# Thread: Vertices and Symmetries

1. ## Vertices and Symmetries

Hi folks,

We are posed a puzzler: Considering the center as the origin, and a vertex given as (1,0). For each of the following regular polygons, find and draw the shape with all the vertices. Describe the vertices in terms of sin and cos of appropriate angles. Describe all the rotations and reflections in terms of any chosen rotation and reflection.

PENTAGON
HEXAGON
HEPTAGON

Ok, so, I have noted that to find the vertices we can break them down into triangles. I am fairly confident with this. I can find the coordinates of the vertices but I don't know what it means by describe them in terms of sin and cos? I mean, they are points not angles?

For the symmetries, I know how to find them all but how to describe them, I don't know? For the rotations, do we just say rotation x degrees about origin? Something like that? It says you can't just name all the symmetries but describe them in terms of one chosen one?

2. ## Re: Vertices and Symmetries

Originally Posted by Adilg
We are posed a puzzler: Considering the center as the origin, and a vertex given as (1,0). For each of the following regular polygons, find and draw the shape with all the vertices. Describe the vertices in terms of sin and cos of appropriate angles. Describe all the rotations and reflections in terms of any chosen rotation and reflection.
PENTAGON
HEXAGON
HEPTAGON
If the regular polygon has N vertices, then the coordinates are given as $\displaystyle \left( {\cos \left( {\frac{{2\pi k}}{N}} \right),\sin \left( {\frac{{2\pi k}}{N}} \right)} \right),\,k = 0,1, \cdots ,N - 1$.

3. ## Re: Vertices and Symmetries

Plato, having read the question does it sound like that is the form I am supposed to give the answer in? I will take your word for it and give thanks.

As for the symmetries, I was thinking of chosing a simple one to start with (reflection in the line from the origin to (1,0) and rotation through one vertex, perhaps) and then try to describe the others as products of them?