pattern formation :Crystals and Tesselations : Spiral Growth of Patterns in the Growth of Plants (part 1)

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Pattern Formation (Intro Page)

Slide Show
	•	Introduction

Reflections and Rotations
	•	Symmetry Operations, Abstract and in Nature
	•	Symmetry Identification Exercise

Links, Chains, & Spirals
	•	On the Spiral Growth of Shells
	•	Spiral Growth and Plants • part 1 • part 2
	•	Chains, Double Chains, & Spirals

Packing and Tesselations
	•	Bubbles in a bowl activity
	•	Crystal Structures in Two Dimensions
	•	Symmetry in Tesselations
	•	Group Reports

Platonic Solids
	•	Platonic Solids activity
	•	Ionic and Other 3-D Crystal Structures

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Breaking Symmetry
	•	Breaking Symmetry in Inorganic and Living Systems
	•	Breaking Symmetry Activity

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Background: Spiral growth patterns are beautifully expressed in pine cones, on the surface of pineapples and by the composite florets of the sunflower. In fact, plant shoots generally exhibit patterns of this sort. On the handout provided, you can see cross-sections cut through the apex of each of three shoots, from the main stem, a primary branch and a secondary branch of the star pine, Araucaria excelsa. In each diagram, leaf primordia (starting points for the growth of individual leaves) are numbered from the youngest one, formed at the apex of the shoot, at the center of the spiral pattern.

Attachments:
• Variation in phyllotactic patterns.

How many spirals are present in each set, in each of the three diagrams? Look carefully at the three pairs of numbers you have counted. How are these numbers related?

They are members of a celebrated series, with a remarkably simple algorithm. Enter the individual numbers, represented in these pairs, in order in the spaces marked (•) in the table below. What numbers precede those you have observed, in the series? What numbers follow them?

You have discovered, represented in these leaf patterns, the Fibonacci series. This series was first recognized around 1200 AD by Leonardo of Pisa, who is better known as Fibonacci. Let's take a look at some properties of this series.

First, calculate the ratio of each number, n to its predecessor, n - 1 in the table. Then, plot both the Fibonacci numbers and the ratios on two graphs, one above the other, against their numerical positions, 1, 2, 3 .... in the series.

What general form does your graph for the Fibonacci numbers take? Why does the ratio behave as it does, as the Fibonacci number increases?

Examine the flowers and pine cones provided.

Can you find other expressions of the Fibonacci series here?

Sketch or diagram any good examples you can find.

Why do plant growth patterns closely approximate this sequence? Numerous attempts have been made to provide a satisfactory functional explanation for this relationship. The most compelling of these, which has recently gained strong support from model experiments (Patterns, p. 107-108), is simply that new elements (leaf primordia) are always added to the pattern where most space is available.

Mathematically, the pattern is defined by the fact that successive leaf primordia are inserted at points that are almost exactly 137.5° away from their predecessors, around the stem. This angle was known to the ancient Greeks as the Golden Angle, long before its role in plant growth was recognized. This angle is that which divides a circle into two sectors, such that

Figure 1: The Golden Angle

What is this ratio?