Phyllotaxis:

Spiral Pattern Formation in plants – a drawing activity (draft)

 Objectives:

To learn how to construct spirals using polar graph paper

To investigate spiral pattern formation in plant growth (Phyllotaxis)

To create a realistic model of a pattern of parastichies spirals using drawings (and later clay models)

Connection Previous topics:

Fibonacci numbers

            Golden ratio

            Golden Spiral

Materials: Paper, scissors, protractor, ruler, sheets of concentric circles or polar graph paper

Activities:

1.      Using the sheet of concentric circle generate a spiral by placing one dot on each circle starting from the inner circle and moving outward circle by circle. Consider what geometric rule you might follow in placing your dots so that the spiral you generate is regular. When you have finished connected the dots in the order.

 There are several ways to generate spirals. One way is to place the dots so that the angle between dots on adjacent circles (as measured from the center of the circle) is constant. This angle is called the divergence angle. When the divergence angle is small the generating spiral is easy to see visually. When larger divergence angles are used, the generating spiral is not as evident, however, often other spirals will emerge as your eyes make connection between nearest neighbor dots. These spirals are called parastichies and they will be investigated in the next activity

 2.      Different wedge shaped pieces of card with a variety of angles have been made which can be used to generate spirals with specific divergence angles. Choose one of the wedges (or create your own) and place dots on the circles as described above. Do not connect the dots in the order you placed them, but instead look for the most obvious visual spirals that emerge (the parastichies). Connect the dots that form these spirals. State a “connect the dot rule” that creates these spirals.

3.      Compare your pattern with the patterns that other people in the class have created and observe the different features of these patterns. Answer the following questions

• Which divergence angles result in parastichies that are straight lines? Find an explanation for why they are straight.
• Which divergence angles result in the most efficient packing – ie in which case are the gaps between dots most evenly distributed.

4.      Create a spiral pattern using the divergence angle that is most efficient for packing. The dots can represent seeds, petals, leaves, or some other plant structure. Replace the dots in your diagram with some representation of a plant structure of your choice. Consider altering the size and shape of your structure as you move radially outward to make your spiral more realistic.

Preliminary Activities:

• Construct a variety of angles with a compass
• Draw various plant which exhibit pastiches and analyse their structure

Follow-up Activities:

• Computer animation of phyllotaxis to explore the consequences of having more seeds and different divergence angle . This will help to explain why the golden angle is the most efficient one for packing  (see www.math.smith.edu). Students will count the number of parastichies and should observe that they come in pairs of Fibonacci numbers. (a subsequent topic of continued fractions will explain why Fibonacci numbers occur in this case.)
• Create a phyllotaxis pattern with divergence angle Phi pattern using clay: Students make a sequence of balls of clay of increasing size and place them on a polar grid in such a way as to create a spiral pattern of parastichies. They will also  decorate and glaze these patterns (previously students will have previously worked with clay in a measurement and scaling activity)