Reflections+in+Vertices+Investigation

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This open-ended investigation involves repeatedly reflecting a point through the vertices of various polygons and as far as we know has no complete solution. We will introduce the question using the case of triangles Construct any triangle ABC. D is any point in the plane. Reflect point D in vertex A (a 180 °  rotation about vertex A) to get point D'. Reflect D' in vertex B to get D and reflect D in vertex C to get D'''. The video below repeats this question and provides details of the construction within GeoGebra.

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As suggested in the video we could continue the reflections in the second cycle i.e. E reflected in B to get E', and E' reflected in C to get E''. What is the result? Investigate this situation with other triangles and seed points, D. What happens and why? Investigate with other polygons. For the triangle case you can pick up from where Jill left off by using the GeoGebra applet below. Edit this page to share your ideas and work by posting text, Jing videos and GeoGebra applets. Information on how to do this posting can be found at the @Collaborative Online Investigation information page.

media type="custom" key="13086290" We encourage you to join this collaborative investigation by posting your thoughts. If you are continuing on the theme of a page, for instance what happens in the triangle case, we suggest that you just Edit the page and add your text and embed your video and GeoGebra applet. If you wish to take the investigation off in a different direction we suggests linking to a new page. For instance if you wish to post ideas concerning what happens with quadrilaterals you might do this on a page called Reflections in Vertices of Quadrilaterals.

Information on how to add and link to pages and embed Jing videos and GeoGebra applets is on the @Collaborative Online Investigation information page.

Is it OK to post here? and now? I am intrigued! Great investigation... I was not surprised to find that eventually the image coincided with the object - but why? I noticed that by joining up some of the images, there are similar triangles, scale factor 2. media type="custom" key="13159138"

I like this investigation a lot, also. Lots of room for conjecturing. The sketch I made allows for tracing the reflection points or the triangle vertices. I also made a quad out of the preimage and images as I thought the midpoint relationships were neat. At GeoGebraTube, the teacher page lets you download the sketch and the student page lets you play in the browser. Here's an image. Thanks - John Golden.

So as Kathryn notes the cycle closes after two rounds of reflections. But as she also says "Why?" I have picked up on Kathryn's idea of joining up every other point in the cycle, i.e. D to D, D' to D etc. and John's idea of joining D to D or D' to E'' to make quadrilaterals out of each round of reflections. This does give multiple similar and congruent triangles and parallel and equal line segments. This suggests we might think of the reflections as translations along vectors. Below I have posted a video expanding on the above and also the related GeoGebra applet. Geoff

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GeoGebra is a great tool for inductive reasoning, but can get in the way when one switches to deduction. With the vector approach above we need to show that the effect of the first round of reflections, the translation along the vector DD is equal and opposite to the effect of the second round of reflections, the translation along the vector DE''. So we need to prove the vectors DD and DE'' are equal in magnitude and parallel. The problem is with GeoGebra the "truth" of this fact is obvious from an inductive demonstration point of view. The points D and E'' lie on top of each other. In exploring a deductive proof it would help if the points were separated. Below I have reproduced the GeoGebra applet above except for the way point E'' is generated. Below E'' is a point on a ray through E' and C, but does not fall on top of D. I am hoping this new diagram might help in the development of a proof. Geoff

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Really frustrated because I haven't had time to work out how to screencast and it would have been so useful to talk through the 'proof' I think I've worked out. I've used vectors, as suggested by Geoff, and end up implying (although I haven't finally wrapped up the proof) that the vector from the final reflected point to A is equal to the vector from the initial point to A. You can step through (I hope) the steps in the proof. However, I still found it too hard to work on the computer and used pencil and paper to think it through before letting GeoGebra do a nice neat diagram for me! media type="custom" key="13326648"

So I have worked out how to use the SMART recorder to do a screencast, but I can't upload it here, because its a .wmv file, I think! So I have uploaded it onto my college Moodle Geogebra course, which has guest access. Here is the link: Reflection in vertices It seems to be working OK, although I only have the laptop internal mic, so it's a bit quiet.

I (Geoff) have converted Kathryn's .wmv file to .flv and embedded it below.

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Neat proof Kathryn! Looks like a totally vector approach is the way to go rather than my blended vector and Euclidean attempt.

I keep returning to this investigation! I'm fairly convinced (but haven't tried to prove anything yet) that for odd numbers of vertices, the reflection coincides with the object after two complete cycles. For an even number of vertices, the reflections end up in rows. It doesn't require a polygon - it is true for one point and two. I shall be teaching vectors, including the conditions for collinearity, after Easter, so I'm wondering if there might be some mileage in this as a problem solving activity for that class!

An example of Kathryn's observation that "For an even number of vertices, the reflections end up in rows", can be seen and explored on the Reflections in Vertices of Quadrilaterals page.