$Rotational symmertries of octahedron, $R(O_3)$$

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$Rotational symmertries of octahedron, $R(O_3)$$

Show that octahedron has 24 rotational symmetries. Show this by showing the automorphisms from R($O_3$) is obtained by which of the eight faces, and how it sends the face with vertices (1,0,0), (0, 1, 0), (0, 0, 1). (please use this method).

Note that the octahedron is on $R^3$ and its vertices are (1,0,0), (0, 1, 0), (0, 0, 1), (-1,0,0), (0, -1, 0), (0, 0, -1).

R($O_3$) denotes the group of rotational symmetries of octahedron.

Combinatorics Game Theory Abstract Algebra

Sarah M

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There are $8$ possibilities for the image of the face with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ (it can go to any of the other $8$ faces). Once that is fixed, then we can choose any of the vertices of the image face to be the image of $(1,0,0)$, and since we can do it in $3$ ways (as every face has $3$ vertices), we described $8 \times 3 = 24$ rotations.

Now, since we are only considering rotations, the orientation is preserved and so the images of $(0,1,0)$ and $(0,0,1)$ are also fixed: if, when looking at the original face, the vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ appear in counter-clockwise order, then they must appear in counter-clockwise order in the image as well, and so there is no choice.

Since we fixed the images of three vectors that form a basis of $\mathbb{R}^3$, and the transformation is linear, then we fixed the image of the whole space, so these are all the possible rotations.

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$Rotational symmertries of octahedron, $R(O_3)$$

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