friedman number.
- 216 is the smallest number n, for which n−3, n−2, n−1, n+1, n+2, n+3 are all semiprimes.
Simak untuk keistimewaan² lainnya.
- Zhi-Wei Sun conjectured in March 2008 that 216 is the only number not of the form p + k(k+1)/2, with p = 0 or p prime. [Hasler]
- The numbers 6*216+1, 12*216+1, and 18*216+1 are all prime and hence their product is Carmichael. [Patterson]
- The number of decompositions of 216 into integer summands without regard to order is prime (15285151248481). [Patterson]
- 216 is the smallest number that may be expressed as the difference of the sums of the squares of successive twin primes (13² + 11²) - (7² + 5²). [King]
- The smallest untouchable number which is a cube. [Gupta]
- The smallest and only known cube sandwiched between two triplets of semiprimes (i.e., 213=3*71, 214=2*107, 215=5*43 and 217=7*31, 218=2*109, 219=3*73). [Gupta]
1 + 7 = 8 = 2 x 2 x 2 = 2³ » 23
-----+-----+-----+-----+-----+ ---
19¨ | 3¨ | 4¨ | 6¨ | 6¨ | 4¤ |
-----+-----+-----+-----+-----+ |
17¨ | 5¨ | 3¨ | 2¨ | 7¨ | 4¤ |
+-----+-----+-----+-----+ |
12¨ | 6¨ | 6¨ | 2¤ (M dan F) |
+-----+-----+-----+ 17¤
11¨ | 3¨ | 3¨ | 5¨ | 3¤ |
-----+-----+-----+-----+-----+ |
19¨ | 4¨ | 4¨ | 5¨ | 6¨ | 4¤ |
+-----+-----+-----+-----+ ---
{18¨}| 5¨ | 5¨ | 8¨ | 3¤ |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+ 12¤
43¨ | 3¨ | 5¨ | 5¨ | 5¨ | 3¨ | 7¨ | 5¨ | 3¨ | 7¨ | 9¤ (C1 dan C2) |
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+ ---
|----- 13¨ -----|-------------- {30¨} --------------|
|----- {13¨} -----|------ 15¨ ------|------ 15¨ ------|
| 1 2 3 | 4 5 6 | 7 8 9 |
Δ Δ Δ - 10 x 10 x 10 = 10³ = 1000
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