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)
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 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.