Math E-320: Spring 2012
Teaching Math with a Historical Perspective
Mathematics E-320: The Butterfly Theorem
Instructor: Oliver Knill
Office: SciCtr 432

The butterfly theorem is mentioned in the book "The Mathematician's Brain by D. Ruelle. Read the page and see that this mathematical theorem has relations to relatively recent history.

The background music to the animation to the left is from Celtic Harp Volume IV, "O'Carolan's Dream, The Butterfly".

The following pages are from the book by Ruelle. It is an interesting story since the theorem is an example of a "murdorous problems". Ruelle outlines also the proof of the theorem. Proofs are difficult to find. And it needs some knowledge in complex analysis, in particular that Moebius transformations preserve the cross ratio. Below is a proof found in Coxeter-Greitzer.

We will look at this theorem in the lecture, but focus on an other aspect of it. When I was a astudent at ETH, I have had Ruelle as a professor in a "Nachdimplom Vorlesung" (=graduate course) on zeta functions in dynamical systems. Ruelle is famous in particular his mathematical work on Chaos theory. Ruelle and Takens coined the term "Strange attractor". Chaos theory knows the Butterfly effect: "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"

Butterflies appear also in other places of mathematics. An example is the Hofstadter butterfly.

David Ruelle

Proof from Coxeter-Greitzer's book

Pages from the book by Ruelle

Please send questions and comments to
Math E320| Oliver Knill | Spring 2012 | Extension School | Harvard University