The Golden Ratio and the Fibonacci Sequence
Learn more about the Golden Ratio, the Fibonacci Sequence, and how they are related.
The Golden ratio is a special ratio. Phi 
(lowercase) is used to refer to this ratio, and the value is 
, which is approximately 1.618.
The uppercase phi (
) is used for the reciprocal of the golden ratio, which is 1/.
Phi can be expressed using a line segment. We make it such that the ratio of B to A+B is equal to the ratio of A to B.
Many designs – used in buildings, sculptures and paintings use the Golden Ratio for their dimensions. Architects and artists tend to use them often as they are considered very pleasing to the eye.
For example, the Parthenon uses the golden ratio for its construction.
Another example is the Mona Lisa painting.
The golden ratio also determines how attractive a person is.
For example, the American pop singer and actress Jessica Simpson is attractive because the proportion of her face fits geometrically on the human face mask which conforms to some aspect of the Golden Ratio.
Golden Rectangle
This rectangle is a special rectangle where the ratio of the length to the width is the Golden Ratio, which is 
.
When a square is cut off from the golden rectangle, the new rectangle is still similar to the original rectangle.
Below is an illustration:
and so on…
Golden Triangle
The golden triangle is a special isosceles triangle. The top angle is 36° while the bottom two angles are 72° each. We then bisect one of these base angles. The resulting blue triangle:
is also a golden triangle! Thus, we can keep bisecting the base angle to get a set of whirling triangles:
From this, we can draw a logarithmic spiral similar to the Fibonacci spiral:
Pentagram
A pentagram is a star-shaped figure which is made out of the five diagonals on a pentagon:
As can be seen, a pentagram has five sides. From three of these sides, we can get a couple of different golden triangles:
From the figure below:
we can also conclude that two non-consecutive sides of a pentagram divide each other in the Golden Ratio.
Fibonacci number
Fibonacci number
The Fibonacci numbers are part of a sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. As can be seen, each number is the sum of the two numbers before it, after the first two numbers. A general equation can therefore be formed:
The Fibonacci sequence is named after Leonardo of Pisa, an Italian mathematician also known as Fibonacci.
Origins
Fibonacci first thought of the sequence as a solution to a problem he posed in a book he wrote, Liber Abaci. The problem was to find out the number of rabbits produced in a rabbit population which starts from a single newly-born pair. It is assumed that each pair produces another pair every month, they each become productive from the second month onwards and the rabbits never die.
Relation to the Golden Ratio
There are many different ways in which the Fibonacci sequence is related to the Golden Ratio. Firstly, the further you go to the right of the sequence, the more the ratio of one term to the one before it estimates the Golden Ratio. The table below shows the first few numbers and their ratios:
| First number | Second number | Ratio |
| 0 | 1 | - |
| 1 | 1 | 1.0000 |
| 1 | 2 | 2.0000 |
| 2 | 3 | 1.5000 |
| 3 | 5 | 1.6667 |
| 5 | 8 | 1.6000 |
| 8 | 13 | 1.6250 |
| 13 | 21 | 1.6154 |
| 21 | 34 | 1.6190 |
| 34 | 55 | 1.6176 |
| 55 | 89 | 1.6182 |
| 89 | 144 | 1.6180 |
To describe this graphically, we can use squares. Start with a 1 by 1 square. Then, add another square of the same size. Subsequently, add squares whose sides are equal to the longest side of the existing rectangle. After we do this four times, we get a rectangle similar to the following:
The rectangle gets increasingly closer to the golden rectangle, where the ratio of the width to the height is the Golden Ratio. If we draw a quarter circle in each of the squares, we get a Fibonacci spiral as such:
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36 Responses to “The Golden Ratio and the Fibonacci Sequence”
On September 17, 2007 at 1:36 pm
Beautifully presented. Still don’t completely comprehend but thank you.
On September 18, 2007 at 4:10 pm
Very nicely done. This helps!
Thank you
On September 18, 2007 at 6:03 pm
The stuff about Leonardo and the Parthenon is bull, as is the suggestion it’s the basis for a beautiful face. For an illustration, answer this question: why have you chosen to put the right vertical of the rectangle on the Mona Lisa in its current position, other than that is the right hand side of a golden rectangle? Read Markowsky – Misconceptions about the Golden Ratio.
But if you want to know something *really* interesting about phi, then look up Penrose Tiling on Wikipedia…
On September 19, 2007 at 3:35 am
Thanks for everyone’s views, making this article the top in the hottest content list! I’m really grateful
On September 19, 2007 at 7:34 am
Look, don’t take this in the negative way, I’m just keeping you informed, an article like this won’t stay in the Hot Content (let alone First Place) for long.
On September 19, 2007 at 8:21 am
There were some fascinating facts regarding this formula in the Da Vinci code. These were mostly referred to as the “devine formula” how these ratios are consistent with human body design etc.. I admit I am not a math person I only look at these postings to test myself.
On September 19, 2007 at 6:57 pm
I have been reading The DaVinci Code and it mentions the Fibonacci Code, which is so freakin’ awesome how PHI pertains to everything from the Mona Lisa to Nautilus and on and on it goes–where it ends, nobody knows!!!!!!!
On September 20, 2007 at 11:00 pm
You have an error
At the top of the page in line 10
It is not ….. = 1.618
It is ….. = 1/1.618
On September 21, 2007 at 5:19 am
Thanks for pointing out that mistake, Federman! I’ll get it fixed straight away
On September 21, 2007 at 10:52 am
go to quazen.com and on the yahoo bar, type “da boss” and go to the first link to see funny stuff
On September 22, 2007 at 6:50 am
You are a genius. This is an amazing article, and i never would have thought of that. Thnak you for sharing this wonderful information with the world.
On September 22, 2007 at 6:53 am
I spelled “Thank You” wrong on my last comment… Sorry!
On September 22, 2007 at 4:33 pm
Maybe some info about the platonic solids & Metatrons cube would help?
On September 22, 2007 at 4:34 pm
that last one was me
On September 23, 2007 at 5:07 am
Thisi s crap non-sense, simply copy-pasted from wikipedia.
On September 23, 2007 at 1:17 pm
ahhhhhhhhhhh i dont get it!!!!!!! plz expain again in stupid people terms
On September 24, 2007 at 1:44 am
Small correction: In the first example of the Golden Ratio on this page where a line segment is divided into two parts A and B. The figure shows that the length of segment A is smaller than the length of segment B. However it should be the reverse; i.e. segment A should be the larger segment when compared with segment B. Only then can A/B or (A+B)/B would be approximately equal to 1.618. In the example given if A is smaller than B, then A/B would be less than 1.
On September 24, 2007 at 7:41 am
You are correct,Abhishek. I meant that A/B was approximately equal to 1/1.618. This was also pointed out by Federman (look above at the earlier comments). I have already submitted a fix to the article, but it hasn’t been authorised yet. Thanks for your feedback!
On September 29, 2007 at 4:03 am
Jessica Simpson is so not attractive!
On September 29, 2007 at 4:03 am
Jessica Simpson is so not attractive!
On September 29, 2007 at 4:03 am
Jessica Simpson is so not attractive!
On October 6, 2007 at 6:08 am
This is a great work. This is not a simple copy paste. But he did his home work. This encourages any writer to do his home work as it is worth doing.
Keep up the good work.
On October 6, 2007 at 9:11 pm
Fibonacci Sequence has really caught my attention after going through these pages.
On January 3, 2008 at 4:20 pm
it catches your attention…and its really detailed…i’m a little confused still but i understand a lot more than i did…good job
On January 28, 2008 at 8:15 am
Thanks, this helped alot!
On June 20, 2008 at 12:01 am
doing as school assigment hasn’y helped me a loy but great interest anyway!
On September 26, 2008 at 10:32 am
dont understand it really. my math teacher wont be so pleased!
On November 1, 2008 at 2:52 pm
this helped with some things; but im stil confused! you need to explain it in simpilar termsss.!
On January 12, 2009 at 3:21 pm
Very helpful, although probably better for someone a little older than me – I was just doing this to find information for school, but all the same thanks!
On February 10, 2009 at 10:56 am
Really amazing work on Fibonacci series. You have combined creativity with mathematics and illustrated with pictures.
On March 18, 2009 at 6:53 am
they forgot The Parthenon, Notre dame of laon, the Great Mosque of Kairouan, , The Castle of Chichen Itza and the Temple of Zeus were all deigned using the golden ratio.
On April 12, 2009 at 9:59 pm
thanks for the help. great explanation. i understand it so much better now thank-you very much
On April 21, 2009 at 8:19 am
Doing the Fibonacci series and golden ratio for a school project. this article has been very informative and interesting and also a great help. thanks!
On May 3, 2009 at 4:08 pm
thanks this helped a lot for my math project. THANKS!
On September 30, 2009 at 4:23 pm
Oh gosh, I got a project due next saturday on this and until now i couldnt find good info on the fibanacci sequence being related to the golden ratio
On October 20, 2009 at 11:44 pm
Man, im a big tool fan. and Maynard uses the Fibonacci Sequencein the song Lateralus, and that happens to be my fav song. Im no mathmatician, but basicly its a spiral into nothingness???
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