Spirals and the Golden Ratio

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:

12 + 12 + 22 + 32 + 52 = 5 x 8

 12 + 12 + . . . + F(n)2 = F(n) x F(n+1)

Fibonacci spiral based on the Fibonacci series in each expansion

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:

Golden rectangle based on phi, the golden ratio, in each expansion

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.)

Fibonacci number spirals in a sunflower seed pod

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.

pine cone Fibonacci spirals

Golden spirals in sea shells

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

Golden ratio proportions in seashell

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 vs a Golden spiral

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.

Nautilus shell showing Golden Ratio proportions

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


  1. greg hope says

    having 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.

  2. Amazing Potential says

    It’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

  3. Cnugg says

    Yes. 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.”

  4. ron phillips says

    Very 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!

  5. BartleyBurnside says

    It 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

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