Algebra Solutions: Find the number of orientation-preserving isometries of a dodecahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of a dodecahedron in $\mathbb{R}^3$ . Show that $|G| = 60$ .

For reference, consider the following diagram of a dodecahedron.

Let $\theta$ be an orientation-preserving isometry of the dodecahedron; that is, if the vertices of a face, read clockwise from outside the figure, are $VWXYZ$ , then $\theta(V) \theta(W) \theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 20 possibilities for $\theta(1)$ . Once $\theta(1)$ is chosen, there are 3 possibilities for $\theta(2)$ . Once these are chosen, the rest of $\theta$ is determined uniquely by orientation. Thus there are $20 \cdot 3 = 60$ possibilities for $\theta$ , all distinct.

Algebra Solutions

Find the number of orientation-preserving isometries of a dodecahedron

No comments:

Post a Comment