Symmetric Trace

Before we begin we need to check something - it's about symmetry.

A pattern continues forever in both directions.

Imagine it's on a roll of paper and two strips are torn off, one of which is turned upside-down and placed underneath the other.

It is not possible to shift the lower strip horizontally so that it lines up and matches the upper strip.

On the other hand for the next pattern. . .



Even with the second piece upside-down the two pieces can still be made to line up and match.

Now to start the real problem.

This problem is about that kind of symmetry.

The pattern is a trace from a point on a rolling wheel.

Before starting, you may find it useful to explore How far does it move? .

Point 1 is on the circumference of the wheel and its trace looks like this:

Trace One

Forget the wheel for a moment and just concentrate on the trace pattern.

If this trace was turned upside-down you would certainly not be able to line it up with itself.

Point 2 is somewhere inside the wheel and its trace looks like this :

Trace Two

Would "Trace Two" line up with itself upside-down?

Justify your answer, if you can.

The third trace is made where a horizontal line from Point 1 intersects with a vertical line through the centre of the wheel. It looks like this :

Trace Three

Can "Trace Three" line up with itself upside-down?

Justify your answer this time.