How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get the following sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on.
This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries. Its properties illuminate an array of surprising topics, from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants (not to mention populations of rabbits!). Consider, for example, the following diagram:
Here we have taken squares with sides whose lengths correspond to the terms of the Fibonacci sequence, and arranged them in an “outwardly spiraling” pattern. Notice that the rectangles which result at each stage are all roughly the same shape, that is, that the ratio of length to width seems to “settle down” as we build the pattern outward. Notice also that the ratio of length to width is at every step the ratio of two successive terms of the Fibonacci sequence, that is, the ratio of the greater one to the lesser. These ratios may be thought of as forming a new sequence, the sequence of ratios of consecutive Fibonacci numbers:
This sequence converges, that is, there is a single real number which the terms of this sequence approach more and more closely, eventually arbitrarily closely. We may discover this number by exploiting the recursive definition of the Fibonacci sequence in the following way. Let us denote the nth term of the sequence of ratios by xn, that is,
Then using the recursive definition of F(n) given above, we have:
Now supposing for the moment that the sequence converges to a real number x (a fact which requires proof, but we'll leave that aside), we may observe that both xn and have the same limit, that is,
Consequently, the real number x to which the sequence of ratios converges must satisfy the following equation:
This is a simple equation to solve for x: it is really a quadratic equation, and its positive root is the value we are looking for:
This number was known to the ancient Greeks and was called by them the Golden Mean. It is usually denoted by the Greek letter f (phi), and sometimes by m (mu). They believed that the proportion f:1 was the most most pleasing, indeed the aesthetically perfect proportion, and all of their artwork, sculpture, and especially architecture made use of this proportion. A rectangle whose sides had this proportion was called the Golden Rectangle. (And that is the shape being more and more closely approximated by our “spiralling rectangles” above.)
Whether or not you agree with the Greeks’ aesthetic judgment, it's a safe bet that Nature herself does:
The growth of this nautilus shell, like the growth of populations and many other kinds of natural “growing,” are somehow governed by mathematical properties exhibited in the Fibonacci sequence. And not just the rate of growth, but the pattern of growth. Examine the crisscrossing spiral seed pattern in the head of a sunflower, for instance, and you will discover that the number of spirals in each direction are invariably two consecutive Fibonacci numbers.
The Fibonacci sequence makes its appearance in other ways within mathematics as well. For example, it appears as sums of oblique diagonals in Pascal’s triangle:
For other web sites with many more details and properties regarding the Fibonacci sequence, be sure to search the (use keyword “fibonacci”).
Spiral Galaxy -
Evolutionarily speaking, the best way to ensure success is to have as many offspring as possible (ergo the Baldwin brothers). The sunflower naturally evolved a method to pack as many seeds on its flower as space could allow. Amazingly, the sunflower seeds grow adjacently at an angle of 137.5 degrees from each other, which corresponds exactly to the golden ratio. Additionally, the number of lines in the spirals on a Sunflower is almost always a number of the Fibonacci sequence.
Like the sunflower, the pine cone evolved the best way to stuff as many seeds as possible around its core. Also, in what was surely an accident, it evolved into perhaps the best substitute for toilet paper when in a pinch. The golden ratio is the key yet again. As with the sunflower, the number of spirals almost always is a Fibonacci number.
The golden ratio is found throughout your body, all the way to your DNA.
Here's one you can see for yourself, dear reader, if you're still with us. If you use your fingernail length as a unit of measure, the bone in the tip of your finger should be about 2 fingernails, followed by the mid portion at 3 fingernails, followed by the base at about 5 fingernails. The final bone goes all the way to about the middle of your palm, which is a length of about 8 fingernails. Again, it's Fibonacci at work and the ratio of each bone to the next comes very close to the golden ratio.
Continuing with the length of your hand to your arm is, again, the golden ratio.
Fibonacci applies even down to what makes you, you. A DNA strand is exactly 34 by 21 angstroms.
The Fibonacci sequence is truly a wonder. The examples are vast, and go way beyond the scale of this article. The patterns in which a tree grows branches, the way water falls in spiderwebs, even the way your own capillaries are formed can all be linked to Fibonacci. Science is just beginning to understand the implications of this simple sequence and some of the most amazing discoveries may be yet to come.
“Art is the imposing of a pattern on
and our aesthetic enjoyment
is recognition of the pattern.”
(Alfred Whitehead North)
Maybe the most important subject of sacred geometry is the Golden Mean. The Golden Mean is a very special ratio and is expressed by the Greek letter Ф called Phi.
= ½ * v5 + ½ = 1.618
Phi like Pi is an irrational number, meaning you can never calculate its exact value, you can only approximate it.
The Phi ratio is expressed in the Golden Section. The Golden Section is a the length of let’s say a rope when it is divided such that the ratio of the longer part of the rope to the whole is exactly the same ratio as the shorter part of the rope is to the longer part. (Read it again)
When the Phi ratio is applied to a rectangle whereas B = 1 and A has length Ø, the rectangle is called a Golden Rectangle.
The Golden Rectangle can be used to create a spiral, the Golden Spiral. Starting with one Golden Rectangle, a second Golden Rectangle can be attached to the first using the longest side of the rectangle, side A as the shortest side B of the next rectangle. To this end the second rectangle is constructed 90 degrees perpendicular to the first rectangle. If this process is continued, called the spiralling of the Golden Rectangle, a curved line can be drawn through the corners of the rectangles creating the Golden Mean spiral. The spiralling of the Golden Mean spiral continues indefinitely in inward and outward directions, it’s getting smaller and smaller spiralling inwards and getting bigger and bigger spiralling outwards.
Golden Mean spiral
A variant of the Golden Mean spiral is the Fibonacci spiral. The difference with the Golden Mean spiral is that is does not spiral in indefinitely but starts with a Golden Rectangle of which one side has length 1 and the other length Phi.
Gradually when the Fibonacci spiral, spirals outward, there will be no distinction noticeable any more between a true Golden Mean spiral and the Fibonacci spiral. The Fibonacci spiral is based on the progression of the Fibonacci sequence.
Crop circle, fractal spirals, Milk Hill, Wiltshire august 12th 2001
Leonardo Fibonacci (1175 AD), a great mathematician of the Middle Ages discovered the Fibonacci sequence by studying nature. He studied the growth of rabbit populations and the growth of leaves and petals and discovered a well-defined mathematical sequence in all of this.
This is the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 etc.
Each number in the sequence is the sum of the two preceding numbers starting with the root number 1. The Fibonacci sequence progresses towards the Golden Mean if we divide two successive numbers in the sequence.
1/1 = 1
2/1 = 2.0
3/2 = 1.5
5/3 = 1.667
8/5 = 1.60
144/89 = 1.618
The Fibonacci sequence propagates towards Phi (Ø) but never reaches it since it is an irrational or transcendent number.
Fibonacci spirals and Golden Mean ratios appear everywhere in the universe. The spiral is the natural flow form of water when it is going down the drain. It is also the natural flow form of air in tornados and hurricanes. Here’s another beautiful example of a Fibonacci spiral in nature, it’s the Nautilus shell and every book about sacred geometry contains one:
The Golden Mean ratio is all over the human body, in the ratios between the bones, the length of your arms and legs. The Golden Mean is also the ratio in the distance from the navel to your toe and the distance from your navel to the top of your head. Michelangelo has beautifully hidden these Golden Mean ratios in his fresco on the ceiling of the Sistine Chapel in Rome:
Michelangelo and the Phi ratios in the human hand
(Courtesy of Dan Winter, www.soulinvitation.com)
Thank you so much AyAnna :-)
Somehow the text doesn't show for me - but the link is visible and working well :-)
I feel that all this is part of my understanding MORE - which I cannot yet grasp ... it's like one part of a big, big jigsaw puzzle and by studying it in detail I get a glimpse of where it might belong - lol!
Sending you LOVE and LIGHT and JOY from my heart, dearest AyAnna,
Thank you my Dear AyAnna and Sonja!
Interesting discussion about how to build PYRAMIDS and the Fibonacci Sequence:
Thank you, Paul!
Thank you dear Sonja,
Please see attached of video of Charles Gilchrist. He is excellent at explaining sacred geometry and you can feel his absolute passion for it. (There are many more videos to watch on the right hand side)
THANK YOU, Paul!!!
Wouldn't you like to post this as a separate discussion? Maybe introducing Charles Gilchrist a bit to us? That would be absolutely fabulous!
From the time it took me to reply you can detect that if I am to do it it will take pretty long, lol ;-)
Sending you a love filled hug, dear Paul,
I have just passed on the video to Peter's latest post - most probably very interesting for you, as well - do you now Peter Melchizedek?
I have just read the post and is very interesting, top of my head was pulsating whilst reading it. However it has been doing that on and off pretty much since I built those pyramids. I think I will print them out and put them on the wall.
Dear Kerrie (Zoolithe) pointed me to your wonderful site here and Peter has just become known to me on the FOL website. The usual magic of stuff showing up in Divine timing :-)
Dear Sonja Myriel,
That is a great idea. I want to watch some of his videos again and will do just that afterwards. So no worries :-)
A big hug right back at you my friend.
Ah - Great :-)
Thank you so much, Paul - and YES - I can feel the flow and magic of how we are all working together in such mystical ways, too, very strongly these days - and the power is magnifying :-)))
Thank you for your presence,
Sonja Myriel Aouine
This video was posted by Dr. SohiniBen - Thank you :-)