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I have trouble understanding what exactly a dihedral group is. I read about how they are rotations and reflections along the faces of a polygon. But then what does that mean? Whatever you do to a polygon you get the same polygon right? Could someone give a dihedral group for dummies explanation?

And also what does it mean for a shape to have a certain dihedral symmetry?

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  • $\begingroup$ Think of it as taking a $n$-sided polygon where the sides are numbered $1, 2, \ldots , n$. The "new" polygons are different because $1$ will be somewhere else or $7$ will be somewhere else. The trick is that the group itself is comprised of the transformations. $\endgroup$ – AJY Sep 23 '16 at 14:55
  • $\begingroup$ Youconsider all movements of the polygon that do not change the polygon as such. But the $n$ vertices of it are permuted. $\endgroup$ – Hagen von Eitzen Sep 23 '16 at 14:57
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Cut out a regular 5-gon from a paper sheet with a blade, and get 5-gon and paper sheet with the 5-gonal hole. Write your name on the 5-gon with blue pen on the one side, and with red pen on the other side. Then try to fit it back in the hole. How many different ways to do it? Two ways are different if your name have different color or different position. The answer to your question: each "way" is the element of the dihedral group $D_5$.

The final note is that $5$ may be any natural number.

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