# Understanding the Fibonacci Sequence and Golden Ratio

The Fibonacci sequence is possibly the most simple recurrence relation occurring in nature. It is 0,1,1,2,3,5,8,13,21,34,55,89, 144… each number equals the sum of the two numbers before it, and the difference of the two numbers succeeding it. It is an infinite sequence which goes on forever as it develops.

**The Golden Ratio/Divine Ratio or Golden Mean –**

The quotient of any Fibonacci number and it’s predecessor approaches Phi, represented as ϕ (1.618), the Golden ratio. The Golden Ratio is best understood geometrically by the golden rectangle. A rectangle unevenly divided resulting into one square and one rectangle, the square’s sides would have the ratio of 1:1, and the new rectangle would be exactly proportionate to the original rectangle – 1:1.618.

This iteration can continue both ways, infinitely. If you plot a quarter circle inside each of the squares as they reiterate, the golden spiral is formed. The golden spiral is possibly the most simple mathematic pattern that occurs in nature like shells of snails, sea shells, horns, flowers, plants. Numbers are only what we use to organize quantitative information.

The Golden Ratio can be applied to any number of geometric forms including circles, triangles, pyramids, prisms, and polygons. The golden ratio is formed by thirds within thirds, sixths, the connection between two and three, including every even and odd number itself. The ratio itself represents the transcendence of numbers, understanding our world is not numbers, but what numbers represent.

Through the spiral, the ratio illustrates how the numbers, all quantities, are quality. Eventually, all quality can be represented through quantity. Properties qualitative and quantitative are just labels of information, our gathered indisputable fact.

If you graph any number system, eventually patterns appear. In mathematics, numbers and their patterns do not only continue infinitely linear, but in all directions. For example, considering infinite decimal expansion, even the shortest segments have an infinite amount of points.

Our universe and the numbers not only go on infinitely linear, but even it’s short segments have infinite points.

(A beautiful short film on Fibonacci sequence in Nature – ‘Nature by Numbers’)

The golden ratio is not the only mathematical pattern that reaches infinity, there are many other patterns as well that reach infinity. Knowing this, ask yourself, how could infinity occur twice? If something were to happen infinitely, how could it happen twice?

The answer is simple, infinity represents what is eternal, what is truly whole. For example, if infinity were to be used as a variable in mathematics like all other numbers, it would be denoted as 1∞, 2∞, 3∞, 4∞, etc.

The oneness of everything factual is what you know, what you perceive, what you are aware of, is all the universe looking at itself. This is the universe, even you are the universe, us and everything we know is all the same thing.

Since the numbers are everywhere, everything is a part of a pattern. Reflections of reflections, wheels within wheels. Life itself is a Fractal.

(Another video on the Egyptian ‘Temple of Man’ is a symbolic representation of the Fibonacci series of balanced organic expansion and an expression of the underlying principles of life in the universe)

**Image source**

Josh Arythe beauty lies in its simplicity

Matthew LambergSo Cool! I think I may need a tutor though!

Matthew LambergSo Cool! I think I may need a tutor though!

AnonymousThe Fibonacci sequence is taught in physics and advanced geometry classes at the high school level.

AnonymousThe Fibonacci sequence is taught in physics and advanced geometry classes at the high school level.

Mădă Rdsjinski<3

Mădă Rdsjinski<3

Melissa JobeWhat does the image (spiral) represent? Is it the Fibonacci sequence OR Golden ration OR both? All websites I have investigated mention both, making it unclear to me as to what the answer to this question. Thanks!

Melissa JobeWhat does the image (spiral) represent? Is it the Fibonacci sequence OR Golden ration OR both? All websites I have investigated mention both, making it unclear to me as to what the answer to this question. Thanks!

Jerri Lynn SilverI became fascinated with all of this by watching the TV show “Touch”. I don’t pretend to have anything but a tiny fraction of understanding of fractals or the Golden Ratio. I am 66 so I guess I will have to wait until I am on the other side to be given all the answers. I was wondering if there is anything in the human body that encompasses fractals or the Golden Ratio??

Annie GaddisThat’s GOD’s math!

BillYou forgot the “1” in front of .618….It’s 1.618……The formula is: (1+square root of 5) divided by 2.

JAY S MEHTABro,, U took the reciprocal of the number !!.., U must do 3090/1910 !! 🙂 and not 1910/3090 !! 🙂

Rest of it really works !! 🙂 Awesomest thing that I have seen !! 🙂

David AThe real gem here is the Golden Proportion. The Fibonacci Sequence is the most well known sequence, but it’s beauty is merely a reflection of the Golden Proportion, also known as phi=.6180…. The Fibonacci Sequence starts with 0 and 1, but to illustrate this, you can start with any combinations of numbers. Give it a try. I’ll use my birthday day – 10th – and my age, 50. 10 +50 =60. Applying the additive sequence: 10, 50, 60, 110, 170, 280, 450, 730, 1180, 1910, 3090….and 1910/3090 = .6181. We are nearly there already. I’ll call this the DavidA Sequence! 😀

Britney Sheahttp://ts1.mm.bing.net/th?id=H.4764361205877400&w=301&h=188&c=7&rs=1&o=3&pid=1.7

Donal MinihaneSchool????

The fodder makers for the fuctories

Celestea Deanesbeautiful time of year to dwell on this, wonder about these things. I’m not sure I understand it, but will keep trying

Natalie BellSimplicity rocks!

Patrick Mcbrideinteresting =)

Ben Ageeif enough “articles” attain enough “patterns,” all will be shown to the awakened. it is kind of calming, reading something like this, and saying to myself “wait, what???” the distraction from logic is beautiful. having said that, i hope you crazy-assed people know i love you! unless you hurt someone else, and then i guess i’ll transfer some equation chakra to you or something. it’s cool. i’m like your nonexistent existential emergency hotline. “am i real?” HELL YES YOU’RE REAL (not counterfeit) NEW PAPER HUNDIES. WHOOOOOOOOO!

Dante Dominic DawsonBookmarked,love it!

Dante Dominic DawsonWhy don’t they teach the information in school. Information that can bring humanity a better understanding of reality.

Munkhjin Enkhtaivan(y)

Randy DavisThis is becoming so understood in our so Irtysh, even WordPress offers themes based on this.

Lucy WhettonBeautiful

Neil Whitehousereally helpful summaries thanks

Kazimierz ZielińskiMatematyka jest językiem natury Wszechświata przetłumaczona na język liczb, punktów i odcinków jakie łączą dwa punkty. Fizyka to jest inne podejście do tego samego problemu.

W centrum mamy podwójną cząsteczkę energii, która ma określony program, strategię wiedzy, która umożliwia oszczędne gospodarowanie energią przepływającą przez tunel czasoprzestrzenny tej cząsteczki. Jeżeli potrafi oszczędnie gospodarować i w wyniku czego promieniuje nadmiarem tej energii wokół siebie to przyciąga do siebie inne cząsteczki. Ponieważ cząsteczka wiruje to tworzy się wir cząsteczek. Wzrost koncentracji energii jest ograniczony, powstają coraz większe cząsteczki, potem atomy, cząsteczki atomowe i tak dalej. Jeżeli te cząsteczki połączymy więzami miłości czyli relacją wzajemnej wymiany energii to na każdą taką cząsteczkę oddziaływają dwie różne siły dośrodkowa i odśrodkowa zależnie od masy cząsteczki i prędkości. Masa gwiazdy centralnej rośnie do pewnej granicy albo jak w przypadku słonecznika czy ślimaka zależy od jego programu życia, dostępu do zasobów energii. Jeżeli źródło energii jest ubogie to program dostosowuje się do warunków i rozprasza się na mniejsze gwiazdy oparte na tym samym programie. Słonecznik produkuje nasiona coraz większe aż do czasu gdy warunki dostępu do energii życia nie pozwalają na dalszy wzrost. Na wiosnę cykl życia się powtarza. Gwiazdy galaktyki rozpraszają się po Wszechświecie i tworzą własny nowy „organizm” planetarny a potem być może nowy gwiazdozbiór. Tak może być także z naszym układem słonecznym. Jeżeli stracimy jako układ właściwości pozytywne, będziemy dzięki prawicowym korporacjom tracili energię nasz układ będzie miał minusowy bilans energetyczny i nie mamy szans na ewolucję w galaktyce, będziemy bezpłodni, stracimy szanse na reprodukcję.

Ale skoro prawica ma Wszechmogącego Boga to może on im pomoże ale ja w cuda nie wierzę ani w nieograniczony wzrost PKB. Raczej uważam, że jest to efekt ograniczonej wyobraźni prawicy.

Sam Getzwhat wonderful site this is

Donna Pugh“BEAUTIFUL…~Dona deLa Boheme~

Rebecca SorensenSo Thankfull for this sight and all the awesome articles and Facts of life !

Carolyn BunkleyI wish stuff like this was available when I was in school trying to learn this! I struggled and still could not get it, yet this is so clear!