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.

Formulas for .618 (p.1 and 2)

These are the unified formulas for .618. Due some glitch they were sent out of order, page 2 is before page1. You should begin at 1. The Numbers Related to .618. In my next post I'll explain these a bit

Life and .618 (3)

Unified Understanding Needed
We need a unified understanding of log spirals, Fibonacci numbers and the Golden Angle- the particular unification we are seeking being the one expressed in plants. We also need to be able to measure curves which, as far as I know, can only be done by measuring squares and rectangles and relating these measurements to curves. In other words, we need to know the relationships in numbers of circles, spirals, rectangles and squares.
The Physical Idea - DNA and .618
Tproposition is that spiral forms in living Nature ultimarely flow from from the spiral form of DNA; and the presence in Nature of forms based on .618 flows from the presence in DNA of forms based on .618.
DNA is a form of spiral called a helix - the spiral one sees in a screw. But DNA does not spiral evenly about its center or pole as does a screw. The two sides of A DNA helix have their sugars and phosphates in reverse order - 3, 5 is joined to 5, 3. This causes the helix to turn unevenly about its pole, swinging out now to one side, now the other. The distance it swings out is .618 matched by .382. Thus there are two "notches" in the helix which recur every 34 Angstroms. They, like leaf phyllotaxy, are based on .618 and, I think, leaf phyllotaxy is based on them.

So this why I keep working on unifying log spirals, the Golden Angle and Fibonacci numbers - because these ideas can never be really tested without better ways of measuring spirals and without a better understanding of how spirals, Fibonacci numbers and the Golden Angle are all interrelated mathematically. Ultimately I believe that all the features that have been discovered about these forms and numbers will turn out to explain one or another feature of organic life.

So now I will post what I know so far about how these are united. Or I will as soon as I figure out how to post a picture since this blog's types don't include mathematical features.

Life and .618 (2)

In Life and .618, I laid out a way of looking at plant growth which suggests that leaf phyllotaxy, stele growth, branch "phyllotaxy", and flower phyllotaxy are all interrelated in growth. This requires us to view plant growth as a matter of precisely aligned cell positioning - a set of proportions based on Fibonacci numbers, the Golden Angle and log spirals. At present cell growth is seen as a matter of doubling allied with a sort of jostling for position akin to people stepping on an elevator and taking positions as far from each other as possible - a sort of collision-based packing. This is incompatible with the sort of precise alignment actually present.

But to analyze the alignment as it is we need the tools that would work on these different proportions. We need a unified understanding of log spirals, Fibonacci numbers and the Golden Angle. (TBC)

Life and .618

I start from the point of view that log spirals, the Golden Angle and Fibonacci numbers are are joined in nature so they must be joined mathematically. This mathematical join if found would express a physical law or laws basic and particular to life, to organic nature.
Looking for the mathematical join , I look at leaf phyllotaxy - the clearest example in nature of the joining of Fibonacci numbers, the Golden Angle and log spirals. And what is joined to leaf phyllotaxy? -- the position of the vascular bundles, the branches, the flowers and the growth of the stem. When we see this we see why we need a unified mathematical explanation of the spirals, angles and numbers involved in their growth.
Gathering all we now know about the formation of leaf primordia and all we once knew about plant morphology in the nineteenth century (see Asa Gray) and all that the combination implies into one compact bundle we form the following picture:

Leaf primordia from around the shoot apical meristem (SAM) ,eacg separated in space by 137.5 degrees, the Golden angle from the preceding primordia. The number of primordia that form a Fibonacci number though the reason for saying this is not evident until the primordia line up on the stem when it may then be seen that that the number of leaves and the number of turns around the stem before a leaf is positioned directly above a preceding leaf is one of these possibilities: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21. But returning to the SAM, the primordia may be conceived as being on a strip around the SAM between its outer edge and the inner edge of the stele. The stele, of course, is not evident. The strip is as wide as a primordia and is divided into "tracks" equal in width and equal in number to the number of primordia aound the SAM. This accounts for overlaps such as we see in the pinecone while the Golden Angle and Fibonacci number explain the spirals on the cone.
Fully formed primordia are moved outward in a straight line by cell growth to the edge of the plant on the outside edge of the stele. The stele still does not extend upward to where the primordia is but as a consequence of the cell growth described above the primordia is now positioned adjacent to the place where a vascular bundle will shortly be. When the vascular system does grow up to the level of the primordia a vein grows out from the vascularbundle into the primordia joining the growing tip of the plant, the leaf, to the root system. The vein joins the primordia at its midpoint, becoming the "midrib" of the leaf and causes a system of veins to branch through the primordia cell mass patterning in ways characteristic of the plant species.

The stem of the plant legthens and thickens without disturbing the relationship between the leaf, its veins and the vascular bundles.

So that leaf phyllotaxy is also a guide to the arrangement of vascular bundles.

Branches begin as axillary buds at the base of each leaf so leaf phyllotaxy is also a guide to branch phyllotaxy

Flowers begin as primordia separated from each other by 137.5 degrees. The follow a pattern similar to leaves in that they form about the SAM and move out to a position on the stem which aligns them with the vascular bundle system - a alignment not disturbed by the lengthening and thickening of the flower bud and the stem. But whereas the flat surface of the leaf shows how the veins patterned the primordia, the flat surface of the fower shows groups of cells moving in log soiral patterns, the number of groups being a Fibonacci number.

We propose that these forms flow from the the form of DNA - evolution being a change in the number of DNA (TBC)