Description
The photograph shows the flowers of a five-petalled weed common in southern Australia, where it is called a soursob. The symmetry in this plant, and in many other naturally occurring examples, contributes to the human understanding of 'beauty'. The mathematical interest in the pentagon and its correlate, the pentagram, can lead to some quite complex mathematics and many apparent 'connections' between living objects.
Activities
1. Explain why you think someone might say 'These flowers show fivefold rotational symmetry'.
2. If you rotate the flower it will soon look the same as when you started. This is what rotational symmetry is all about. What angle do you need to rotate the flower through for it to look the same? After how many degrees will it again look the same? And again?
3. For any one flower, join the (estimated) midpoint of the tip of each petal to its neighbour, What shape does this form? Join each midpoint to the two remaining (opposite) midpoints that it is not already connected to. Find out what you can about this resulting 'star' shape.
4. In the image below, measure the lengths of the orange (AB), purple (AD), green (BC) and blue (CD) line segments. Find the ratios of pairs of these measures in order of size. What do you notice?
(a) There is an internal pentagon inside the pentagram. Join the vertices of this internal pentagon.
(b) Write about the resulting shape and its size. Demonstrate how the pattern continues inside the original pentagram. Demonstrate how the original pentagram is inside a larger one. Write instructions for another student to follow so that they can draw the original pentagram inside a larger one.
5. What are the next three numbers in this number pattern: 0, 1, 1, 2, 3, 5, 8, 13 ...? These numbers have a special name: they are known as the Fibonacci numbers. Can you find out what that name is all about?
6. The Fibonacci numbers are everywhere! Examine your own hands and fingers ... Look for Fibonacci numbers when counting their various parts. Spooky, or what?
7. Can you write the growth pattern of the Fibonacci numbers in words, or in symbols? (Let
be the nth term, eg 
= 3) Can you write it as a formula, using an Excel spreadsheet?
8. Using a spreadsheet or a graphics calculator, examine the ratios of successive Fibonacci numbers, for the first 100 or so. What do you notice?
(a) Does this prove that
?
(b) Numerical values x and y are in golden ratio,, when:
. Can you use this relationship to show
and thus find an exact numerical value of
9. The use of the golden section in design, art and architecture is common in some cultures. Explore some instances of its use in modern times.
