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Rules Of Population
Introduction
Much more obscure than the Rule Of 70 are the Rule Of 110 (for trebling and thirding) and the Rule Of 140 (for quadrupling and quartering). Other rules exist for other factors, but these two will be sufficient to confirm some natural patterns in nature.
Rule Of 110
Taking a growth rate of r, the accurate value X is calculated as:
X = Ln 3 / Ln (1 + (r / 100))
Taking a growth rate of r, the accurate value Y is calculated as:
Y = Ln 3 / Ln (1 - (r / 100))
Curiously enough this difference seems to equate to the Natural Logarithm of 3 (Ln 3 = 1.098612), which is where we get the Rule Of 110 in the first place. This confirms a similar finding for the Rule Of 70.
Ln 3 = X + Y
This can be written:
X = ABS( Y) + Ln 3
| Trebling Period | Thirding Period | X + Y | |||
| Growth Rate | Rule Of 110 |
Accurate Value (X) |
Rule Of 110 |
Accurate Value (Y) |
|
| 1 | 110 | 110.409624 | 110 | -109.3110026 | 1.098621 |
| 2 | 55 | 55.47810764 | 55 | -54.37945872 | 1.098649 |
| 3 | 36.66666667 | 37.16700967 | 36.66666667 | -36.06831495 | 1.098695 |
| 4 | 27.5 | 28.01102276 | 27.5 | -26.91226388 | 1.098759 |
| 5 | 22 | 22.51708531 | 22 | -21.41824388 | 1.098841 |
| 6 | 18.33333333 | 18.85417668 | 18.33333333 | -17.75523427 | 1.098942 |
| 7 | 15.71428571 | 16.23757367 | 15.71428571 | -15.13851178 | 1.099062 |
| 8 | 13.75 | 14.27491459 | 13.75 | -13.17571468 | 1.0992 |
| 9 | 12.22222222 | 12.74822067 | 12.22222222 | -11.6488641 | 1.099357 |
| 10 | 11 | 11.52670461 | 11 | -10.42717266 | 1.099532 |
Rule Of 140
Taking a growth rate of r, the accurate value X is calculated as:
X = Ln 4 / Ln (1 + (r / 100))
Taking a growth rate of r, the accurate value Y is calculated as:
Y = Ln 4 / Ln (1 - (r / 100))
Curiously enough this difference seems to equate to the Natural Logarithm of 4 (Ln 4 = 1.386294), which is where we get the Rule Of 140 in the first place. Again, a natural pattern is confirmed.
Ln 4 = X + Y
This can be written:
X = ABS( Y) + Ln 4
| Growth Rate | Quadrupling Period | Quartering Period | X + Y | ||
| Rule Of 140 |
Accurate Value (X) |
Rule Of 140 |
Accurate Value (Y) |
||
| 1 | 140 | 139.3214338 | 140 | -137.9351279 | 1.386306 |
| 2 | 70 | 70.00557756 | 70 | -68.61923698 | 1.386341 |
| 3 | 46.66666667 | 46.8995445 | 46.66666667 | -45.51314613 | 1.386398 |
| 4 | 35 | 35.34597537 | 35 | -33.95949604 | 1.386479 |
| 5 | 28 | 28.41339817 | 28 | -27.02681467 | 1.386583 |
| 6 | 23.33333333 | 23.79132209 | 23.33333333 | -22.40461117 | 1.386711 |
| 7 | 20 | 20.4895367 | 20 | -19.10267502 | 1.386862 |
| 8 | 17.5 | 18.01293668 | 17.5 | -16.62590083 | 1.387036 |
| 9 | 15.55555556 | 16.08646345 | 15.55555556 | -14.69922992 | 1.387234 |
| 10 | 14 | 14.54508179 | 14 | -13.15762696 | 1.387455 |
Conclusion
Although not as popular as the Rule Of 70, it was worth examining the Rule Of 110 and the Rule Of 140 to confirm the suspicion that the Natural Logarithm of a factor (in this case 3 and 4) appears to be roughly the difference between the time it takes a population to grow by that factor ( multiply by 3 or multiply by 4) and the time it takes to shrink by that factor (divide by 3 or divide by 4).
This brief investigation also confirm that shrinkage time is always the smaller of the two (the time to multiply and the time to divide).