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

We consider the unit circle

${\displaystyle {}S^{1}={\left\{(x,y)\in \mathbb {R} ^{2}\mid x^{2}+y^{2}=1\right\}}\,.}$

The circle can be parametrized by the trigonometric functions. These assign, for an angle ${\displaystyle {}\alpha \in [0,2\pi )}$ (with respect to the ${\displaystyle {}x}$-axis, counter-clockwise), the corresponding point

${\displaystyle (\cos \alpha ,\sin \alpha )}$

on the circle. A uniform subdivision of the interval ${\displaystyle {}[0,2\pi ]}$ in ${\displaystyle {}n}$ pieces of the same length, given by the boundaries

${\displaystyle 0,\,{\frac {2\pi }{n}},\,2{\frac {2\pi }{n}},\,3{\frac {2\pi }{n}},\ldots ,(n-1){\frac {2\pi }{n}},\,n{\frac {2\pi }{n}}=2\pi ,}$

yields a uniform subdivision of the circle with the points

${\displaystyle (1,0),\,\left(\cos {\frac {2\pi }{n}},\,\sin {\frac {2\pi }{n}}\right),\,\left(\cos 2{\frac {2\pi }{n}},\,\sin 2{\frac {2\pi }{n}}\right),\,\left(\cos 3{\frac {2\pi }{n}},\,\sin 3{\frac {2\pi }{n}}\right),\ldots ,\left(\cos(n-1){\frac {2\pi }{n}},\,\sin(n-1){\frac {2\pi }{n}}\right).}$

These points are the vertices of a regular ${\displaystyle {}n}$-gon. The regelular "2-gon“ has the vertices ${\displaystyle {}}$ (1,0) ${\displaystyle {}}$, the regular (equilateral) trianlge has the vertices

${\displaystyle \left(1,\,0\right),\,\left(-{\frac {1}{2}},\,{\frac {\sqrt {3}}{2}}\right),\,\left(-{\frac {1}{2}},\,-{\frac {\sqrt {3}}{2}}\right),}$

the regular ${\displaystyle {}4}$-gon (square) has the vertices

${\displaystyle (1,0),\,(0,1),\,(-1,0),\,(0,-1),}$

We consider a regular ${\displaystyle {}n}$-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 ${\displaystyle {}n}$-gon, a motion of the ${\displaystyle {}n}$-gon maps this center to itself. Because such a motion does not change the length, the point ${\displaystyle {}(1,0)}$ is sent to a vertex; only these points have distance one from the origin. A motion does not change angles; therefore, the adjacent vertex ${\displaystyle {}\left(\sin {\frac {2\pi }{n}},\,\cos {\frac {2\pi }{n}}\right)}$ is sent to an adjacent vertex of the image point of ${\displaystyle {}(1,0)}$. 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 ${\displaystyle {}n}$-gon are the rotations by a multiple of ${\displaystyle {}2\pi /n}$.

If we also allow improper symmetries, then there are also the reflections at an axis; for ${\displaystyle {}n}$ even, the reflection axes run through two opposite vertices, or two opposite edge centers, and bei ${\displaystyle {}n}$ odd, the reflection axes run through a vertex and an opposite edge center.

Let ${\displaystyle {}n}$ be fixed, and set ${\displaystyle {}\alpha =2\pi /n}$, and let ${\displaystyle {}\varphi }$ denote the rotation of the ${\displaystyle {}n}$-gon around ${\displaystyle {}\alpha }$ counter-clockwise. Then every rotation of the ${\displaystyle {}n}$-gon can be written as ${\displaystyle {}\varphi ^{k}}$, with a uniquely determined ${\displaystyle {}k}$ between ${\displaystyle {}0}$ and ${\displaystyle {}n-1}$. Here, ${\displaystyle {}\varphi ^{0}=\operatorname {Id} }$ is the rotation by zero degree (the identical motion). If ${\displaystyle {}\varphi }$ is performed ${\displaystyle {}n}$-times, then we have done one complete rotation. Looking at the result, this is just the identical motion. More generally, when ${\displaystyle {}\varphi }$ is performed ${\displaystyle {}m}$-times, then the result (that is, the effective motion) does only depend on the remainder ${\displaystyle {}m\mod n}$. The inverse motion of ${\displaystyle {}\varphi ^{k}}$ is ${\displaystyle {}\varphi ^{-k}}$, that is, ${\displaystyle {}k}$-times the motion backwards, but this is the same as ${\displaystyle {}\varphi ^{(n-k)}}$. All rotations of a regular ${\displaystyle {}n}$-gon form a cyclic group of order ${\displaystyle {}n}$.