# Symmetry/Rotations at regular n-gon/Group known/Example

We consider the unit circle
    $S^{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\right\}}\,.$
The circle can be parametrized by the trigonometric functions. These assign, for an angle $\alpha \in [0,2\pi )$ (with respect to the $x$-axis, counter-clockwise), the corresponding point
    $(\cos \alpha ,\sin \alpha )$
on the circle. A uniform subdivision of the interval $[0,2\pi ]$ in $n$ pieces of the same length, given by the boundaries
    $0,\,{\frac {2\pi }{n}},\,2{\frac {2\pi }{n}},\,3{\frac {2\pi }{n}},\ldots ,(n-1){\frac {2\pi }{n}},\,n{\frac {2\pi }{n}}=2\pi ,$
yields a uniform subdivision of the circle with the points
    $(1,0),\,\left(\cos {\frac {2\pi }{n}},\,\sin {\frac {2\pi }{n}}\right),\,\left(\cos 2{\frac {2\pi }{n}},\,\sin 2{\frac {2\pi }{n}}\right),\,\left(\cos 3{\frac {2\pi }{n}},\,\sin 3{\frac {2\pi }{n}}\right),\ldots ,\left(\cos(n-1){\frac {2\pi }{n}},\,\sin(n-1){\frac {2\pi }{n}}\right).$
These points are the vertices of a regular $n$-gon. The regelular "2-gon“ has the vertices $$ (1,0) $$, the regular (equilateral) trianlge has the vertices
    $\left(1,\,0\right),\,\left(-{\frac {1}{2}},\,{\frac {\sqrt {3}}{2}}\right),\,\left(-{\frac {1}{2}},\,-{\frac {\sqrt {3}}{2}}\right),$
the regular $4$-gon (square) has the vertices
    $(1,0),\,(0,1),\,(-1,0),\,(0,-1),$
We consider a regular $n$-gon as a rigid object, and we are interested how it can be transformed to itself. The origin is the center (of gravity) of the $n$-gon, a motion of the $n$-gon maps this center to itself. Because such a motion does not change the length, the point $(1,0)$ is sent to a vertex; only these points have distance one from the origin. A motion does not change angles; therefore, the adjacent vertex $\left(\sin {\frac {2\pi }{n}},\,\cos {\frac {2\pi }{n}}\right)$ is sent to an adjacent vertex of the image point of $(1,0)$. For a proper motion (realizable in the plane in the physical sense!), also the ordering (the "orientation“) of the vertices is preserved, so that the only proper motions of a regular $n$-gon are the rotations by a multiple of $2\pi /n$.
If we also allow improper symmetries, then there are also the reflections at an axis; for $n$ even, the reflection axes run through two opposite vertices, or two opposite edge centers, and bei $n$ odd, the reflection axes run through a vertex and an opposite edge center.
Let $n$ be fixed, and set $\alpha =2\pi /n$, and let $\varphi$ denote the rotation of the $n$-gon around $\alpha$ counter-clockwise. Then every rotation of the $n$-gon can be written as $\varphi ^{k}$, with a uniquely determined $k$ between $0$ and $n-1$. Here, $\varphi ^{0}=\operatorname {Id}$ is the rotation by zero degree (the identical motion). If $\varphi$ is performed $n$-times, then we have done one complete rotation. Looking at the result, this is just the identical motion. More generally, when $\varphi$ is performed $m$-times, then the result (that is, the effective motion) does only depend on the remainder $m\mod n$. The inverse motion of $\varphi ^{k}$ is $\varphi ^{-k}$, that is, $k$-times the motion backwards, but this is the same as $\varphi ^{(n-k)}$. All rotations of a regular $n$-gon form a cyclic group of order $n$.