Phyllotaxis and the Golden Ratio

Sophia Darwin (email)

Description

You may have heard of both the Golden Ratio and the Fibonacci Sequence; both are simple to define, and have been studied for centuries. They have surprising and unique properties - for example the Golden Ratio has been described as the "most irrational" number - and are closely related.

Phyllotaxis refers to the patterns we see in plants: the way leaves are arranged on a stem, or scales on a pine cone, or seeds in a sunflower. When we study these patterns, the Golden Ratio and the Fibonacci Sequence keep coming up: in this project you will investigate why this happens. The "most irrational" property turns out to be key.

This topic is suitable for several students.

Prerequisites

No particular topic in maths beyond 1H level is required. You are likely to be using some number theory (modular arithmetic, continued fractions, rational approximations) and possibly some complex analysis.

Resources

  • Ron Knott's Fibonacci site is has lots of fascinating stuff - including Phyllotaxis in the "Nature" section.
  • This Mathworld article contains facts, references and lots of links.
  • Wikipedia provides an introduction to all three:, Fibonacci sequence, Golden Ratio , and Phyllotaxis.
  • The Fibonacci Quarterly is full of odd and interesting facts.
  • Some books in the University Library.
  • There's even some good stuff on Youtube - along with plenty of mystical nonsense, of course.

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