This topic is to share mathematical object generated with Grasshopper/Rhino.

The object of interest is a three-legged stackable stool. This one is particular since it is a collaboration with an interior designer, Valerie Rostaing. Valerie has been curious about my hobby for mathematical design for many years. The stool that followed is our common experiment with parametric design. I did the program with Grasshopper, she did the parameter fine tuning and other design choices.

I am grateful to Grasshopper. If the tool would not have been fun to use, I would have never done this programming in my spare time.

For the mathematical shape of the stool, it is inspired by spiral plant patterns found in sunflowers or pinecones. More generally, it is based on phyllotaxis, the arrangement of botanic elements such as leaves on a stem. The shape is purely biomimetic, and the stools are stacked in the same way that roses stack their petals or pinecones their scales.

In the 19th century, botanists discovered that some plants arrange their organs spirally. The scales of a pinecone form 5 spirals to the right and 8 spirals to the left. The number of spirals rising in opposite directions always match a pair of consecutive numbers in a mathematical sequence connected with the golden ratio, the Fibonacci sequence (1,1,2,3,5, 8,13,21, …). Thus, 5 and 8 are respectively the 5th and 6th Fibonacci number.

When stacked, one particular leg of the Namamo stool form 2 spirals to the right and 3 spirals to the left (2 and 3 are in the 3rd and 4th position of the Fibonacci sequence). Since they are three legs, you can spot on the stack at the rear 6 spirals to the right and 9 to the left.

The wavy seat reflects the outline of plants sprouting petals and is based on the same mathematical equations as its botanic model. The stool is drawn from 4 curves only: two for the seat and two for each leg. These curves are the result of equations that describe the shape of spiral plants.

The stool’s edge ripples 9 times around the seat. The 9 hollows of this wavy line are designed to fit the legs of the stools stacked on top of it:

- The 3 legs of the first stacked stool fit into the 3 deepest hollows of the stool below.
- The 3 legs of the stool stacked on the next level above fit into the3 shallower hollows.
- The 3 legs of the stool stacked on the next level fit into the3 shallowest hollows.

From the upper edge of each Namamo stool, the seat and legs follow a conical shape, prolonged by the stacked stools, leaving no gap between them.

To my knowledge, this shape is an inter-disciplinary innovation in design. The stool mixes complex equations of plant growth. For this project, I am grateful to Stéphane Douady who patiently explained the phylotaxis equations to me. Stéphane is the French physicist who, in 1992, discovered the link between phylotaxis and the Fibonacci sequence.

The stool is on show with Valerie’s other works at the Gradiva Gallery till the 30th of October in Paris (9, quai Voltaire, in the 7th district).