The golden ratio and the Fibonacci sequence appear in various areas of nature (Scott, and Gulick, 2010, p.37). The Fibonacci sequence, for example, can be found when analysing patterns of seeds on a sunflower (Darvas, 2007, p.112). The individual seeds create spiral arms, curving to the right and the left. The number of spirals to the left, is however not equal to those spiraling to the right. The spiral arms to the left and to the right are always two successive numbers of the Fibonacci sequence (Darvas, 2007, p.112).
Looking at these two diagrams, one can see that there are 21 spiral arms curving to the right and 34 spiral arms curving to the left. These two numbers successive numbers in the Fibonacci sequence. Therefore, seeds in a sunflower follow the pattern of the Fibonacci sequence.
The golden angle plays an important role for the creation of this distinct alignment of seeds. The golden angle is approximately 137.5° and seeds in the sunflower are arranged according to it (Prusinkiewicz and Lindenmayer, 1990, p.100). This angle is irrational which means that no seed has a neighboring one at the exact same angle from the centre (Prusinkiewicz and Lindenmayer, 1990, p.101). This creates the spirals in the pattern of sunflower seeds.
Lesson Ideas for Sunflower Topic
- Bring in sunflower or photograph and devise experiment to test whether spiral arms to left and right are numbers of the Fibonacci sequence
- Producing artwork using sunflower seeds
- Link to agriculture and food production
Darvas, G. (2007) Symmetry: Cultural-historical and Ontological Aspects of Science-Arts Relations. Basel: Birkhäuser Verlag.
National Museum of Mathematics: How to count the spirals. Web. Available at: http://momath.org/home/fibonacci-numbers-of-sunflower-seed-spirals/ (Accessed: 20 November 2015)
Prusinkiewicz, P. and Lindenmayer, A. (1990) The Algorithmic Beauty of Plants. New York: Springer.
Scott, J. and Gulick, D. (2010) The Beauty of Fractals: Six Different Views. Washington, D.C.: Mathematical Association of America.
Stakhov, A. and Olsen, S. (2009) The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing Company.