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A Roll of Patterned Paper

Age 14 to 16 Challenge Level:

Single Unit for Roll

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 :

Basic Pattern

Basic Pattern Rotated 180 degrees

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

Basic Pattern reflected in a vertical mirror line

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,

Basic Pattern with a Horizontal Mirror Line image 1

or (below) the bottom half.

Basic Pattern with a Horizontal Mirror Line image 2

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'.