One Reflection Implies Another
When a strip has vertical symmetry there always seems to be a
second place where a mirror line could go. Perhaps you can find a
design that has only one mirror line across it. Or, if you thought
that was impossible, could you explain why ?
Age
14 to 16
Challenge level
Problem
This problem builds on from 'A Roll of Patterned Paper'
For the unit shapes I've tried so far I noticed something :
When a strip maps to itself with a mirror line somewhere across it, there always seems to be a second place where a mirror line would also map the pattern to itself.An illustration may help :
Image
The mirror could be at position 1 or at position 2, either way the reflection will map the pattern onto itself.
Your challenge :
Can you find a design with only one mirror line across the strip?Or perhaps, if you thought that was impossible, can you explain how you could be sure about that?
If you're ready for more, try the problem called 'Rotations Aren't Single Round Here'.
Getting Started
Draw a strip which does have two vertical lines of symmetry.
Choose a particular point within the unit shape and mark that point on each unit along your strip.
Now look where that point reflects to across the first line of symmetry and where it reflects to across the second line of symmetry.
Choose a particular point within the unit shape and mark that point on each unit along your strip.
Now look where that point reflects to across the first line of symmetry and where it reflects to across the second line of symmetry.
Teachers' Resources
The Frieze Symmetries are an important mathematical idea.
The question is about an infinite sequence of translations mapping onto itself by :
Reflection, V (vertical mirror line)
or Reflection, H (horizontal mirror line)
or Rotation, R
or by a combination of H with a half unit shift left or right (G, glide reflection) .
This is an excellent opportunity for children to categorize : R or not R, V or not V, etc.
There are in fact only 7 categories, and there's plenty to explore and discuss as the combination options are investigated.