Eur. Phys. J. D 62, 5-17 (2011)
https://doi.org/10.1140/epjd/e2010-00271-8

How universal are Fibonacci patterns?

¹ Department of Mathematics, Colorado State University, 80523-1874 Fort Collins, USA
² Department of Mathematics, University of California-Irvine, USA
³ Program in Applied Mathematics, University of Arizona, USA
⁴ Department of Mathematics, University of Arizona, USA

^a e-mail: shipman@math.colostate.edu

Received: 22 May 2010
Received in final form: 18 August 2010
Published online: 5 November 2010

Abstract

Pattern patterns, or phyllotaxis, the arrangements of phylla (flowers, leaves, bracts, florets) in the neighborhood of growth tips, have intrigued natural scientists for over four hundred years. Prominent amongst the observed features is the fact that phylla lie on families of alternately oriented spirals and that the numbers in these families belong to subsets  {m_j} of the integers defined by the Fibonacci rule m_j + 1 = m_j + m_j − 1. The corresponding patterns, which we call Fibonacci patterns, are widespread and universal on plants. Our goal in this paper is to ask if they may also be seen in other physical structures and to try to quantify the circumstances under which one may expect Fibonacci patterns to occur.