Appearances in Nature

A Natural Ratio

Although appearances of the golden mean in art, architecture, and human proportions are largely unsubstantiated, the golden ratio genuinely appears in many elements of nature.

Nautilus shells, for example, have a logarithmic spiral pattern. The golden spiral is also a logarithmic spiral which is governed by the golden ratio. For every 90 degree turn, the radius of the spiral grows by a factor of the phi. Similar logarithmic spirals map the Pergrine falcon’s prey snatching flight path, swirl patterns in hurricanes, and the shape of spiral galaxies.

Hurricane - NASA; Galaxy - NASA; Pinecone - "Paddy Patterson" (license); Nautilus - "Chris 73" (license)

Spiraling Outta Control!

The number of spirals that can be outlined in pineapples, cauliflower, pinecones, cactus spines, and the seeds of sunflowers are all Fibonacci numbers. For a nice demonstration of the 8 spirals one way and the 13 spirals the other way in a pinecone, visit phyllotaxis. Ok, but how do we know that all these occurrences in nature are not purely coincidental? These distributions of the respective units and spirals they form can actually be recreated.

Growing Perfection

Spin a source point (or growth point) at a constant speed and release a seed every 1/φ ≈ 0.618 turn of the circle. Using this special angle related to φ, you end up not only with spiral patterns (Fibonacci in number), but also you get a distribution of seeds that is the most evenly spaced distribution possible. It looks exactly like the seeds in the sunflower. There is no overcrowding and no strange spaces or holes. The golden ratio results in a perfectly spaced pattern containing a Fibonacci number of spirals. Since 360° times 1/φ approximately equals 222.5° this is the ideal angle, along with 137.5° (the same angle measured from the other direction). To see this in action, see this animation, where you can simulate the growth pattern by setting the angle to 137.5°, and then increasing the number of seeds.

Coincidence??

There is a brilliant and logical explanation for all this. Nature is governed by efficiency. It wants to pack the most amount of parts in the least amount of space, but not put everything so close that it can’t function. The golden ratio accomplishes this goal by being the key angle of rotation to make the most efficient, equally spaced packing.

The leaf arrangements of plants also have a spiraling pattern based on the golden ratio. If the plant only produces a leaf about every 0.618 of a turn about the stem (1.618 leafs per turn), then we get the best and most even distribution of leaves. The maximum amount of space is allowed between every leaf that is directly above another one on the stem, so shadowing is minimized. The leaves make the best use of the space they have to capture sunlight in.

Counting on Plants

Because a plant’s growth acts this way, with high dependence on the golden ratio, it makes sense that the Fibonacci numbers would pop up. If you look at the number of petals in flowers, more often than not, they are Fibonacci numbers. Lilies, irises, and trillium all have 3 petals. Buttercups, geraniums, pansies, primroses, rhododendrons, and tomato blossoms all have 5 petals, the most common number of petals for a flower to have. Delphiniums have 8, marigold and ragwort have 13, and black-eyed susans ,chicory, and asters all have 21. Finally daisies often have 34, 55, or even 89 petals.