Heather Wang

Stanford University lecturer John Edmark explains the mathematics behind his spiral artwork during the Math Club and Art Club joint workshop. Edmark described the relationship between the golden ratio and the Fibonacci numbers, both of which are present in spirals in nature.

Forty students attended a lecture and hands-on workshop by Stanford professor John Edmark about the geometry of spirals in a workshop hosted by Art Club and Math Club on April 28.

Edmark introduced the concept of the logarithmic spiral, a pattern where each curve maintains its shape while changing size. This type of spiral is related to the golden ratio, an irrational number approximately equal to 1.618. He explained how this spiral appears throughout nature, in structures like sunflowers, pinecones and artichokes, where each new petal grows 137.5 degrees, or the golden angle, away from the previous one. 

“[Edmark’s] work is right on that sweet spot between engineering and art,” visual arts teacher Brian Caponi said. “It’s really fascinating to see precision and design within the context of the broader relationship between art and engineering. This is by far the most precise drawing that will ever happen in this space, and it’s really cool to see a system as it relates to larger systems in the world.”

Students participated in a collaborative drawing activity, using rulers, pencils and a golden angle template to create spiral patterns on a large piece of paper. Edmark then instructed students to connect numbers that are eight or 13 apart. As students connected the points, spiral patterns emerged. 

Edmark explained that different patterns become more visible depending on the expansion factor, which is the ratio between the sizes of consecutive turns in the spiral. A higher expansion factor leads to a more spread-out spiral, while a lower expansion factor results in a tighter spiral. Regardless of the expansion factor, the underlying mathematical structure of the spiral remains constant.

Frosh attendee Chian-Shin Du highlighted how the hands-on nature of the workshop allowed students to explore these mathematical concepts in a tangible way.

“I liked how it was interactive and how we got to actually draw things,” Chian-Shin said. “The fact that the spiral pattern can be found a lot in nature is really cool and something that I thought was interesting.”

Edmark also discussed the presence of Fibonacci numbers, noting that the ratio between consecutive Fibonacci numbers approaches the golden ratio.

To close the event, Edmark described examples from his own work, explaining how he created moving sculptures that mimic natural spirals using a single block of wood and laser-cutting techniques. He showcased one of his personal works entitled “Bloom,” a kinetic piece that displayed a spiral movement as it spun under a strobe light that flashed every time the sculpture turned 137.5 degrees, giving the illusion that it was growing. Edmark noted that creating the structure took nearly five years of development.

“I hope that for those that were here today that maybe weren’t so mathematically inclined, that might have helped them see that there is a beauty and excitement and a creativity in mathematics, this being just one example,” Edmark said. “Math is a great tool for exploring ideas. Without math, without geometry, I couldn’t really explore these ideas.”