I know that |rot(tetrahedron) | = 12 ( i know how we came up with this number ) my question what is the number of symmetries in tetrahedron ? is it 12 or 24? if is it 24 can anyone explain to me how we get this number? Also can u write down all possible symmetries of tetrahedron that carry one flag to another flag?
2025-01-12 23:33:01.1736724781
what are the symmetries and flags of tetrahedron?
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If I number the vertices of a tetrahedron, I can take vertex $1$ to any other vertex I like, and then, by rotating about an axis through vertex $1$ I can take vertex $2$ to any of the other three places - that's $3\times 4$ ways. But the images of those two vertices determine the position of the tetrahedron.
If I am allowed reflections, I can swap the remaining two vertices if I wish - a further factor of $2$ - by a reflection in the plane which bisects and is perpendicular to the edge joining the vertices $1,2$.
In this way I can obtain any of the $24$ permutations of the vertices.