Aloe polyphylla, also known as spiral aloe, naturally grows in this distinctive spiral shape.
Aloe polyphylla, also known as spiral aloe, naturally grows in this distinctive spiral shape. (Photo: picturepartners/Shutterstock)

The universe may be chaotic and unpredictable, but it's also a highly organized physical realm bound by the laws of mathematics. One of the most fundamental (and strikingly beautiful) ways these laws manifest is through the golden ratio.

It's not hard to find examples of this logarithmic phenomenon in nature — whether it's a simple houseplant (like the aloe plant above) or an expansive spiral galaxy (like the spiral galaxy, Messier 83, seen below), they all originate from the same mathematical concepts.

Messier 83, a spiral galaxy located 15 million light-years away from Earth.
Messier 83, a spiral galaxy located 15 million light-years away from Earth. (Photo: William Blair/Hubble Heritage Team/NASA/ESA)

The golden ratio (often represented by the Greek letter ϕ) is directly tied to a numerical pattern known as the Fibonacci sequence, which is a list composed of numbers that are the sum of the previous two numbers in the sequence. Often referred to as the natural numbering system of the cosmos, the Fibonacci sequence starts out simply (0+1= 1, 1+1=2, 1+2=3, 2+3=5, 3+5=8...), but before long, you'll find yourself adding up numbers in the thousands and millions (10946+17711=28657, 17711+28657=46368, 28657+46368=75025...) and it just keeps going on forever like that.

When a Fibonacci number is divided by the Fibonacci number that came before it, it approaches the golden ratio, which is an irrational number that starts out as 1.6180339887... and, once again, goes on forever.

When the golden ratio is applied as a growth factor (as seen below), you get a type of logarithmic spiral known as a golden spiral.

golden ratio spiral

Learn more about the Fibonacci sequence and natural spirals in this fascinating video series by mathematician Vi Hart:

As Hart explains, examples of approximate golden spirals can be found throughout nature, most prominently in seashells, ocean waves, spider webs and even chameleon tails! Continue below to see just a few of the ways these spirals manifest in nature.

Chameleon tails

A chameleon tail is famous for its tight spiraling shape.
A chameleon tail is famous for its tight spiraling shape. (Photo: Ryan M. Bolton/Shutterstock)

Seashells

A seashell is one of the most well-known examples of the golden ratio spiral in nature.
A seashell is one of the most well-known examples of the golden ratio spiral in nature. (Photo: Tramont_ana/Shutterstock)

Fern fiddleheads

The curled up fronds of a young fern are known as fiddleheads.
The curled up fronds of a young fern are known as fiddleheads. (Photo: Zamada/Shutterstock)

Ocean waves

Despite their tumultuous nature, ocean waves are another example of the golden ratio manifesting in nature.
Despite their tumultuous nature, ocean waves are another example of the golden ratio manifesting in nature. (Photo: irabel8/Shutterstock)

Flower buds

Even when a flower hasn't yet bloomed, it's easy to see evidence of the golden ratio.
When a flower hasn't yet bloomed, it's easy to see evidence of the golden ratio. (Photo: Romantsova Olga/Shutterstock)

Snail shells

Snail shells are a lovely example of the golden ratio in nature.
Snail shells are a lovely example of the golden ratio in nature. (Photo: vvoronov/Shutterstock)

Romanesco broccoli

A close-up of the fractal spirals of Romanesco broccoli (Brassica oleracea).
The fractal spirals of Romanesco broccoli (Brassica oleracea). (Photo: Sergey Skleznev/Shutterstock)

Whirlpools

Whirlpool exhibiting a golden ratio spiral.
A whirlpool exhibiting a golden ratio spiral. (Photo: Elaine Davis/Shutterstock)

Comfrey flowers

Blooms and buds following the spiral of this Comfrey flower's stalk.
You can see the spiral of this Comfrey flower's stalk. (Photo: Mikhail Melnikov/Shutterstock)

Pine cones

Golden ratio spirals as seen from the bottom of a pinecone.
Golden ratio spirals as seen from the bottom of a pinecone. (Photo: bzanchi/Shutterstock)

Sunflower seed head

The golden ratio of growth seen in a sunflower head.
The golden ratio of growth seen in a sunflower head. (Photo: Portogas D Ace/Shutterstock)

Hurricane Isabel (2003)

The spiral of Hurricane Isabel in 2003.
The spiral of Hurricane Isabel in 2003. (Photo: Mike Trenchard/NASA)

Calla lilies

The spiraling bloom of a Calla Lily
The spiraling bloom of a calla lily. (Photo: Lynn Watson/Shutterstock)

Conch shells

A spiky shell with a golden ratio spiral.
A spiky shell with a golden ratio spiral. (Photo: Fontana/Shutterstock)

Spiral aloe

Spiral aloe plant in gardening pot.
A spiral aloe plant sits in a gardening pot. (Photo: PhotoSky/Shutterstock)

Spiderwebs

A spider form formed using a spiral shape.
This spider web uses a spiral shape. (Photo: mycteria/Shutterstock)

Flower petals

Curly flower petals
There are many golden ratios among these curly flower petals. (Photo: Ian Grainger/Shutterstock)