The picture on the right shows the actual effect of reflection along the line

The picture on the right shows the actual effect of

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The picture on the right shows the actual effect of reflection along the line La. All points on the line La remain fixed. But points not on the line exchange places with their respective counterparts in the other half. Thus the vertex A remains fixed while the vertices B and C get exchanged. Note that to find the counterpart of a point P, we draw a perpendicular line from the point P to the line La and extend it to an equal distance on the other side. The point where the line ends now, say P’, is the counterpart of the point P. Thus a `reflection along line La’ is `a function’, which takes P to P’ and vice-versa. See the picture below illustrating this. figure 10
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From this example we can see that reflection takes place along a line. All points on that line of reflection remain fixed . In non-identity rotations , as seen earlier, there is only one fixed point namely the point through which the axis of rotation passes. This fixed point is called the rotocenter . This is the main difference between a reflection and a rotation for finite figures lying in a plane. Are there any other reflection or mirror symmetries of an equilateral triangle ? The answer is yes . The pictures below show two more reflection symmetries that are possible. The effect of the reflection on the vertices is shown in each of the pictures on the right. figure 11 Let us give names to the reflections we have seen above. Let us call reflection along line La as ϕ a , the reflection along line Lb as ϕ b , and the reflection along line Lc as ϕ c . The table below gives the effect that these reflection symmetries have on the vertices of the equilateral triangle. What happens if you apply a reflection twice? ( Q2 .) As we did in the case of the rotations, can we define the order of a reflection? ( Q3 .) Table 2 From Table 2, it is clear that each reflection fixes exactly one vertex, namely the vertex through which the line of reflection passes and just interchanges the other two vertices. The lines La, Lb and Lc are also called lines of symmetry or axes of symmetry for the equilateral triangle ABC. The three reflections described above are the only ones possible for an equilateral triangle. So we have described up to now six different symmetries of an equilateral triangle. These are the three rotations I, ψ , and ψ 2 and the three reflections ϕ a , ϕ b , and ϕ c . From the lesson on permutations you will learn that there are only 6 Reflection Vertices A A B C ϕ a C B A C B B ϕ b C A A B B A ϕ c C C
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permutations possible of the letters A, B, C. Since any symmetry permutes the vertices it has to be one of the six possible permutations and so there can only be at most six symmetries of an equilateral triangle. But we have produced six symmetries already.
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  • Fall '09
  • Math, Symmetry group

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