---
tag: 263GGJFA21
kind: example
name: Rotation group SO(3)
msc: ["22E15", "22C05", "20G20"]
instance_of: [9TFC6ADC1C]
uses: [0CGYNDCWRR, 0N5N15HWZ7, 1DDYB6VVSR, 7JBRGCVJRB]
sources:
  - url: https://en.wikipedia.org/wiki/3D_rotation_group
same_as:
  wikidata: Q1256500
  wikipedia: 3D rotation group
---

## Example

The *rotation group* is the [[special orthogonal group|0CGYNDCWRR]] in dimension $3$:
$$SO(3) = \set{R \in M_3(\R) : R^T R = I_3, \ \det R = 1}.$$
Each $R \in SO(3)$ is a rotation by some angle $\theta \in [0, \pi]$ about some line through the origin.
For example, the rotation by $\theta$ about the $z$-axis is
$$R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
Check: its columns are orthonormal, so $R_z(\theta)^T R_z(\theta) = I_3$, and its determinant is $\cos^2\theta + \sin^2\theta = 1$.

$SO(3)$ 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 $M_3(\R)$.

## Remarks

$SO(3)$ is connected and has dimension $3$.
Its 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.
As a manifold, $SO(3)$ is diffeomorphic to the [[real projective space|1DDYB6VVSR]] $\R P^3$.
Its fundamental group has order $2$, so it is not simply connected.
Its universal cover is $SU(2)$, the group of unit quaternions, and the covering map has kernel $\set{\pm 1}$.

## Explanations

### L1 human

The rotation group is the set of all ways to turn a rigid object in space while keeping its centre fixed.
Doing one rotation after another is again a rotation, and every rotation can be undone.
Unlike rotations in the plane, rotations in space do not commute.
A quarter turn about one axis followed by a quarter turn about another axis gives a different result in the other order.
A loop of rotations that turns once by $360$ degrees cannot be shrunk to "doing nothing".
The loop that turns twice, by $720$ degrees, can be shrunk.
This is the meaning of the fundamental group of order $2$.