Formulas for .618 Explained - Numbers
1. The first section gives a list ofthe squares and square roots related to .618.
From this list it can be seen that pi - or a number very close to pi - is 4 times the square root of phi (4 X .786 = 3.144) when we take phi as .618. This is an example of the reason why in a unified system it is best to take phi as .618. If phi is taken as 1.618, then it is very hard to relate it to any other number since it is difficult to square it or multiply it.
The first number related to phi is .382 which is .618 squared. The two numbers added together equal 1 while .382 divided by .618 equals .618. These two properties together explain the unique properties of .618. There is no other fraction in which a square and its root equal 1.
Any number, dimension or angle multiplied by .618 is divided into two numbers which added together equal the whole (i.e. 1 or unity or "the whole") but which divided by each other equal .618. The two numbers are really .618X and .382X; X cancels out and then .618 divided into its square gives .618.
The second number related to phi is .786, the square root of phi.
I think it very likely that the square root of phi or .786 is the actual length of a one quarter of the circumference of a circle whose diameter is 1 and whose circumference is 3.14 or pi. Here's two main reasons why.
First: If we take the present value of pi reached by approximation, we have to assume that it is very slightly different from the true value of pi. Now very close to the present value of pi (4 times .78537 = 3.1415) is a number, (4 times .786 = 3.144). Because .786 is the square root of phi, it has the same scaling properties which are characteristic of the circle. Is this just a coincidence or does this number enter into the basic construction of the circle as the explanation of its properties?
Second: if we take the present value of pi as 3.1415 and divide by 4 we get .78537. But if we take the present value of pi as 3.142 and divide by 4 we get .7855. Rounding again we get .786. That's how close the present value of pi is to being based on .786. It's right on the halfway mark. So since this halfway mark is an approximation slightly less than reality perhaps we can say that that the real value of pi is beyond the halfway mark. Then wouldn't it be true that .786 times 4 (3.144) is closer to pi than .785 (3.14) or even than .78537 (3.1415)?
The next number is the fourth root of .618. If .786, the square root of .618, is one quarter of the circumference of a circle, then this number squares the circle.
The next number is .146, the fourth power of .618, . This number is hard to explain without a diagram but quite interesting. If you have a circle whose diameter is 1, its radius is .5. If this radius lies along the hypotenuse of a square whose side is .5, it is more than half the hypotenuse and the difference is .1465 (.5 - .3535 = .1465). I can't think how to prove it, yet it seems that this again suggests that the circle is related to the square root of phi.
Furthermore, the four right angle triangles made by the crossing of the two hypotenuses are each exactly the same triangle as is formed by dropping a line from the point where the circle crosses the hypotenuse to the side of the square below or above (but not sideways). This gives a way to relate a moving radius such as occurs in a log spiral (at any given moment, it is a circle with a given radius) to similar circles and squares.
Next post, I will explain how these numbers simplify such concepts as the Golden Mean, the Golden Angle, and the relation of of high Fibonacci numbers to .618.
From this list it can be seen that pi - or a number very close to pi - is 4 times the square root of phi (4 X .786 = 3.144) when we take phi as .618. This is an example of the reason why in a unified system it is best to take phi as .618. If phi is taken as 1.618, then it is very hard to relate it to any other number since it is difficult to square it or multiply it.
The first number related to phi is .382 which is .618 squared. The two numbers added together equal 1 while .382 divided by .618 equals .618. These two properties together explain the unique properties of .618. There is no other fraction in which a square and its root equal 1.
Any number, dimension or angle multiplied by .618 is divided into two numbers which added together equal the whole (i.e. 1 or unity or "the whole") but which divided by each other equal .618. The two numbers are really .618X and .382X; X cancels out and then .618 divided into its square gives .618.
The second number related to phi is .786, the square root of phi.
I think it very likely that the square root of phi or .786 is the actual length of a one quarter of the circumference of a circle whose diameter is 1 and whose circumference is 3.14 or pi. Here's two main reasons why.
First: If we take the present value of pi reached by approximation, we have to assume that it is very slightly different from the true value of pi. Now very close to the present value of pi (4 times .78537 = 3.1415) is a number, (4 times .786 = 3.144). Because .786 is the square root of phi, it has the same scaling properties which are characteristic of the circle. Is this just a coincidence or does this number enter into the basic construction of the circle as the explanation of its properties?
Second: if we take the present value of pi as 3.1415 and divide by 4 we get .78537. But if we take the present value of pi as 3.142 and divide by 4 we get .7855. Rounding again we get .786. That's how close the present value of pi is to being based on .786. It's right on the halfway mark. So since this halfway mark is an approximation slightly less than reality perhaps we can say that that the real value of pi is beyond the halfway mark. Then wouldn't it be true that .786 times 4 (3.144) is closer to pi than .785 (3.14) or even than .78537 (3.1415)?
The next number is the fourth root of .618. If .786, the square root of .618, is one quarter of the circumference of a circle, then this number squares the circle.
The next number is .146, the fourth power of .618, . This number is hard to explain without a diagram but quite interesting. If you have a circle whose diameter is 1, its radius is .5. If this radius lies along the hypotenuse of a square whose side is .5, it is more than half the hypotenuse and the difference is .1465 (.5 - .3535 = .1465). I can't think how to prove it, yet it seems that this again suggests that the circle is related to the square root of phi.
Furthermore, the four right angle triangles made by the crossing of the two hypotenuses are each exactly the same triangle as is formed by dropping a line from the point where the circle crosses the hypotenuse to the side of the square below or above (but not sideways). This gives a way to relate a moving radius such as occurs in a log spiral (at any given moment, it is a circle with a given radius) to similar circles and squares.
Next post, I will explain how these numbers simplify such concepts as the Golden Mean, the Golden Angle, and the relation of of high Fibonacci numbers to .618.
posted by wildswan at 12/26/2006 08:52:00 AM
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