On .618
On .618
Past History of .618
There is a number .618, which from the time of the Egyptians, has been considered a constant in nature like Pi (3.14). The Egyptians used it in setting proportions when they built the Pyramids and the Greeks used it in building the Parthenon. The Renaissance rediscovered it and Leonardo used it in his pictures. Kepler tried to use it to build a cosmography. It was swept into the rubbish bin when Kepler showed that the form of the planetary orbits was an ellipse not a circle because .618 builds forms that are circles, spheres or spirals with inscribed triangles or squares, not ellipses.
Fall from Grace of .618
Ejected from the heavens, it was not allowed to remain on earth because it was assumed that astronomical forms and organic forms were basically the same. This has been assumed since earliest times. The basis of nature worship is the idea that life and the stars emerge from a common matrix (aka Mother Nature) and develop their forms in a common way.
Renaissance of .618 Explained
But I don't think that this is so. There is an observable difference between organic and non-organic forms which relates organic forms but not inorganic forms to .618. The difference is that in a symmetrical non-organic form such as a snowflake or a crystal or even a virus the opposite sides of the symmetry, The points of the snowflake, are joined by a lines passing straight through the center whereas in an organic form which appears to have a similar symmetry similar lines do not pass through the center or even intersect somewhere else. They form sets of triangles. But the lines from the points of a organic form, for instance from the petals of a flower, do pass through a center if they are considered as built up in time one after another - each petal separated from the previous petal by the Golden Angle. The Golden Angle is 222.5 degrees or .618.
So since this difference exists it seems to me that .618 is part of organic nature but not inorganic.
Need to Unify .618
In trying to understand how forms could be built up, I encountered the difficulty that the fame of .618 had caused it to be considered in very different ways. For instance, as the Golden Angle, it is seen as .618x360. But as the Golden Section it is seen as 1 + .618. There is a construction called The Flying Squares in which subdivisions are created by a formula involving the square root of 5, yet if you simply look at the form of the Flying Squares you are looking at Fibonacci numbers so that subdivisions could be created without the square root of 5. After 34 all adjacent Fibonacci numbers are in the ratio .618. But how does it all fit together to build natural form? .618 - E Pluribus Unum
To see how it fits together to build organic form we could start with how it fits mathematically and then look for a physical law. I fit it together in this way.
There is a number and only one number and that number is .382 which meets these two conditions: X divided by the square root of X equals the square root of X and X plus the square root of X equals 1.
.618 is the square root of .382.
.618 + .382 = 1 and .382/.618 = .618
Everything else flows from this unique relationship and can be defined in terms of it.
The Golden Angle and .618 Newly Understood
The Golden Angle is a version of X plus the square root of X = 1 in which X times a number plus the square root of X times the same number equals the number. The two decimals, .382 and .618, equal one so any number multiplied by them is divided into two fractions of itself which, added together, equal itself.
X times 360 degrees plus the square root of X times 360 degrees = 360 degrees
.382 times 360 = 137.5; .618 times 360 = 222.5; 137.5 + 222.5 = 360
The other relationship X divided by the square root of X = the square root of X also holds so that the two angles 137.5 and 222.5 have the relationship that 137.5/222.5 = .618 is in the same ratio as 222.5 / 360 = .618. The smaller angle is to the larger as the larger is to the whole - the definition of the Golden Angle.
The Golden Section and .618 Newly Understood
The Golden Section is a version of X divided by the square root of X = the square root of X.
X times Length divided by the square root of X times Length equals the square root of X because the Length is in the numerator and denominator and cancels out.
For instance a line is ten feet long:
.382 X 10/ .618 X 10 = 3.82/6.18 = .618
6.18/ 10 = .618
The other relationship X + the square root of X = 1 holds so that the length is divided into two fractions equal to the whole length. The larger fraction is the length multipied by .618 so naturally this fraction divided by the whole length equals .618. The two fractions of the length as a ratio equal .618 and the larger fraction divided by the whole has the same ratio .618. This is the definition of the Golden Section (and the mean and extreme ratio.)
Coming Next - Fibonacci and .618
Star Attraction - Fibonacci and the Flying Squares
Next I will show how Fibonacci numbers fit into this but first I have to figure out how to get pictures into this blog because Fibonacci numbers require the Flying Squares to understand them.
Past History of .618
There is a number .618, which from the time of the Egyptians, has been considered a constant in nature like Pi (3.14). The Egyptians used it in setting proportions when they built the Pyramids and the Greeks used it in building the Parthenon. The Renaissance rediscovered it and Leonardo used it in his pictures. Kepler tried to use it to build a cosmography. It was swept into the rubbish bin when Kepler showed that the form of the planetary orbits was an ellipse not a circle because .618 builds forms that are circles, spheres or spirals with inscribed triangles or squares, not ellipses.
Fall from Grace of .618
Ejected from the heavens, it was not allowed to remain on earth because it was assumed that astronomical forms and organic forms were basically the same. This has been assumed since earliest times. The basis of nature worship is the idea that life and the stars emerge from a common matrix (aka Mother Nature) and develop their forms in a common way.
Renaissance of .618 Explained
But I don't think that this is so. There is an observable difference between organic and non-organic forms which relates organic forms but not inorganic forms to .618. The difference is that in a symmetrical non-organic form such as a snowflake or a crystal or even a virus the opposite sides of the symmetry, The points of the snowflake, are joined by a lines passing straight through the center whereas in an organic form which appears to have a similar symmetry similar lines do not pass through the center or even intersect somewhere else. They form sets of triangles. But the lines from the points of a organic form, for instance from the petals of a flower, do pass through a center if they are considered as built up in time one after another - each petal separated from the previous petal by the Golden Angle. The Golden Angle is 222.5 degrees or .618.
So since this difference exists it seems to me that .618 is part of organic nature but not inorganic.
Need to Unify .618
In trying to understand how forms could be built up, I encountered the difficulty that the fame of .618 had caused it to be considered in very different ways. For instance, as the Golden Angle, it is seen as .618x360. But as the Golden Section it is seen as 1 + .618. There is a construction called The Flying Squares in which subdivisions are created by a formula involving the square root of 5, yet if you simply look at the form of the Flying Squares you are looking at Fibonacci numbers so that subdivisions could be created without the square root of 5. After 34 all adjacent Fibonacci numbers are in the ratio .618. But how does it all fit together to build natural form? .618 - E Pluribus Unum
To see how it fits together to build organic form we could start with how it fits mathematically and then look for a physical law. I fit it together in this way.
There is a number and only one number and that number is .382 which meets these two conditions: X divided by the square root of X equals the square root of X and X plus the square root of X equals 1.
.618 is the square root of .382.
.618 + .382 = 1 and .382/.618 = .618
Everything else flows from this unique relationship and can be defined in terms of it.
The Golden Angle and .618 Newly Understood
The Golden Angle is a version of X plus the square root of X = 1 in which X times a number plus the square root of X times the same number equals the number. The two decimals, .382 and .618, equal one so any number multiplied by them is divided into two fractions of itself which, added together, equal itself.
X times 360 degrees plus the square root of X times 360 degrees = 360 degrees
.382 times 360 = 137.5; .618 times 360 = 222.5; 137.5 + 222.5 = 360
The other relationship X divided by the square root of X = the square root of X also holds so that the two angles 137.5 and 222.5 have the relationship that 137.5/222.5 = .618 is in the same ratio as 222.5 / 360 = .618. The smaller angle is to the larger as the larger is to the whole - the definition of the Golden Angle.
The Golden Section and .618 Newly Understood
The Golden Section is a version of X divided by the square root of X = the square root of X.
X times Length divided by the square root of X times Length equals the square root of X because the Length is in the numerator and denominator and cancels out.
For instance a line is ten feet long:
.382 X 10/ .618 X 10 = 3.82/6.18 = .618
6.18/ 10 = .618
The other relationship X + the square root of X = 1 holds so that the length is divided into two fractions equal to the whole length. The larger fraction is the length multipied by .618 so naturally this fraction divided by the whole length equals .618. The two fractions of the length as a ratio equal .618 and the larger fraction divided by the whole has the same ratio .618. This is the definition of the Golden Section (and the mean and extreme ratio.)
Coming Next - Fibonacci and .618
Star Attraction - Fibonacci and the Flying Squares
Next I will show how Fibonacci numbers fit into this but first I have to figure out how to get pictures into this blog because Fibonacci numbers require the Flying Squares to understand them.
posted by wildswan at 9/22/2006 06:56:00 AM
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