"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    ⇒     37Φ = Phi = 1.618034    73Φ = 554.985662 = 555φ = Φ -1 = phi = 0.618034    73 π φ = 665.972672 = 666 Circle = 360º = 2 π radians   ⇒   1º = 2 π/360 radians or 1º = π/180 radians   then: 30º = 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

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

3.       Synthesis of the results

 NumericalRoot(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 DividerDividend 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=27 ⇒ 2+7=9 111+222+333+444+555+666+777+888+999=4995 ⇒ 4+9+9+5=27 ⇒ 2+7=9 65934 ÷ 4995 = 13.2 ≈ 13        13.2 ⇒ 1 + 3 + 2 = 6 65934 ÷ 99 = 666        65934 ÷ 66 = 999 4995 ÷ 65934 = 0,0 757575 757575 757575 757575 ... 757575 ⇒ 757 + 575 = 1332 ⇒ 1332 ÷ 2 = 666    1 + 3 + 3 + 2 = 9 4995 ÷ 7,5 = 666        4995 ÷ 5 = 999 65934 + 4995 = 70929 ⇒ 7 + 0 + 9 + 2 + 9 = 27 ⇒ 2 + 7 = 9 65934 - 4995 = 60939 ⇒ 6 + 0 + 9 + 3 + 9 = 27 ⇒ 2 + 7 = 9 70929 + 60939 = 131868 ⇒ 1 + 3 + 1 + 8 + 6 + 8 = 27 ⇒ 2 + 7 = 9 131868 ÷ 4995 = 26.4 = 2 x 13.2 ≈ 26 131868 ÷ 198 = 666         131868 ÷ 132 = 999

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

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

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 Rootr2n=α Final Rootrα 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)