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

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.

When you have a good understanding of how to do that, move on to try reflections.

There are two main 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'.