PRESENTATION: Mirrored Art
One of the most simple ideas in symmetry that most students will already be aware of is the idea of mirroring. Mirrors are something they encounter every day and so they are very familiar with the concept of a mirror image.
This presentation introduces the idea of mirror images by first looking at the uses of mirrors. It then gives the students three opportunities to demonstrate their understanding of mirrors in different ways – a kinaesthetic activity they can do with their partners, drawing mirrored art by hand and then exploring more intricate mirrored art using some computer software (the link is on the last slide).
WORKSHEET: Fold & Cut Theorem
The Fold & Cut Theorem states that any shape with straight lines, including those with holes and made of separate parts, can be cut from a piece of paper with a single cut, given that the paper is folded in a particular way. In order to do this, the majority of more simple shapes can be cut out using symmetries.
In this Numberphile video, they explain how this can be done with squares and then show that it is possible to cut out every letter of the alphabet with one, single cut. They also give some examples of where this has been used in real-life, telling a supposedly true story from American history. The idea of using symmetry to make processes more efficient can be applied to many different areas of engineering and construction, for example.
The above worksheet, created by JD Hamkins, gives students the opportunity to attempt to cut out some shapes with a single cut themselves. You could extend this by getting the students to try to recreate the Numberphile video by cutting out letters of the alphabet as an additional challenge.
VIDEO: Ambiguous Cylinder Illusion
The Ambiguous Cylinder Illusion was an entry into the 2016 Best Illusion of the Year Contest by Kokichi Sugihara from Meiji University, Japan. It caught the public’s attention at the time as the set-up is so simple but the result it utterly mind-boggling. The illusion, which involves mirrors and therefore symmetry, can be found in this video:
If you wanted to show an explanation as to how this illusion works (which may involve ruining the magic of it!) you can show this video, by an engineer who managed to 3D-print the shape needed for the illusion.