Do Plants Know Math - Companion site

Do Plants Know Math?

The Exhibit

About the Exhibit

As a math professor at Smith College, I have an unusual job: I research the spirals found in plants. I count the spirals on pinecones, which actually wind in two directions. Along with my students, I observe the seeds on strawberries, which form complex patterns you might never have noticed. The photographs in this exhibition invite you into our world. Look closely. Count everything—petals, seeds, spines, scales. Open your eyes to these beautiful patterns. See where you can find the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13 ...), which is surprisingly prevalent in the plant world.

You may want to simply look at the photos. But if you’d like to delve deeper into the patterns, scroll down for details about each image. If you're game, draw the spirals and answer the questions.

This exhibition accompanies a new book, Do Plants Know Math?: Unwinding the Story of Plant Spirals, from Leonardo da Vinci to Now (Princeton University Press). I could not have written it alone. The study of plant spirals is interdisciplinary, so I brought in three friends—Stéphane Douady, a physicist in France; Jacques Dumais, a biologist in Chile; and Nancy Pick, a science writer based in Amherst, MA. The larger images on the wall, most of which appear in the book, are the work of Texas photographer Victor Mozqueda, as is the spiraling fern on the book cover. We hope we have whetted your appetite for more.

—Christophe Golé, Department of Mathematical Sciences, Smith College

Winner of the 2025 PROSE Award in Popular Science and Popular Mathematics!

Mockup of exhibit at McConnell Hall, Smith College, Fall 2024. See photos here

Pinecone
(Photo by Alberto Castro, from "Spirals in Plants" exhibit, 2003, Smith College)

The scales of this pinecone form visible spirals, winding in opposite directions. How many spirals do you see unrolling clockwise? What about the more elusive counter-clockwise spirals?

Try tracing these spirals and counting them yourself. See instructions at the bottom.

Click on the picture for more.

Strawberry
(Photo by Victor Mozqueda)

The seeds in a strawberry also form rows winding around the fruit... But they are somewhat irregular, and we'd have to rotate the strawberry to count them. There are ways, however, and our studies show suprising results.

Click on the picture for more.

Romanesco Broccoli
(Photo by Victor Mozqueda)

This plant is what inspired co-author Stéphane Douady to study plant patterns. As a physicist, he first noticed the Romanesco's fractal-like structure: florets within florets within florets... up to seven stages! And all in spirals. Can you count some of them?

Click on the picture for more.

First Modeling of Botanical Spirals
(Drawing by Alexander Braun, 1832)

In the 1830s, German botanists Alexander Braun and Karl Schimper were the first to systematically study spirals in plants. They modeled them as points on a disk, moving out at a certain rate while rotating at a constant angle.

For this image, Braun drew equidistant concentric circles and placed points on successive circles at an angle of 8/21 of a turn (or 137.14 degrees). Joining points to their nearest neighbors at left and right, he identified 8 clockwise spirals (blue) and 5 counterclockwise spirals (red). This kind of pattern is now called "(5, 8) phyllotaxis." (The word phyllotaxis—phyllo (leaf) + taxis (order)—was coined by Schimper, who was a poet as well as a scientist.)

Click on the picture for more.

Embryonic Artichoke Heart
(Scanning Electron Micrograph by Jacques Dumais)

This picture shows the beautiful arrangement of nascent hairs on a tiny 1/5"- wide embryonic artichoke heart. To get the image, coauthor Jacques Dumais had to carefully open the minute flower bud and make an imprint of the heart, using a gel designed for dentists. He then sprayed the imprint with metallic powder, so that the electrons beamed by the microscope could bounce off and reflect a picture.

The spiral structure is complex here, and varies as you move from the border of the heart to the center.

Click on the picture for more.

Pineapple, Unrolled

(Digitally recomposed photo by Stéphane Douady)

To analyze cylindrical patterns in plants, it is easier to unroll them on the plane. Traditionally, this was done by rolling the plant on clay. Co-author Stéphane Douady, a physicist in Paris, invented a clever method of digital unrolling. The plant gets attached to a slowly rotating platform and filmed. The video frames are then digitally sliced and stitched together.

Can you count the numbers of distinct rows in the two directions? (They correspond to spirals on the pineapple.)

Click on the picture for more.

Lace Hedgehog Cactus

(Photo by Victor Mozqueda)

At first sight, this beautiful lace hedgehog cactus (Echinocereus reichenbachii) seems to have spines arranged radially. But the spines also form spirals turning in both directions. To trace the clockwise spirals, choose a bushy spine cluster near the center, and moving outwards, connect it to the closest cluster to its right. Continue moving out and joining clusters to their righthand neighbors, until you reach the cactus's outer border.

Draw all the clockwise spirals in this way—they should cover all the clusters of spines. Repeat the process using another copy of the image, but this time connecting neighbors to the left, to obtain the counter-clockwise spirals.

How many spirals do you count in each direction?

Click on the picture for more.

African Daisy

(Photo by Victor Mozqueda)

This eye-popping African Daisy (Osteospermum ecklonis) also has spirals that are hard to trace. As a hint, each petal connects to a short counter-clockwise spiral of 3 to 4 florets, ending at its base. The clockwise spirals are more tightly wound.

Click on the picture for more.

Eriosyce Cactus

(Photo by Victor Mozqueda)

This Eriosyce cactus blossom looks almost like a water lily. To trace the spirals, pay attention to where each petal touches the one below, to its right and to its left. You may find numbers of spirals that surprise you...

Click on the picture for more.

Visitor Info

Through Dec 14, 2024 | Free admission

Smith College

McConnell Hall

2 Tyler Court, Northampton, MA

Open to the general public

Monday-Friday, 7am-6pm
Closed Saturday-Sunday

Open to Smith OneCard holders

Monday-Friday, 7am-11pm
Saturday-Sunday, 7am-11pm

Tracing Spirals

At the exhibit site, we provide laminated 2-sided copies of the exhibit photos for you to trace on. They're located on the counter to the left of the exhibit, together with some instructions.

If you're purely online,

Choose one of the pictures in the exhibit to trace spirals on and take a screen shot of the photo.
Recover this photo in your photo program
Using the edit function, click on the markup (or drawing) function. Choose a bright color for contrast and use your finger to trace all the clockwise spirals. Tip: don’t try to draw spirals across the whole flower, only where they are most obvious—usually toward the outside. The center can get messy.
Count the spirals, writing the number on your picture. Take a screenshot.
Revert to the original photo. Now, using a different bright color, trace all the counterclockwise spirals.

Thanks

We'd like to thank Marguerite Suozzo-Golé for helping to design this website, as well as the poster and text panel for the exhibit. Thanks to Jim Gipe, from PivotMedia who did a wonderful printing job on canvas. Fraser Stables, from the Smith Arts Department brilliantly lead us in the hanging of the photographs. Leigh Fagin and her Smith Office For the Arts not only provided financial assistance, but also crucial help in hanging the photos. Finally, this exhibit would not have taken place without the wonderful pictures by Victor Mozqueda and Alberto Castro.

This exhibit and website were created while Chris Golé was taking part of the Smith College Kahn Institute project "Vegetal Forms". He would like to acknowledge the support of the participants, and especially of its organizing fellows Jimmy Grogan and Colin Hoag. The Kahn Institute also contributed funds to this project.

More ...

Interviews of Nancy and Chris on the podcast Breaking Math https://www.breakingmath.io/ (link on Spotify) and on the NEPM show the Fabulous 413

Review of the book by Siobahn Roberts in the Wall Street Journal

Photographer Victor Mozqueda' website of his nature pictures.

To learn more about our research on fronts, check out "Fibonacci or Quasisymmetry: Why?"

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