Nature's Symmetry

'Explore symmetry in nature and produce beautiful symmetry images of your own' 

Our survival as human beings has relied on our ability to recognise symmetry.  In the past, seeing a symmetrical object in the distance probably meant that there was something to eat, or something that could eat us, approaching!  Today flowers rely on this symmetry to pollinate and survive.  Bees with their terrible eyesight (have you ever noticed how they fly around bumping into things?) can just about pick out striking patterns through the blur.  Symmetrical objects stand out really clearly for them.  So flowers have cleverly evolved to be really quite perfect symmetrical shapes to attract the bees and use them to spread their pollen and survive.  In this activity you will examine just how good the symmetry of flowers and other objects in nature is using photographs and dynamic geometry software on your computer.

Resources & Description

Below you should see a photograph of a flower.  Change the angle «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#945;«/mi»«/math» and see what happens.  The flower has almost perfect rotational symmetry.  What angle do you need to demonstrate this rotational symmetry?  Are there any other angles which will re-create the same symmetry of images?  What is the relationship between the angles?

Below you should see some photographic images taken from nature.  Your job is to create dynamic geometry files like the one above and explore their symmetry.  For each image you should describe fully the symmetry of the images which means the order of rotational symmetry or the equation of the line of symmetry. Careful, not all the images have rotational symmetry; some may have reflective (line) symmetry. 

A video tutorial giving you some hints how you can do this with geogebra can be viewed below.

Credits: california poppy by njhdiver Credits: clover by misterteacher Credits: African Iris by Tatters Credits: butterfly by Duff Axsom

You may wish to discuss with your teacher the best way of presenting your work.  You could take screenshots of your files, create a wikispace and embed the dynamic geometry file into it. (Here is a quick video tutorial of how you can do that with Geogebra from youtube

Now try to find some images of your own.  A good source of images can be found on flickr . Ensure that the photographs have been taken square on, otherwise the symmetry will be skewed.  Try to find flowers with a variety of symmetries: rotational symmetry order 3, 4 lines of reflective symmetry …

Finally create some flowers of your own using dynamic geometry with perfect symmetry.  Create the initial shape and use the software to rotate or reflect the shape so that you end up with the full flower (see example below)

  1. Flower one should have 4 lines of reflective symmetry (you will have to think carefully about the equations of the lines of symmetry that will make this shape).
  2. Flower two should have rotational symmetry of order 6 but no reflective symmetry.
  3. Flower three should have 8 lines of reflective symmetry.
  4. Use your imagination to create flower four!

 Example of a flower with four lines of symmetry

 Help with Geogebra

You may find this video demonstration of how to use geogebra to make reflections and rotations useful


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