## Fibonacci numbers and Phi are related to spiral growth in nature.

If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number. This property results in the Fibonacci spiral, based on the following progression and properties of the Fibonacci series:

1^{2} + 1^{2} + 2^{2} + 3^{2} + 5^{2} = 5 x 8

1^{2} + 1^{2} + . . . + F(n)^{2} = F(n) x F(n+1)

A Golden spiral is very similar to the Fibonacci spiral but is based on a series of identically proportioned golden rectangles, each having a golden ratio of 1.618 of the length of the long side to that of the short side of the rectangle:

The Fibonacci spiral gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively)

Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi. Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of Equiangular spirals. An Equiangular spiral itself is a special type of spiral with unique mathematical properties in which the size of the spiral increases but its shape remains the same with each successive rotation of its curve. The curve of an equiangular spiral has a constant angle between a line from origin to any point on the curve and the tangent at that point, hence its name. In nature, equiangular spirals occur simply because they result in the forces that create the spiral are in equilibrium, and are often seen in non-living examples such as spiral arms of galaxies and the spirals of hurricanes. Fibonacci spirals, Golden spirals and golden ratio-based spirals often appear in living organisms.

## Alternate spirals in plants occur in Fibonacci numbers.

The most common appearances of a Fibonacci numbers in nature are in plants, in the numbers of leaves, the arrangement of leaves around the stem and in the positioning of leaves, sections and seeds.

Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)

Pinecones and pineapples illustrate similar spirals of successive Fibonacci numbers, with the example below showing the alternating pattern of 8 and 13 spirals in a pine cone.

Click on an image below to see the full size versions of each image above:

## Golden spirals in sea shells

Golden ratios are also sometimes found in the proportions of successive spirals of a sea shell, as shown below.

## The Nautilus shell spiral is not a Golden spiral but often still has Golden Ratio proportions.

The nautilus shell is often shown as an illustration of the golden ratio in nature, but the spiral of a nautilus shell is NOT a golden spiral, as illustrated below. The golden spiral overlay is provided by PhiMatrix golden ratio software:

Nautilus shell spiral compared to a Golden Spiral

The Nautilus spiral, however, while not a Golden spiral, often displays proportions its dimensions that are close to a golden ratio, appearing in successive rotations of the shell as the Nautilus grows. As with all living organisms, there is variation in the dimensions of individuals, so the appearance of the golden ratio is not universal.

See also other examples and explanations of the golden ratio in the nautilus spiral.

greg hope says

December 9, 2012 at 3:24 pmhaving to do with the way energy nests itself into matter: note the mammalian ear cochlea; and how the lower tones which carry the farthest are sensed in the tightest spirals, and the higher tones, nearest, in the larger. Fascinating. Thank Fibonacci; otherwise, we’d have to be fascinated all over.

Amazing Potential says

December 25, 2012 at 11:58 pmIt’s fascinating how this pattern occurs through all of creation. We have just posted a video by Drunvalo Melchizedek that touched breifly on the fibonacci and the golden mean, which lead me to do some more research which lead me here. Thanks

laura robbins says

February 28, 2013 at 11:47 amAwesome site! Thanks for the help and inspiration.

Cnugg says

March 18, 2013 at 10:33 pmYes. Netscape….IM me on the MySpace!…seriously, does the golden ratio have anything to with our gravity specific to earth? I’ m thinking water spiraling through drain/toilet?? Btw, I’m not THAT educated, this could have already been covered?

Maybe I’ll ogle it…google it.

-“comedy Devine, idea mine.”

belinda says

July 1, 2013 at 10:04 amamazing!

mary jazzar says

August 21, 2013 at 8:02 pmSo much great info, thanks.

N K Srinivasan says

April 29, 2014 at 9:27 amGreat pictures and explanations on PHI —thanks

rwef says

June 5, 2014 at 4:17 amhelped a bit tar

sverre heltvedt says

August 7, 2014 at 2:42 amit´s fascinating:-)

Patricia Cashman says

August 31, 2014 at 4:55 pmappreciate the images and moving demonstrations!

ron phillips says

October 12, 2014 at 4:37 amVery interesting, thank you, but I would like to be able to stop animation of the pine cone, so that I can examine it more closely. Just my preference!

Julie-Ann says

August 21, 2018 at 5:12 pmI totally agree that the animated pine cones would have been enjoyable to look at and linger over as still images

Gary B Meisner says

August 23, 2018 at 10:26 amThe still images are there for your viewing pleasure too. Just click on any of the three images below the animated image and the stills will open in full size in a gallery.

BartleyBurnside says

December 15, 2014 at 3:33 pmIt has to do with the slow growth rate of organisms and gravitational pull and rotation of the earth. Nature has to have a starting and stoping point. Our species is about at the end of our growth and technology rates… Prepare accordingly, we are all gonna die soon. Mothman Prophecy

David Koski says

March 7, 2015 at 9:35 pmhttps://www.youtube.com/watch?v=LkWp_KpcFNQ

harrison says

September 10, 2015 at 2:36 amVery good site!! Thanks for the moving demonstrations! Really helped for my maths assignment!

Lucy Henehan says

March 24, 2016 at 11:53 amThanks this cleared up my confusion between the Fibonacci series and the Golden Spiral.

Scott says

June 24, 2016 at 5:38 amI’m having problems drawing this in a CAD program…. the arc commands won’t replicate the spiral precisely. I am literally trying to draw it in the FibanaccI rectangle pattern and can’t make it work! Any ideas?

Gary B Meisner says

June 25, 2016 at 3:58 pmYou’re likely experiencing two issues:

The spiral created by squares of progressive Fibonacci numbers is only an approximation of the spiral created by adjacent golden rectangles. You’ll be closer if you build the spiral using golden rectangles.

Even then, using a circular arc from the corners of the squares in a golden rectangle is only an approximation of a true golden ratio spiral. To get it perfect you would need to have graphing capabilities and the formula for the golden ratio spiral. See https://en.wikipedia.org/wiki/Golden_spiral for details.

thermo says

November 12, 2016 at 7:10 pmThese spirals appears in stable solitons on superfluids, check this Sculptures….

thermo says

November 12, 2016 at 7:34 pmThe Density Wave Theory explains the Spiral movement of Galaxies:

Dave Zelenka says

March 18, 2017 at 2:18 pmHere’s an example of how some interesting features emerge from interference patterns produced by spirals:

Jorge Xerxes says

December 14, 2017 at 6:40 amVery nice article about spirals, golden ratio and Nature. Ser below a kind of contribution… https://jorgexerxes.wordpress.com/2017/12/12/golden-ratio-sequence/

Julie-Ann says

August 21, 2018 at 5:26 pmI love this.

Innate my

Attraction.

Infinite

And formal.

Sexy as hell.