‘Explore the symmetries in beautiful Escher tilling patterns using Geogebra’
You’ve probably seen and admired some of the work of the amazing artist M.C. Escher. Whilst visiting the Alhambra Palace in Grenada, Spain, he became inspired by all the beautiful and ornate tiling patterns that filled the different rooms. He started studying these tiling patterns and the hidden mathematics behind them and this gave rise to a new direction and style in his drawings. You are going to study some of his work and discover the different symmetry patterns behind them. This might inspire you to achieve great things too!
This activity requires access to a computer.
Dynamic geometry software will be used. A free version of Geogebra can be downloaded from geogebra.org
You may wish to record the results of your investigation here Escher symmetries
A help video is provided below if you need some technical assistance using Geogebra.
The teacher may wish to read the follow teacher notes Escher Symmetry for further insight into this task.
The task is outlined below.
After a class discussion, you should explore some images using dynamic geometry software.
You are given a lot of help on the first few then you must gradually make decisions for yourself and create some symmetries for yourself.
You should discuss with your teacher the best way of presenting your findings and answers.
Below you should see three images. The butterfly pattern and the bird pattern are both pieces of artwork from Escher. The other is a computer generated pattern. Look at the three patterns and discuss what you notice. Which one do you like best? Why? Are there any similarities/differences?
2. Comparing Symmetries
Below you will be able to watch a short video explaining ALL the symmetry of the butterflies pattern:
Here is an applet to help you analyse the symmetries of the geometry pattern. In your answers you should explain clearly all the different symmetries of this pattern. You may wish to take screenshots of the images to help your explanation.
You should have noticed that the two patterns have exactly the same symmetry. In fact if you look carefully at the bird pattern it also has rotational symmetry of order 6 about one centre, rotational symmetry order 3 about and second centre and rotational symmetry of order 2 about a third centre of rotation – the same symmetry signature. We could say that mathematically all three patterns are exactly the same!
3. Different Symmetries
You may now be thinking that all patterns have the same symmetry. They don’t! Have a look at these two patterns. Is it possible to describe them in the same was as the patterns in part 2?
You may have noticed that the two patterns have reflective as well as rotational symmetry. The following applet will help you find the symmetry signature of these two patterns. Use the check boxes to display all the different symmetries then describe them fully.
4. Glide Reflections
This pattern has 3 different symmetries. There are two reflections and one glide reflection. You may not have heard of glide reflections before, but you have almost certainly seen them. A glide reflection could be described as a reflection followed by a translation (or the other way around!).
Use this Geogebra file Glide Reflections Click on the first checkbox below to see the reflection then use the arrow to control how far it is translated. Can you make the pattern match up?
Once you have got the glide reflection right hide it by unchecking the boxes, then plot the lines of symmetry and get the applet to reflect the ‘image pattern’. A help video is provided below if you need some technical assistance using Geogebra.
5. Rotational Symmetries
This pattern has only rotational symmetries, but hopefully you will see that it is different from the butterflies pattern in part 1. There are 4 different centres of rotational symmetry. Use this Geogebra file 4 Rotational Symmetries. You can use the checkboxes to view three of the centres with their symmetries. Describe them. Can you find the 4th ? Hide the other rotations and carefully plot this 4th point on the image, then use geogebra to make the rotation.
This pattern has 3 different rotational symmetries.
Use this Geogebra file 3 Rotational Symmetries to explore them. This time you will have to do the transformations on Geogebra for yourself.
Describe fully the symmetry.
6. Final Challenge
Can you find all the symmetries in the following pattern?
Clue: there are 4 different ones in total.
Use this Geogebra file Final Challenge
If you need some technical help using geogebra to produce these symmetries you may find the following video screencast useful:
‘Recreate this space age animation to discover the mathematics behind it!’
This task is really easy to explain! Watch the videos below and try to recreate them using dynamic geometry software. Of course, it will not be quite so easy to complete! Making geometry dynamic really helps to understand the properties of a situation. In the videos below, only one feature of the construction is changing at any one time. The task is to look at how the images are changing and try to figure out what could be causing those changes. In the first instance you need to gather evidence, describe what you can see and describe the changes you are witnessing. It is equally important to consider what is not changing. With this evidence and in discussion with the class you should begin to reconstruct what you can see and hopefully recreate an animation with similar properties for yourselves. Good Luck!
Access to computers and dynamic geometry software is needed for this activity. There are some questions just below the videos to help approach the task. Teachers can prepare for this activity by reading these teacher notes Dr Who.
Dr Who (part 1)
This first video shows what happens when one feature of the construction is changed.
Dr Who (part 2)
This video shows the same situation as the first but with a different feature being changed.
Dr Who (part 3)
This video has ‘the twist in the tail!’ to really get you thinking. The same feature is being changed as in the first video but this one takes the changes a bit further.
These are the questions to think about whilst looking at the videos. Answers to these questions should come from investigation and discussion with the class. They should then help with the recreation of the animations.
Why are there lots of shapes?
How are they changing?
What is not changing?
What mathematics describe the way they are changing?
What are the possible variables in the scenario?
Where does the second pattern come from and why?
The following describes a brief overview of the task.
Watch the first video and have and set the problem
Have discusion about what is going on. What Changes? What stays the same?
Repeat the above with the second and third videos
Students try to recreate their own animations with similar properties.