"Everywhere there is number, there is beauty."
(PROCLUS, 410-485 before J.-C.)


Isomorphous Triplets

1.       Structure of Isomorphous Triplets

The 9 Isomorphous Triplets
37 x 3 = 111 = (12 x 9) + 3
37 x 3 = 1x2x3x13 + (1/2)x3!x23 + (1/2)x3!x33
12=1!x2!x3! ; 9=1!+2!+3! ; 3=(1/2)x3!
12=3 4 times ; 9=3 3 times ; 3=3 1 time  
3 x 37  37 x 3 x 1  111  1+ 1 +1 = 3 
6 x 37  37 x 3 x 2  222  2+ 2 +2 = 6 
9 x 37  37 x 3 x 3  333  3+ 3+ 3 = 9 
12 x 37  37 x 3 x 4  444  4+4+4 = 12 
15 x 37  37 x 3 x 5  555  5+5+5 = 15 
18 x 37  37 x 3 x 6  666  6+6+6 = 18 
21 x 37  37 x 3 x 7  777  7+7+7 = 21 
24 x 37  37 x 3 x 8  888  8+8+8 = 24 
27 x 37  37x 3 x 9  999  9+9+9 = 27 



Golden Ratio Phi (Φ), Pi (π) and Isomorphous Triplets Links


73 = 343        3 4 + 3         37
Φ = Phi = 1.618034    73Φ = 554.985662 = 555
φ = Φ -1 = phi = 0.618034    73 π φ = 665.972672 = 666
Circle = 360 = 2 π radians     1 = 2 π/360 radians
or 1 = π/180 radians   then: 30 = π/6
111   =   (1/5)73Φ   =   (1/6)73πφ
222   =   (2/5)73Φ   =   (2/6)73πφ
333   =   (3/5)73Φ   =   (3/6)73πφ
444    =   (4/5)73Φ   =   (4/6)73πφ
555   =   (5/5)73Φ   =   (5/6)73πφ
666   =   (6/5)73Φ   =   (6/6)73πφ
777   =   (7/5)73Φ   =   (7/6)73πφ
888    =   (8/5)73Φ   =   (8/6)73πφ
999   =   (9/5)73Φ   =   (9/6)73πφ
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2.       Generation of Isomorphous Triplets


Number 99: generator of Isomorphous Triplets
1 99 = 0.0101010 10 10 ... 101 + 010 = 111
2 99 = 0.0202020 20 20 ... 202 + 020 = 222
3 99 = 0.0303030 30 30 ... 303 + 030 = 333
4 99 = 0.0404040 40 40 ... 404 + 040 = 444
5 99 = 0.0505050 50 50 ... 505 + 050 = 555
6 99 = 0.0606060 60 60 ... 606 + 060 = 666
7 99 = 0.0707070 70 70 ... 707 + 070 = 777
8 99 = 0.0808080 80 80 ... 808 + 080 = 888
9 99 = 0.0909090 90 90 ... 909 + 090 = 999
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3.       Synthesis of the results


Numerical
Root
(n)
Sums
of
terms
(Dividends)
Isomorphous Triplets
(Dividers)
Number
of
cases
111
(2n)
222
(n)
333 444 555 666 777 888 999
3 666 6 3 2     1       4
4 888 8 4   2       1   4
5 1110 10 5     2         3
6 1332 12 6 4 3   2       5
7 1554 14 7         2     3
8 1776 16 8   4       2   4
9 1998 18 9 6     3     2 5
10 2220 20 10   5 4         4
11 2442 22 11               2
12 2664 24 12 8 6   4   3   6
13 2886 26 13               2
14 3108 28 14   7     4     4
15 3330 30 15 10   6 5       5
16 3552 32 16   8       4   4
17 3774 34 17               2
18 3996 36 18 12 9   6     4 6
19 4218 38 19               2
20 4440 40 20   10 8     5   5
21 4662 42 21 14     7 6     5
22 4884 44 22   11           3
23 5106 46 23               2
24 5328 48 24 16 12   8   6   6
TOTAL 65934 22 22 8 11 4 8 3 6 2 86


Links between Isomorphous Triplets and Cumulative Sum 65934
Divider
Dividend
111 222 333 444 555 666 777 888 999
65934 594 297 198 148.5 118.8 99 84.86 74.25 66
Numerical Root of 65934: 6+5+9+3+4=272+7=9
111+222+333+444+555+666+777+888+999=4995 4+9+9+5=272+7=9
65934 4995 = 13.213        13.21 + 3 + 2 = 6
65934 99 = 666        65934 66 = 999
4995 65934 = 0,0 757575 757575 757575 757575 ...
757575757 + 575 = 13321332 2 = 666    1 + 3 + 3 + 2 = 9
4995 7,5 = 666        4995 5 = 999
65934 + 4995 = 709297 + 0 + 9 + 2 + 9 = 272 + 7 = 9
65934 - 4995 = 609396 + 0 + 9 + 3 + 9 = 272 + 7 = 9
70929 + 60939 = 1318681 + 3 + 1 + 8 + 6 + 8 = 272 + 7 = 9
131868 4995 = 26.4 = 2 x 13.226
131868 198 = 666         131868 132 = 999
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4.       Inventory of the nonisomorphous triplets


Construction of the nonisomorphous triplets
Basis Numerical Combinaisons Nb. Cases
Basis 0 012 023 034 045 056 067 078 089 36
013 024 035 046 057 068 079  
014 025 036 047 058 069    
015 026 037 048 059      
016 027 038 049        
017 028 039          
018 029            
019              
Basis 1 123 134 145 156 167 178 189   28
124 135 146 157 168 179    
125 136 147 158 169      
126 137 148 159        
127 138 149          
128 139            
129              
Basis 2 234 245 256 267 278 289     21
235 246 257 268 279      
236 247 258 269        
237 248 259          
238 249            
239              
Basis 3 345 356 367 378 389       15
346 357 368 379        
347 358 369          
348 359            
349              
Basis 4 456 467 478 489         10
457 468 479          
458 469            
459              
Basis 5 567 578 589           6
568 579            
569              
Basis 6 678 689             3
679              
Basis 7 789               1
TOTAL   120
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5.       Classification of the nonisomorphous triplets according to the numerical root


Classification of the nonisomorphous triplets
Basis
Root
Basis 0 Basis 1 Basis 2 Basis 3 Basis 4 Basis 5 Basis 6 Basis 7 TOTAL
3 012               1 
4 013               1 
5 014 023               2 
6 015 024 123             3 
7 016 025
034
124             4 
8 017 026
035
125 134             5 
9 018 027
036 045
126 135 234           7 
10 019 028
037 046
127 136
145
235           8 
11 029 038
047 056
128 137
146
236 245           9 
12 039 048
057
129 138
147 156
237 246 345         10 
13 049 058
067
139 148
157
238 247
256
346         10 
14 059 068 149 158
167
239 248
257
347 356         10 
15 069 078 159 168 249 258
267
348 357 456       10 
16 079 169 178 259 268 349 358
367
457       9 
17 089 179 269 278 359 368 458 467       8 
18   189 279 369 378 459 468 567     7 
19     289 379 469 478 568     5 
20       389 479 569 578     4 
21         489 579 678   3 
22           589  679   2 
23             689   1 
24               789 1
TOTAL 36 28 21 15 10 6 3 1 120
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6.       Links between numerical root and sum of the circular shifts


Structure of the sums of circular shifts
Numerical Root
n
Double Root
2n
Median Root
r2n
Final Root
rα
Derived Sum
Sn
3 6 6   666
4 8 8   888
5 10 ⇒ 1+0=1   1110
6 12 ⇒ 1+2=3   1332
7 14 ⇒ 1+ 4=5   1554
8 16 ⇒ 1+ 6=7   1776
9 18 ⇒ 1+8=9   1998
10 20 ⇒ 2+0=2   2220
11 22 ⇒ 2+2=4   2442
12 24 ⇒ 2+4=6   2664
13 26 ⇒ 2+6=8   2886
14 28 ⇒ 2+8=10 ⇒ 1+0=1 3108
(3=2+1)
15 30 ⇒ 3+0=3   3330
16 32 ⇒ 3+2=5   3552
17 34 ⇒ 3+4=7   3774
18 36 ⇒ 3+6=9   3996
19 38 ⇒ 3+8=11 ⇒ 1+1=2 4218
(4=3+1)
20 40 ⇒ 4+0=4   4440
21 42 ⇒ 4+2=6   4662
22 44 ⇒ 4+4=8   4884
23 46 ⇒ 4+6=10 ⇒ 1+0=1 5106
(5=4+1)
24 48 ⇒ 4+8=12 ⇒ 1+2=3 5328
(5=4+1)
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666: Bivalent Myth