Arrange whatever pieces come your way

(with apologies to Virginia Wolff)

A simple, classic puzzle is to give two shapes and ask if there is a way to cut one up so the pieces can be rearranged into the other. This game might seem to become silly if both shapes are the same;  if we insist that the new arrangement must be different the game becomes interesting again. Think about it, can you come up with ways to cut up a square so that the pieces can be formed into two different squares? Here is an example, not with a square, but with a rhombus:Having the same shape has an advantage. Think about the letter p below, it is part of the blue trapezium, when we rearrange the tiles the p moves with the shape. As the two shapes are the same we can think of this new p within the original rhomb. We can now repeat the process as many times as we want. In this case, it might be a little unsatisfying, however, as the next step for our p would cut it into two different pieces, as it lies on the edge. So where is it safe to put a p so that it will never get cut up? To answer this we have to follow the cutting lines, and a beautiful pattern emerges:The p would be safe within any of the pentagons, but if it crosses any of the edes it will, eventually be cut apart.

Puzzle: Can you work out the difference between the green and the blue pentagons? (Hint: it relates to the dotted and solid lines in the earlier pictures).

Studying what happens when we can move points or objects around in a space (in this case moving p around a rhomb) is studied in a part of mathematics called Dynamical systems the particular example here is called a Piecewise Isometry  (see this paper for a more formal account of their history and study). I have studied these systems myself, and recently submitted a paper looking at the behaviour and number theory that occurs within the pentagon generating system shown above (take a look! It has lots of pictures as well as more formal mathematics).

As you might have guessed from my preoccupations part of my interest in these systems is the pretty images that they produce; this system is particularly rich. This leads to the image at the top. You can take any rhombus and cut it up in a similar way. Take any rhomb (as shown below) and rotate until the side of the rhomb lines up with the top. This will leave a triangle and a trapezium that can be moved back on top of the original rhomb:Additionally this gives a system where the rotation on the two parts is the same, just around different points. You have to be a little careful, but you can use this to give a system for any angles. For any of these systems we can ask the question: Where is it safe to write p? Every angle gives a different pattern, and tiny changes in the angle leads to large changes in the pattern, however the patterns do relate to one another in some ways, as you can see in this video: