# How to find the numer of symmetries of a geometric figure

Is there any strategy to find out the number of symmetries of any geometric figure?

Can you please explain applying to the following figure where all triangles are equilateral:

By exhaustion, I have found 6 rotatations and 6 reflections wich sum up to 12 symmetries. Is this correct?

• It belongs to $D_{3h}$ group. Apr 4, 2017 at 8:54

In general, when you need to find the symmetry group of a figure, the best way to deal with the question is to find a nice set on which this isometry group acts. For instance :

• the isometry group of the tetrahedron acts on the four vertices of the tetrahedron. This eventually leads to the isomorphism between the isometry group of the tetrahedron with $S_4$ (the symmetric group over four elements).

• the isometry group of the cube acts on the four big diagonals of the cube. This eventually leads to the isomorphism between the direct isometry group of the cube with $S_4$.

Let $X$ be your figure $O$ be its center and $G$ be its isometry group. Clearly, $G$ acts on the faces of $X$, furthermore a square is sent by $G$ on a square. As a result, denoting $F_1=(2,3,6,5)$, $F_2=(2,4,7,5)$ and $F_3=(3,4,7,6)$, the group $G$ acts on $\mathcal{F}:=\{F_1,F_2,F_3\}$ and therefore you have a group morphism from $G$ to $S_{\mathcal{F}}$.

I claim that this group morphism is onto. Furthermore, I claim that the kernel of this group morphism is of cardinal $2$ (to do this, you can remark that if $C_i$ the center of the face $F_i$ then if $g$ is in the kernel then $g\cdot OC_i=OC_i$ for all $i=1,2,3$).

Therefore you have $2\times |S_3|=12$ elements in your group.