number phi and fibonacci

[Italo]

I have a project about math in nature, it is actually really important because it is my graduation project.

I had to state a question in order to do my hypothesis so I could work on my project but when I had my first tutoring they told me my whole problem was wrong because the answer about the divine proportion and the fibonacci series was already given in books. The told me to state the question again

I have been thinking about it for a couple of days and Im having trouble with finding such a question that can lead to an investigation, something that I have to find out for myself and isn't already given.

if anyone has an idea I would really apreciate it because I am in deep trouble with this and I need to hand it in again next week.

thanx a lot!

Ana L.

[Epsilon=One]

they told me my whole problem was wrong because the answer about the divine proportion and the fibonacci series was already given in books. The told me to state the question againI will refer to the divine proportion as the Golden Ratio (http://www.physicsmathforums.com/showthread.php?t=105) (GR). What isn’t usually given in books is that any simple additive sequence, no matter what two numbers you start with, no matter what order these starting two numbers are, the terms of the series will converge to the GR when a term is divided by the prior term. Thus, in this respect concerning the GR, the Fibonacci sequence (FS) is not unusual from any other simple additive sequence.

Im having trouble with finding such a question that can lead to an investigation, something that I have to find out for myself and isn't already given.The following mathematical comments are nowhere but on this site. You alone may use them freely. If you state that: “you agree with them”; your work will be original as no one else has ever so stated.

if anyone has an idea I would really apreciate it … I need to hand it in again next week.If you are going to “hand it in” you must understand what I write; so, please ask questions about what is not clear.

To begin; you might put something like this statement as a question to investigate or a postulate (axiom) to begin from: Mathematics does not describe Nature; Nature describes mathematics. As simple as this seems, few mathematicians would agree; and, I know of none that can prove it. In fact, if you research Kurt Gödel (http://www.americanscientist.org/template/BookReviewTypeDetail/assetid/45922;jsessionid=baaaoJKX8J0LQG), you’ll find that he “proved” the opposite in the early ‘30s; and, no world-class philosopher or mathematician has since proved otherwise.

Much overlaps with the proof. For now I will demonstrate where numbers come from and how this is Naturally related to: that the FS is incomplete; the GR is directly related to (not just convergent to successive terms) the revised (complete) Fibonacci sequence (http://www.physicsmathforums.com/showthread.php?t=103) (rFS) and the circle; that all ellipses have a constant (http://www.physicsmathforums.com/showthread.php?t=107); et cetera (the list is near endless).

There is a series, the Brunardot Series (http://www.physicsmathforums.com/showthread.php?t=99) (BS) (You will only find it here.), that is the crux of all the above. And, the salient (most important) component of the BS is the Natural function (http://www.physicsmathforums.com/showthread.php?t=121) (NF), which is: x^2 – x.

I will explain all of this in more detail later in the day as time allows. In the meantime follow the above URLs (links) for some background. Try to find the reason that the FS and the GR are so ubiquitous in Nature.

Have you studied geometry???

Of course the title of this post (and its explanation) may be all you need . . .

[Epsilon=One]

What are numbers? How does nature use them?

Numbers, as used in arithmetic, are usually referred to as “integers.”

Natural integers are symbols that represent multiples of particular a “unit.” The “unit” can be an apple, a marble, an inch, or most anything; or, it can just be the symbol that is called “one.” Though Zero is for many purposes considered an integer, it is not a Natural integer because it is not a multiple of a “unit” as the other integers are.

If the generic “one” is used as the first natural integer, when it is doubled the next integer is two; and so on, for as many multiples as you like; or, numbers that you can name or count.

The important thing is that to define numbers, a “unit” must be selected for which all the other numbers are multiples of that “unit.”

To find some beginning numbers and their “unit”:

Construct an equilateral triangle with each side having length, “ x.”

(If no image appears below, "Click" your browser "Refresh" icon.)
http://www.CQthus.info/PT/Ellipses/et190a.gif

Label the base, points B and D; the apex F. (The arrangement of labels is to tie this exercise to other illustrations that are elsewhere.)

(If no image appears below, "Click" your browser "Refresh" icon.)
http://www.CQthus.info/PT/Ellipses/et250b.gif

Construct an ellipse with points B and D as the foci and lines BF (r1) and DF (r2) the varying radials (as demonstrated by (r1 and r2 in Mathworld image at: http://mathworld.wolfram.com/Ellipse.html).

(If no image appears below, "Click" your browser "Refresh" icon.)
http://www.CQthus.info/PT/Ellipses/et250c.gif

For reference, bisect the ellipse horizontally and vertically. The horizontal bisection line, that coincides with the base BD, is the major diameter; the vertical bisection line is the minor diameter.

Construct a line parallel to the minor diameter and perpendicular to the major diameter through point D, upward, that intersects the ellipse at point A.

Draw a line AB through points A and B.

Triangle ABD is a right triangle.Line BD = x; let line AD = y.

Line EG = 2x (The major diameter equals the sum of r1 and r2).

Line AB + Line AD = 2x (The major diameter equals the sum of r1 and r2).

Line AB = 2x – y (Substituting).

x^2 + y^2 = (2x – y)^2 (Pythagorean Theorem).

x^2 + y^2 = 4x^2 – 4xy + y^2 (Expanding).

3x^2 = 4xy (Consolidating).

3x = 4y (Dividing each side of the equation by x).

It can be seen that the smallest values for x and y that are both integers will be:

x = 4
y = 3Thus, line BA = 5 (Pythagorean Theorem).

Extend Line BD to the ellipse at points E and G.

Line EB = 2 and Line DG = 2 (Major Diameter Minus x divided by half)

(If no image appears below, "Click" your browser "Refresh" icon.)
http://www.CQthus.info/PT/Ellipses/et250d.gif

EB = DG = perigee, p, = 2
BC = CD = soliton, s, = 2
AD = radius, r, = 3
BF = DF = vector, v, = 4
BD = wave (focal length), w, = 4
BA = hypotenuse, h = 5
ED = BG = apogee, o, = 6

(If no image appears below, "Click" your browser "Refresh" icon.)
http://www.CQthus.info/PT/Ellipses/et250e.gif

The ellipse that is so constructed is the most circular ellipse that can be drawn; all other ellipses are generated from triangles with a larger angle at the apex than the 60 degrees of the equilateral triangle..

All the illustrated lines are exact multiples of a particular "unit"; that "unit" is referred to as epsilon.

Thus, for the most circular ellipse, epsilon = one; and, epsilon is represented by line DI that is the hypotenuse, h, minus the wave, w. In fact, epsilon = one for any elliptical shape if the vector, v, is set as the square of the perigee, p (v = p^2). Because of this relationship, epsilon = one is referred to as the Elliptical Constant.

DI = h – w = epsilon = one.

For any ellipse when epsilon = one, and the perigee, p, = x, the following relationships apply:

Line DI = epsilon, e, = one;
Line BE = DG = perigee, p, = x
Line BF = FD = vector, v, = x^2;
Line BC = CD = soliton, s, = x^2 - x. (This is referred to as the Natural function);
Line BD = wave, w, = 2s;
Line BA = hypotenuse, h, = w + e;
Line AD = radius, r, = 2x – e;
Line DE = BG = apogee, o, = p + w
Line IG = key, k, = p – e;
Radius oP = hypotenuse radius, Hr, = k (Inscribed Pythagorean circle)

(Note: the radius of an inscribed Pythagorean circle = the product of the sides divided by the sum of the sides.



To be continued.

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