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  "body": "\n## Example\n\nThe *rotation group* is the [[special orthogonal group|0CGYNDCWRR]] in dimension $3$:\n$$SO(3) = \\set{R \\in M_3(\\R) : R^T R = I_3, \\ \\det R = 1}.$$\nEach $R \\in SO(3)$ is a rotation by some angle $\\theta \\in [0, \\pi]$ about some line through the origin.\nFor example, the rotation by $\\theta$ about the $z$-axis is\n$$R_z(\\theta) = \\begin{pmatrix} \\cos\\theta \u0026 -\\sin\\theta \u0026 0 \\\\ \\sin\\theta \u0026 \\cos\\theta \u0026 0 \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}.$$\nCheck: its columns are orthonormal, so $R_z(\\theta)^T R_z(\\theta) = I_3$, and its determinant is $\\cos^2\\theta + \\sin^2\\theta = 1$.\n\n$SO(3)$ is a [[compact Lie group|9TFC6ADC1C]]: it is a closed subgroup of $GL_3(\\R)$, so a [[Lie group|7JBRGCVJRB]], and it is closed and bounded in $M_3(\\R)$.\n\n## Remarks\n\n$SO(3)$ is connected and has dimension $3$.\nIts Lie algebra $\\mathfrak{so}(3)$ is the space of real skew-symmetric $3 \\times 3$ matrices, which is isomorphic to $\\R^3$ with the [[cross product|0N5N15HWZ7]] as bracket.\nAs a manifold, $SO(3)$ is diffeomorphic to the [[real projective space|1DDYB6VVSR]] $\\R P^3$.\nIts fundamental group has order $2$, so it is not simply connected.\nIts universal cover is $SU(2)$, the group of unit quaternions, and the covering map has kernel $\\set{\\pm 1}$.\n\n## Explanations\n\n### L1 human\n\nThe rotation group is the set of all ways to turn a rigid object in space while keeping its centre fixed.\nDoing one rotation after another is again a rotation, and every rotation can be undone.\nUnlike rotations in the plane, rotations in space do not commute.\nA quarter turn about one axis followed by a quarter turn about another axis gives a different result in the other order.\nA loop of rotations that turns once by $360$ degrees cannot be shrunk to \"doing nothing\".\nThe loop that turns twice, by $720$ degrees, can be shrunk.\nThis is the meaning of the fundamental group of order $2$.\n",
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