Rotations Are Not Single Round Here
I noticed this about streamers that have rotation symmetry : if
there was one centre of rotation there always seems to be a second
centre that also worked. Can you find a design that has only one
centre of rotation ? Or if you thought that was impossible, could
you say why ?
Age
14 to 16
Challenge level
If you haven't seen the problem 'A Roll of Patterned Paper' you may want to try that first.
In that problem I'd been finding designs for streamers - this problem follows on from that.
I noticed this about the streamers that had rotation symmetry :
In the ones I tried, if there was one centre of rotation there always seems to be another centre that worked as well.Image
Your challenge :
Can you find a design with only one centre of rotation?
Or if you thought that was in fact impossible, could you say why that's true ?Taking it further :
I noticed that if there are mirror lines across the strip, and if there is also a horizontal mirror line, then the strip always seems to have rotation symmetry.
Can that be explained?
Draw a strip which does have two different centres for rotation
symmetry.
Choose a particular point within the unit shape and mark that point on each unit along your strip.
Now look where that chosen point rotates to around the first centre and where it rotates to around the second centre.
Choose a particular point within the unit shape and mark that point on each unit along your strip.
Now look where that chosen point rotates to around the first centre and where it rotates to around the second centre.
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.