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The Line and Its Strange Pair

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

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Mapping the Wandering Circle

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

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Like a Circle in a Spiral

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

Symmetric Trace

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This is a Three Star Challenge so you won't be expecting a quick fix.

But as a hint : split the trace into chunks that mean something.

Firstly an obvious chunk would be the trace for one revolution of the wheel because that's clearly going to repeat (the period).

You might notice that the first half and the second half of the period match each other in a particular way - how would you describe that kind of mathematical matching and, most importantly, can you account for it?

All three traces have this property but the "same when upside-down" quality isn't there for all of them.

What makes that "same when upside-down" work?