A roll of patterned paper

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection
Age
14 to 16
Challenge level
filled star empty star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



A Roll Of Patterned Paper


In what follows I'm going to call the design above the "unit shape" and imagine it repeated endlessly along a line - rather like a stream of paper coming off a roll.



Here are two pieces from the roll :

A Roll Of Patterned Paper


A Roll Of Patterned Paper


The second piece has then been turned around (rotated 180 $^\circ$).

Your challenge:



Try to design a new unit shape (probably simpler than mine) to make a strip which looks the same after a 180 $^\circ$ rotation.

In other words, make it so that you could not say whether the torn off strip had or had not been rotated.

There are two possibilities: across and along

First the original strip could have a mirror across it.

This illustration shows the right side as the reflection of the left side



A Roll Of Patterned Paper


Can you create a unit shape so that the strip has reflection symmetry across a vertical mirror line ?

And where would the mirror line need to be to be placed ?



Now try a mirror 'along' the torn off strip

The mirror could be reflecting the top half,

A Roll Of Patterned Paper


or (below) the bottom half.

A Roll Of Patterned Paper


Can you make a strip that has reflection symmetry across a horizontal mirror line ?

If you'd like more of this sort of thing, try the problem called 'One Reflection Implies Another'.