Physics Photo of the Week

February 25, 2011

Mona Lisa and Fibonacci Pinecones

Cover of Math and the Mona Lisa, by Bülent Atalay, Smithsonian Books, Copyright 2006

Recently Professor Bülent Atalay of Mary Washington University presented a fascinating public Lecture at Warren Wilson College on Leonardo's Universe featuring much of the mathematics that Leonardo da Vinci incorporated in many of his studies, drawings, and paintings - including the Mona Lisa

Much of the mathematics that Leonardo da Vinci used was discovered about 3 centuries earlier by "The Other Leonardo": Leonardo Fibonacci di Pisa.  Leonardo Fibonacci is most noted for the simple series that bears his name: 1, 1, 2, 3, 5, 8, 13, 21, ... in which each succeeding number is the sum of the previous two numbers.  These numbers are found in many instances of nature - replication of animals, tree branches, and the spirals of pine cone segments, sunflower seed heads, and pineapples.  The ratio of any Fibonacci number to the previous number in the sequence has a magnificent property.  As the numbers become larger, this number approaches a limit: 1.618034....  For example, the ratio of the 19th Fibonacci number (4181
) divided by the 18th (2584) equals  1.618034....  This number, φ = "phi", is also called "The Golden Ratio", "The Golden Mean", and "The Divine Proportion".  In the book Math and the Mona Lisa, Dr. Atalay points out many features in art, architecture, nature, and especially the Mona Lisa painting many instances of the divine proportion.  It seems that Leonardo mastered the use of this simple irrational number to make his art take on a universal appeal.

The spirals of pinecones and sunflower heads involve Fibonacci numbers.  In the enlarged photo of the cone of a
common pitch pine (found in one of my neighbor's yards) the spirals have been highlighted for easy counting.  There are 8 spirals bending to the left as they open out indicated in red.  Likewise there are 13 spirals opening to the right indicated by blue traces.  8 and 13 are consecutive Fibonacci numbers.  Several pine cones of the same species also exhibited the 8 and 13 spirals.  Cones of other pine species exhibit 5 and 8 spirals - two other consecutive Fibonacci numbers.

Another amazing feature about the spirals in nature is that they are logarithmic spirals.  Notice that the spirals become further apart as they open up further away from the center.  Contrast this with an Archimedes spiral that resembles the spiral of a rolled-up carpet or snowball.

Several students sent me summaries of Dr. Atalay's talk that are incorporated in this summary.  Thanks to Rashad Ali, Evelyn Breziner, Elan Gabel-Richards, Kathryn Kipfer, Gordon Jones, Kaitlyn Varnot, Emilene Whidbee, Kendra Marcus, Mary Reding, Maya Rios, and Skye Rios.  Special thanks are extended to the Lyceum Committee, Dr. Paul Magnarella, the WWC Departments of Art, History/Political Science, Peace and Justice, Chemistry, Physics, and Biology for financial support.  We extend another warm thank you to Dr.
Bülent Atalay for his inspiring presentation and visit.

Physics Photo of the Week is published weekly during the academic year on Fridays by the Warren Wilson College Physics Department.  These photos feature interesting phenomena in the world around us.  Students, faculty, and others are invited to submit digital (or film) photographs for publication and explanation.  Atmospheric phenomena are especially welcome.  Please send any photos to 

All photos and discussions are copyright by Donald Collins or by the person credited for the photo and/or discussion.  These photos and discussions may be used for private individual use or educational use.  Any commercial use without written permission of the photoprovider is forbidden.

Click here to see the Physics Photo of the Week Archive.

Observers are invited to submit digital photos to: