Friday, November 30, 2007

The golden ratio

After a long time I am writing for my blog, after the post on antichrist was busy working on some secret society and wanted to write about them in this blog. But now dropped the idea of writing about them, due to some ethical reasons…..

I have also told that ill write about 666 in some post, I think my next post will be about 666, but today this post is about the golden mean.

I have always been attracted towards mathematical constants, yes speed off light, gravitational constant..etc.. I read about the golden mean in Da Vinci code by Dan Brown, I was fascinated by the number how can this number occur so frequently in nature. Then it is not the natures addiction for this number but the natural selection process. Before we get into how this number occurs in nature so frequently, we will see, what is golden mean and where it occurs…

Two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. From the diagram it is clear that the golden ratio is (a+b)/a if and only if it is also equal to a/b.

It is represented by phi of greek alphabet. It is also related to golden mean, golden section, divin ratio, divine proportion etc..

This also closely related to fibbonacci series. The fibbonacci series is the sum of previous two numbers of the series, the series starts with 1, the series is 1,1,2,3,5,8,13,21,34,55,89,…

The ratio of the two consecutive numbers in fibbonacci series tends to zero on 1.618034, the inverse of this number is 0.618034. The formula to calculate the phi is (1+ sprt(5))/2.

Another method to find the golden ratio is

  1.  Enter the number one.
  2.  Add one, take its reciprocal
  3.  Add one, take its reciprocal.. continue this to get the accurate golden ratio, phi has no exact value, it is endless….

The value of golden number to 1000th place of its decimal is given here…

1.6180339887 4989484820 4586834365 6381177203 0917980576
2862135448 6227052604 6281890244 9707207204 1893911374
8475408807 5386891752 1266338622 2353693179 3180060766
7263544333 8908659593 9582905638 3226613199 2829026788
0675208766 8925017116 9620703222 1043216269 5486262963
1361443814 9758701220 3408058879 5445474924 6185695364
8644492410 4432077134 4947049565 8467885098 7433944221
2544877066 4780915884 6074998871 2400765217 0575179788
3416625624 9407589069 7040002812 1042762177 1117778053
1531714101 1704666599 1466979873 1761356006 7087480710
1317952368 9427521948 4353056783 0022878569 9782977834
7845878228 9110976250 0302696156 1700250464 3382437764
8610283831 2683303724 2926752631 1653392473 1671112115
8818638513 3162038400 5222165791 2866752946 5490681131
7159934323 5973494985 0904094762 1322298101 7261070596
1164562990 9816290555 2085247903 5240602017 2799747175
3427775927 7862561943 2082750513 1218156285 5122248093
9471234145 1702237358 0577278616 0086883829 5230459264
7878017889 9219902707 7690389532 1968198615 1437803149
9741106926 0886742962 2675756052 3172777520 3536139362..

The golden rectangle is the one whose sides are related to phi. This in simple terms is given as the ratio of the two sides of the rectangle will give the golden ratio, that is if one side is 1 then the other side will be 1.618034…

A golden rectangle has the interesting property that, if you create a new rectangle by swinging the long side around one of its ends to create a new long side, the new rectangle is also golden.

The logarithmic spiral is what we get when we start a curve from one end of the rectangle and draw a quarter circle to the opposite end of the rectangle and then draw another rectangle by swinging the long side.. this swing follows a logarithmic spiral.

In the picture, the yellow line follows a logarithmic spiral.

The relation between the golden ratio and the number of beast(666).

-phi/2=sin666=cos (6*6*6)
this can also be written as

Occurances of golden ratio, golden rectangle, and the golden spiral

1। The proportions of different plant components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species।

2। A rectangle that is one mile long by one kilometer wide is within 1% of a golden rectangle, with a mile being exactly 1।609344 km।

3. It is sometimes claimed that the number of bees in a beehive divided by the
number of drones yields the golden ratio।

4। Great artists like leonardo da vinci, and piet mondrian, salvador dali used this geometric miracle I ntheir works। Someof the famous works based on this number is the Vitruvian man, Monalisa and the sacrament of last supper।

5। The approach of a hawk to its prey। Their sharpest view is at an angle to their direction of flight; this angle is the same as the golden spiral's pitch।

6। The arms of spiral galaxies। Our own galaxy, the Milky Way, is believed to have four major spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way। In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees।

7। The arms of tropical cyclones, such as hurricanes।

8. Many biological structures including spider webs and the shells of mollusks.

In the internet you will find many such instances were this sacred geometry occurs.
But why is our nature so obsessed with this number??

First why it occurs in plants
If there are 1.618... leaves per turn it gives flowers or petals the best possible exposure to insects to attract them for pollination or exposure to sunlight. The whole of the plant seems to produce its leaves, flower head petals and then seeds based upon the golden number.

And about the logarithmic spiral
Start with any irregularly shaped two-dimensional figure F0. Expand F0 by a certain factor to get F1, and place F1 next to F0, so that two sides touch. Now expand F1 by the same factor to get F2, and place it next to F1 as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other.

And almost all the above occurances are optimal solutions of their kind, to ask why nature likes this number is like asking why all the heavenly bodies are spherical and not cube or any cylindrical…।

There is nothing mystical about this number but still this number amazes the scientists and artists.