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There is a problem on Invariance in Arthur Engel's Problem Solving Strategies.

Problem: The following operation is performed with a nonconvex non-self-intersecting polygon P. Let A,B be two nonneighboring vertices. Suppose P lies on the same side of AB. Reflect one part of the polygon connecting A with B at the midpoint O of AB. Prove that the polygon becomes convex after finitely many such reflections.

My Attempt: I could easily see that perimeter, the length of the sides are both invariant while the area of the polygon always increased. But then I thought that the order of the sides changes in each step. So I could not convince myself that I had solved it. I saw Engel's solution.

Solution from Engel's book: The permissible transformations leave the sides of the polygon and their directions invariant. Hence, there are only a finite number of polygons. In addition, the area strictly increases after each reflection. So the process is finite.

Remark. The corresponding problem for line reflections in AB is considerably harder. The theorem is still valid, but the proof is no more elementary. The sides still remain the same, but their direction changes. So the finiteness of the process cannot be easily deduced. (In the case of line reflections, there is a conjecture that 2n reflections suffice to reach a convex polygon.)

My Doubt: It should be that the line reflection preserves the order while the reflection about the midpoint of AB changes the order.

Is'nt that true? Please help me understand.

Edit: I found this website where they define reflection about midpoint O of AB as a "flipturn". It is clearly stated in proof of "Joss and Shannon's 2nd Theorem" that in a flipturn, "sides differ only by their cyclic order". Doesn't this confirm that order is indeed different when reflected about O?

TryingHardToBecomeAGoodPrSlvr
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  • What exactly happens when you reflect across $O$? And what does the order of the sides do? – Akababa Jun 06 '17 at 20:17
  • Reflection about O is same as rotation by $180^o$. So if the part of the polygon to get reflected is ARTB, and if the vertices before A and after B are X and Y respectively, then after reflection, the polygon would change to ....XBTRAY.... where as before reflection, it was .....XARTBY....... So now, the sequence of sides changes from XA, AR, RT, TB, BY to XB, BT, TR, RA, AY. So basically the sequence of sides changes and hence the order. Where am I going wrong? – TryingHardToBecomeAGoodPrSlvr Jun 06 '17 at 22:15
  • it seems like the version of line reflections is here: https://math.stackexchange.com/questions/4303227/convexifying-a-concave-polygon-by-reflections – cineel Nov 27 '21 at 18:09

1 Answers1

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I think you may be confusing the order of the edges and the directions of the edges. Reflection through $O$ changes the order of the edges around the polygon, but leaves their angles (say with respect to the $x$-axis) fixed. Reflection across $AB$ will leave the order alone but change the directions.

The order of the edges doesn't matter to Engel's argument since we are only arguing that there is a finite number of polygons, multiplying by $n!$ to take every possible order still leaves a finite number.

The direction does matter, because we are arguing that there are only a finite number of possible polygons and we need to complete the argument to say that the angles can't keep changing as we iterate the process, preventing us from getting to a convex polygon.

Michael Biro
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  • It makes a lot of sense now. My thinking was that a polygon can be thought of as a "necklace" made of edges of fixed lengths where two adjacent sides can twist by any angle. A reflection about O changes the "necklace" due to change in order of sides around the polygon. Your comments makes it clear to me now as to why reflection about O has finite number of configurations. However, I would think that with reflection about AB, the order of the edges around the polygon is fixed. Isn't that information good enough to deduce that there are only finite number of "necklace configurations"? – TryingHardToBecomeAGoodPrSlvr Jun 08 '17 at 21:53