Golden Powers in the Fibonacci series
08/06/10 12:26 Filed in: Mathematics
Below is a table showing how taking increasingly
distant intervals in the Fibonacci series converge to
powers of the golden ratio (Phi).
|
|
Phi (converging)
|
Phi-squared
|
Phi-cubed
|
Phi^4
|
Phi^5
|
Phi^6
|
Phi^7
|
Phi^8
|
|
1
|
||||||||
| 1
|
1
|
|||||||
| 2
|
2
|
2
|
||||||
| 3
|
1.5
|
3
|
3
|
|||||
| 5
|
1.666666667
|
2.5
|
5
|
5
|
||||
| 8
|
1.6
|
2.666666667
|
4
|
8
|
8
|
|||
| 13
|
1.625
|
2.6
|
4.333333333
|
6.5
|
13
|
13
|
||
| 21
|
1.615384615
|
2.625
|
4.2
|
7
|
10.5
|
21
|
21
|
|
| 34
|
1.619047619
|
2.615384615
|
4.25
|
6.8
|
11.33333333
|
17
|
34
|
34
|
| 55
|
1.617647059
|
2.619047619
|
4.230769231
|
6.875
|
11
|
18.33333333
|
27.5
|
55
|
| 89
|
1.618181818
|
2.617647059
|
4.238095238
|
6.846153846
|
11.125
|
17.8
|
29.66666667
|
44.5
|
| 144
|
1.617977528
|
2.618181818
|
4.235294118
|
6.857142857
|
11.07692308
|
18
|
28.8
|
48
|
| 233
|
1.618055556
|
2.617977528
|
4.236363636
|
6.852941176
|
11.0952381
|
17.92307692
|
29.125
|
46.6
|
| 377
|
1.618025751
|
2.618055556
|
4.235955056
|
6.854545455
|
11.08823529
|
17.95238095
|
29
|
47.125
|
| 610
|
1.618037135
|
2.618025751
|
4.236111111
|
6.853932584
|
11.09090909
|
17.94117647
|
29.04761905
|
46.92307692
|
| 987
|
1.618032787
|
2.618037135
|
4.236051502
|
6.854166667
|
11.08988764
|
17.94545455
|
29.02941176
|
47
|
| 1597
|
1.618034448
|
2.618032787
|
4.236074271
|
6.854077253
|
11.09027778
|
17.94382022
|
29.03636364
|
46.97058824
|
| 2584
|
1.618033813
|
2.618034448
|
4.236065574
|
6.854111406
|
11.09012876
|
17.94444444
|
29.03370787
|
46.98181818
|
| 4181
|
1.618034056
|
2.618033813
|
4.236068896
|
6.854098361
|
11.09018568
|
17.94420601
|
29.03472222
|
46.97752809
|
| 6765
|
1.618033963
|
2.618034056
|
4.236067627
|
6.854103343
|
11.09016393
|
17.94429708
|
29.03433476
|
46.97916667
|
| 10946
|
1.618033999
|
2.618033963
|
4.236068111
|
6.85410144
|
11.09017224
|
17.9442623
|
29.03448276
|
46.97854077
|
| 17711
|
1.618033985
|
2.618033999
|
4.236067926
|
6.854102167
|
11.09016907
|
17.94427558
|
29.03442623
|
46.97877984
|
| 28657
|
1.61803399
|
2.618033985
|
4.236067997
|
6.85410189
|
11.09017028
|
17.94427051
|
29.03444782
|
46.97868852
|
| 46368
|
1.618033988
|
2.61803399
|
4.23606797
|
6.854101996
|
11.09016982
|
17.94427245
|
29.03443957
|
46.9787234
|
| 75025
|
1.618033989
|
2.618033988
|
4.23606798
|
6.854101955
|
11.09016999
|
17.94427171
|
29.03444272
|
46.97871008
|
| 121393
|
1.618033989
|
2.618033989
|
4.236067976
|
6.854101971
|
11.09016993
|
17.94427199
|
29.03444152
|
46.97871517
|
| 196418
|
1.618033989
|
2.618033989
|
4.236067978
|
6.854101965
|
11.09016995
|
17.94427188
|
29.03444198
|
46.97871323
|
| 317811
|
1.618033989
|
2.618033989
|
4.236067977
|
6.854101967
|
11.09016994
|
17.94427192
|
29.03444181
|
46.97871397
|
| 514229
|
1.618033989
|
2.618033989
|
4.236067978
|
6.854101966
|
11.09016994
|
17.94427191
|
29.03444187
|
46.97871369
|
| 832040
|
1.618033989
|
2.618033989
|
4.236067977
|
6.854101966
|
11.09016994
|
17.94427191
|
29.03444185
|
46.97871379
|
| 1346269
|
1.618033989
|
2.618033989
|
4.236067978
|
6.854101966
|
11.09016994
|
17.94427191
|
29.03444186
|
46.97871375
|
| 2178309
|
1.618033989
|
2.618033989
|
4.236067977
|
6.854101966
|
11.09016994
|
17.94427191
|
29.03444185
|
46.97871377
|