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Nature and Math: The Fibonacci Sequence

Nature and Math:
The Fibonacci Sequence in Nature

Closeup of the spiral center of a sunflower

Where can we find the golden ratio in nature and art?

Discover a mathematical sequence that can be used to create the shape of a spiral. See how this pattern shows up in nature and art!

Each of the images below includes a spiral—whether huge or tiny, hidden or obvious! Look closely at each of the works of art below. How many spirals can you find?

Bi (disc) with incised spiral designs. China, Western Han dynasty, 206 BC–AD 24. Light green translucent nephrite jade, with calcifications. Gift of Leverett S. Woodworth, MD 1926, and Mrs. Woodworth, 81.049.012.

Bottle with spiral designs. Japan, 18th or 19th century. Porcelain with underglaze blue. Anonymous gift, 82.062.028.

A charcoal drawing of a raised spiral

Dana Duff (American, born 1955), Spiral A, 1986. Charcoal. Gift of Michael Rubenstein, Class of 1957, and Juliet Rubenstein, 90.050.

A grid of tiny spiral patterns in varying brown and black tones

Russell Crotty (American, born 1956), Galaxies (detail), 1996. Pencil and ballpoint pen. Gift from the Estate of Janet Liebowitz, 2011.095.

Spirals in art

We can find spirals in art from many different time periods and cultures. The spiral’s curving, winding, squiggly line shows up is visible in so many different places and contexts! What different kinds of spirals can you find in the artworks above? Do any of the works of art have spirals that look similar?

The red spiral lines on the Japanese inro (a small, portable box) curve toward each other. These red spirals make a pattern similar to the blue lines on the ceramic vase. On the jade-green bi disk from ancient China, small spirals are carved in neat, straight lines, just like the metal spirals in the railing of the staircase in the photograph. Another artist has created a large grey spiral in a square print. This spiral looks different from the rest of the artworks.

Why do you think these artists and creators from different cultures and different parts of the world all used the same design? Have you ever drawn a spiral? Where have you seen spirals in real life?

A brown container decorated with red flowers

Spirals in nature

You might have already thought of one place we often find spirals: in nature! Artists—and all humans!—are surrounded by spirals in everyday life. Whether we know it or not, these spirals and other shapes can inspire us. Look! See if you can find the spirals in these nature photographs. Some are trickier to see than others.

A scaly curved green tail with spots of red in front of a green stem

Closeup of the spiral center of a sunflower

Small green leaves in the center of a succulent plant

A swirl of stars in the night sky

Water spiraling down a drain

A spiral shell

What do you notice about these spirals? Did you find any similarities between the different images?

The curve of the chameleon’s tail is just like the shape of the shell (which is a special type of shell called a Nautilus). It almost looks like if you put the two images on top of each other, they would match up. The inside of the sunflower and the leaves of the succulent don’t have spiral lines in the same way, but the seeds and leaves are organized in a similar spiral pattern. In the photos of the galaxy and the water puddle, it looks like many different spirals are layered on top of each other. What other connections can you find?

Each of the spirals in these photographs follows the same mathematical pattern. Read on to find out more about the magical mathematical explanation!

 

The mathematical secret behind nature’s spirals

The natural spirals aren’t identical—some are big, some small, some show up as a line, some as rows of leaves or petals. These spirals are each golden spirals. Do you have any guesses why they might be called that? In order to figure out the answer, we have to learn about something called the Fibonacci sequence. Check out this video to learn more!

 

Odessa Thompson (Cornell Class of 2024), an intern at the Johnson Museum, explores the connections between art, science, and math using the Fibonacci sequence in this special video for 3rd and 4th graders.

The sequence that Fibonacci discovered shows up all over nature. Of course, a chameleon’s tail or a whirlpool of water don’t use a piece of graph paper, or consciously add up these numbers, in order to make themselves into a perfect spiral. The pattern has showed up naturally since before humans existed, and was discovered through observation.

Just like you saw in the video, you can use a piece of graph paper to create your own golden spiral! The special Fibonacci numbers from this sequence can also be combined, divided, and multiplied in ways that are used in many contexts, from geometry to architecture to design.

Keep your eye out for the spirals in the world around you! Your new knowledge can help you identify a special Fibonacci spiral on a nature walk, or appreciate a delicate artistic spiral in a work of art.

—Elizabeth S. Yearsley, Post-Baccalaureate Fellow in Pre-K–12 Museum Education (2022)

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