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3D rotation group

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $$\R^3$$ under the operation of composition.

By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition.

Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a Lie group. It is compact and has dimension 3.

Rotations are linear transformations of $$\R^3$$ and can therefore be represented by matrices once a basis of $$\R^3$$ has been chosen. Specifically, if we choose an orthonormal basis of $$\R^3$$, every rotation is described by an orthogonal 3 × 3 matrix (i.e., a 3 × 3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).

The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin.

Length and angle
Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length (see the law of cosines): $$\mathbf{u} \cdot \mathbf{v} = \frac{1}{2} \left(\|\mathbf{u} + \mathbf{v}\|^2 - \|\mathbf{u}\|^2 - \|\mathbf{v}\|^2\right).$$

It follows that every length-preserving linear transformation in $$\R^3$$ preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on $$\R^3$$, which is equivalent to requiring them to preserve length. See classical group for a treatment of this more general approach, where $SO(3)$ appears as a special case.

Orthogonal and rotation matrices
Every rotation maps an orthonormal basis of $$\R^3$$ to another orthonormal basis. Like any linear transformation of finite-dimensional vector spaces, a rotation can always be represented by a matrix. Let $R$ be a given rotation. With respect to the standard basis $e_{1}, e_{2}, e_{3}$ of $$\R^3$$ the columns of $R$ are given by $(Re_{1}, Re_{2}, Re_{3})$. Since the standard basis is orthonormal, and since $R$ preserves angles and length, the columns of $R$ form another orthonormal basis. This orthonormality condition can be expressed in the form

 * $$R^\mathsf{T}R = RR^\mathsf{T} = I,$$

where $R^$ denotes the transpose of $R$ and $I$ is the $3 × 3$ identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all $3 × 3$ orthogonal matrices is denoted $O(3)$, and consists of all proper and improper rotations.

In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix $R$, note that $det R^ = det R$ implies $(det R)^{2} = 1$, so that $det R = ±1$. The subgroup of orthogonal matrices with determinant $+1$ is called the special orthogonal group, denoted $SO(3)$.

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group $SO(3)$.

Improper rotations correspond to orthogonal matrices with determinant $−1$, and they do not form a group because the product of two improper rotations is a proper rotation.

Group structure
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space $$\R^3$$.

Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x.

The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.

Complete classification of finite subgroups
The finite subgroups of $$\mathrm{SO}(3)$$ are completely classified.

Every finite subgroup is isomorphic to either an element of one of two countably infinite families of planar isometries: the cyclic groups $$C_n$$ or the dihedral groups $$D_{2n}$$, or to one of three other groups: the tetrahedral group $$\cong A_4$$, the octahedral group $$\cong S_4$$, or the icosahedral group $$\cong A_5$$.

Axis of rotation
Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of $$\R^3$$ which is called the axis of rotation (this is Euler's rotation theorem). Each such rotation acts as an ordinary 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation).

For example, counterclockwise rotation about the positive z-axis by angle φ is given by

 * $$R_z(\phi) = \begin{bmatrix}\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end{bmatrix}.$$

Given a unit vector n in $$\R^3$$ and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then

 * R(0, n) is the identity transformation for any n
 * R(φ, n) = R(−φ, −n)
 * R($\pi$ + φ, n) = R(π − φ, −n).

Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space.
 * n is arbitrary if φ = 0
 * n is unique if 0 < φ < π
 * n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical).

Topology
The Lie group SO(3) is diffeomorphic to the real projective space $$\mathbb{P}^3(\R).$$

Consider the solid ball in $$\R^3$$ of radius π (that is, all points of $$\R^3$$ of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through an angle $\theta$ between 0 and π (not including either) are on the same axis at the same distance. Rotation through angles between 0 and −π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through −π are the same. So we identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.

Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space $$\mathbb{P}^3(\R),$$ so the latter can also serve as a topological model for the rotation group.

These identifications illustrate that SO(3) is connected but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how it is deformed, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting (by example) at the identity (center of the ball), through the south pole, jumping to the north pole and ending again at the identity rotation (i.e., a series of rotation through an angle φ where φ runs from 0 to 2π).

Surprisingly, running through the path twice, i.e., running from the north pole down to the south pole, jumping back to the north pole (using the fact that north and south poles are identified), and then again running from the north pole down to the south pole, so that φ runs from 0 to 4π, gives a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The plate trick and similar tricks demonstrate this practically.

The same argument can be performed in general, and it shows that the fundamental group of SO(3) is the cyclic group of order 2 (a fundamental group with two elements). In physics applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin–statistics theorem.

The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S3 onto SO(3) that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel {±1}. Topologically, this map is a two-to-one covering map. (See the plate trick.)

Connection between SO(3) and SU(2)
In this section, we give two different constructions of a two-to-one and surjective homomorphism of SU(2) onto SO(3).

Using quaternions of unit norm
The group $SU(2)$ is isomorphic to the quaternions of unit norm via a map given by $$q = a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = \alpha + \beta \mathbf{j} \leftrightarrow \begin{bmatrix}\alpha & \beta \\ -\overline\beta & \overline \alpha\end{bmatrix} = U$$ restricted to $a^2+ b^2 + c^2 + d^2 = |\alpha|^2 +|\beta|^2 = 1$ where $ q \in \mathbb{H}$, $a, b, c, d \in \R$ , $ U \in \operatorname{SU}(2)$ , and $$\alpha = a+bi \in\mathbb{C}$$, $$\beta = c+di \in \mathbb{C}$$.

Let us now identify $$\R^3$$ with the span of $$\mathbf{i},\mathbf{j},\mathbf{k}$$. One can then verify that if $$v$$ is in $$\R^3$$ and $$q$$ is a unit quaternion, then $$qvq^{-1}\in \R^3.$$

Furthermore, the map $$v\mapsto qvq^{-1}$$ is a rotation of $$\R^3.$$ Moreover, $$(-q)v(-q)^{-1}$$ is the same as $$qvq^{-1}$$. This means that there is a $2:1$ homomorphism from quaternions of unit norm to the 3D rotation group $SO(3)$.

One can work this homomorphism out explicitly: the unit quaternion, $q$, with $$\begin{align} q &= w + x\mathbf{i} + y\mathbf{j} + z\mathbf{k}, \\ 1 &= w^2 + x^2 + y^2 + z^2 , \end{align}$$ is mapped to the rotation matrix $$ Q = \begin{bmatrix} 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\ 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\ 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2 \end{bmatrix}. $$

This is a rotation around the vector $(x, y, z)$ by an angle $2θ$, where $cos θ = w$ and $|sin θ| = \|(x, y, z)\|$. The proper sign for $sin θ$ is implied, once the signs of the axis components are fixed. The $2:1$-nature is apparent since both $q$ and $−q$ map to the same $Q$.

Using Möbius transformations

The general reference for this section is. The points $1⁄2$ on the sphere

 * $$\mathbf{S} = \left \{(x,y,z)\in\R^3: x^2 +y^2 +z^2 = \frac{1}{4} \right \}$$

can, barring the north pole $(x, y, z) = (0, 0, 1⁄2)$, be put into one-to-one bijection with points $z = −1⁄2$ on the plane $(ξ, η)$ defined by $P$, see figure. The map $N$ is called stereographic projection.

Let the coordinates on $M$ be $S(P) = P'$. The line $M$ passing through $z = −1⁄2$ and $S$ can be parametrized as

 * $$L(t) = N + t(N - P) = \left(0,0,\frac{1}{2}\right) + t \left ( \left(0,0,\frac{1}{2}\right) - (x, y, z) \right ), \quad t\in \R.$$

Demanding that the $(ξ, η)$-coordinate of $$L(t_0)$$ equals $L$, one finds

 * $$t_0 = \frac1{z-\frac12}.$$

We have $$L(t_0)=(\xi,\eta,-1/2).$$ Hence the map

 * $$\begin{cases} S:\mathbf{S} \to M \\ P = (x,y,z) \longmapsto P'= (\xi, \eta) = \left(\frac{x}{\frac{1}{2} - z}, \frac{y}{\frac{1}{2} - z}\right) \equiv \zeta = \xi + i\eta \end{cases}$$

where, for later convenience, the plane $N$ is identified with the complex plane $$\Complex.$$

For the inverse, write $P$ as

 * $$L = N + s(P'-N) = \left(0,0,\frac{1}{2}\right) + s\left( \left(\xi, \eta, -\frac{1}{2}\right) - \left(0,0,\frac{1}{2}\right)\right),$$

and demand $z$ to find $−1⁄2$ and thus

 * $$\begin{cases} S^{-1}:M \to \mathbf{S} \\ P'= (\xi, \eta) \longmapsto P = (x,y,z) = \left(\frac{\xi}{1 + \xi^2 + \eta^2}, \frac{\eta}{1 + \xi^2 + \eta^2}, \frac{-1 + \xi^2 + \eta^2}{2 + 2\xi^2 + 2\eta^2}\right) \end{cases}$$

If $M$ is a rotation, then it will take points on $L$ to points on $x^{2} + y^{2} + z^{2} = 1⁄4$ by its standard action $s = 1⁄1 + ξ^{2} + η^{2}$ on the embedding space $$\R^3.$$ By composing this action with $g ∈ SO(3)$ one obtains a transformation $S$ of $M$,

 * $$\zeta=P' \longmapsto P \longmapsto \Pi_s(g)P = gP \longmapsto S(gP) \equiv \Pi_u(g)\zeta = \zeta'.$$

Thus $S$ is a transformation of $$\Complex$$ associated to the transformation $Π_{s}(g)$ of $$\R^3$$.

It turns out that $S$ represented in this way by $S ∘ Π_{s}(g) ∘ S^{−1}$ can be expressed as a matrix $Π_{u}(g)$ (where the notation is recycled to use the same name for the matrix as for the transformation of $$\Complex$$ it represents). To identify this matrix, consider first a rotation $Π_{s}(g)$ about the $g ∈ SO(3)$-axis through an angle $M$,

 * $$\begin{align}

x' &= x\cos \phi - y \sin \phi,\\ y' &= x\sin \phi + y \cos \phi,\\ z' &= z. \end{align}$$

Hence

 * $$\zeta' = \frac{x' + iy'}{\frac{1}{2} - z'} = \frac{e^{i\phi}(x + iy)}{\frac{1}{2} - z} = e^{i\phi}\zeta = \frac{e^{\frac{i\phi}{2}} \zeta + 0 }{0 \zeta + e^{-\frac{i\phi}{2}}},$$

which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if $Π_{u}(g)$ is a rotation about the $Π_{u}(g) ∈ SU(2)$-axis through an angle $φ$, then

 * $$w' = e^{i\theta}w, \quad w = \frac{y + iz}{\frac{1}{2} - x},$$

which, after a little algebra, becomes

 * $$\zeta' = \frac{\cos \frac{\theta}{2}\zeta +i\sin \frac{\theta}{2} }{i \sin\frac{\theta}{2}\zeta + \cos\frac{\theta}{2}}.$$

These two rotations, $$g_{\phi}, g_{\theta},$$ thus correspond to bilinear transforms of $g_{φ}$, namely, they are examples of Möbius transformations.

A general Möbius transformation is given by

 * $$\zeta' = \frac{\alpha \zeta + \beta}{\gamma \zeta + \delta}, \quad \alpha\delta - \beta\gamma \ne 0.$$

The rotations, $$g_{\phi}, g_{\theta}$$ generate all of $z$ and the composition rules of the Möbius transformations show that any composition of $$g_{\phi}, g_{\theta}$$ translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices

 * $$\begin{pmatrix}\alpha & \beta\\ \gamma & \delta\end{pmatrix}, \qquad \alpha\delta - \beta\gamma = 1,$$

since a common factor of $g_{θ}$ cancels.

For the same reason, the matrix is not uniquely defined since multiplication by $x$ has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices $R^{2} ≃ C ≃ M$.

Using this correspondence one may write

 * $$\begin{align}

\Pi_u(g_\phi) &= \Pi_u\left[\begin{pmatrix} \cos \phi & -\sin \phi & 0\\ \sin \phi & \cos \phi & 0\\ 0 & 0 & 1 \end{pmatrix}\right] = \pm \begin{pmatrix} e^{i\frac{\phi}{2}} & 0\\ 0 & e^{-i\frac{\phi}{2}} \end{pmatrix},\\ \Pi_u(g_\theta) &= \Pi_u\left[\begin{pmatrix} 1 & 0 & 0\\ 0 & \cos \theta & -\sin \theta\\ 0 & \sin \theta & \cos \theta \end{pmatrix}\right] = \pm \begin{pmatrix} \cos\frac{\theta}{2} & i\sin\frac{\theta}{2}\\ i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}. \end{align}$$

These matrices are unitary and thus $SO(3)$. In terms of Euler angles one finds for a general rotation

one has

For the converse, consider a general matrix

 * $$\pm\Pi_u(g_{\alpha,\beta}) = \pm\begin{pmatrix} \alpha & \beta\\ -\overline{\beta} & \overline{\alpha} \end{pmatrix} \in \operatorname{SU}(2).$$

Make the substitutions

 * $$\begin{align}

\cos\frac{\theta}{2} &= |\alpha|,   &  \sin\frac{\theta}{2} &= |\beta|,    & (0 \le \theta \le \pi),\\ \frac{\phi + \psi}{2} &= \arg \alpha, & \frac{\psi - \phi}{2} &= \arg \beta. & \end{align}$$

With the substitutions, $α, β, γ, δ$ assumes the form of the right hand side (RHS) of ($θ$), which corresponds under $−I$ to a matrix on the form of the RHS of ($φ$) with the same $g, −g ∈ SL(2, C)$. In terms of the complex parameters $Π_{u}(SO(3)) ⊂ SU(2) ⊂ SL(2, C)$,

 * $$g_{\alpha,\beta} = \begin{pmatrix}

\frac{1}{2}\left( \alpha^2 - \beta^2 + \overline{\alpha^2} - \overline{\beta^2}\right) & \frac{i}{2}\left(-\alpha^2 - \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) & -\alpha\beta - \overline{\alpha}\overline{\beta}\\

\frac{i}{2}\left(\alpha^2 - \beta^2 - \overline{\alpha^2} + \overline{\beta^2}\right) & \frac{1}{2}\left(\alpha^2 + \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) & -i\left(+\alpha\beta - \overline{\alpha}\overline{\beta}\right)\\

\alpha\overline{\beta} + \overline{\alpha}\beta & i\left(-\alpha\overline{\beta} + \overline{\alpha}\beta\right) & \alpha\overline{\alpha} - \beta\overline{\beta} \end{pmatrix}.$$

To verify this, substitute for $z$ the elements of the matrix on the RHS of ($θ$). After some manipulation, the matrix assumes the form of the RHS of ($ψ$).

It is clear from the explicit form in terms of Euler angles that the map

 * $$ \begin{cases}

p:\operatorname{SU}(2) \to \operatorname{SO}(3)\\ \pm \Pi_u(g_{\alpha \beta}) \mapsto g_{\alpha \beta} \end{cases}$$

just described is a smooth, $x$ and surjective group homomorphism. It is hence an explicit description of the universal covering space of $L$ from the universal covering group $xy$.

Lie algebra
Associated with every Lie group is its Lie algebra, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket. The Lie algebra of $x'y'$ is denoted by $$\mathfrak{so}(3)$$ and consists of all skew-symmetric $xy$ matrices. This may be seen by differentiating the orthogonality condition, $L$. The Lie bracket of two elements of $$\mathfrak{so}(3)$$ is, as for the Lie algebra of every matrix group, given by the matrix commutator, $z$, which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula.

The elements of $$\mathfrak{so}(3)$$ are the "infinitesimal generators" of rotations, i.e., they are the elements of the tangent space of the manifold SO(3) at the identity element. If $$R(\phi, \boldsymbol{n})$$ denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector $$\boldsymbol{n},$$ then

 * $$\forall \boldsymbol{u} \in \R^3: \qquad \left. \frac{\operatorname{d}}{\operatorname{d}\phi} \right|_{\phi=0} R(\phi,\boldsymbol{n}) \boldsymbol{u} = \boldsymbol{n} \times \boldsymbol{u}.$$

This can be used to show that the Lie algebra $$\mathfrak{so}(3)$$ (with commutator) is isomorphic to the Lie algebra $$\R^3$$ (with cross product). Under this isomorphism, an Euler vector $$\boldsymbol{\omega}\in\R^3$$ corresponds to the linear map $$\widetilde{\boldsymbol{\omega}}$$ defined by $$\widetilde{\boldsymbol{\omega}}(\boldsymbol{u}) = \boldsymbol{\omega}\times\boldsymbol{u}.$$

In more detail, most often a suitable basis for $$\mathfrak{so}(3)$$ as a $z$-dimensional vector space is

\boldsymbol{L}_x = \begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}, \quad \boldsymbol{L}_y = \begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}, \quad \boldsymbol{L}_z = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}. $$

The commutation relations of these basis elements are,

[\boldsymbol{L}_x, \boldsymbol{L}_y] = \boldsymbol{L}_z, \quad [\boldsymbol{L}_z, \boldsymbol{L}_x] = \boldsymbol{L}_y, \quad [\boldsymbol{L}_y, \boldsymbol{L}_z] = \boldsymbol{L}_x $$

which agree with the relations of the three standard unit vectors of $$\R^3$$ under the cross product.

As announced above, one can identify any matrix in this Lie algebra with an Euler vector $$\boldsymbol{\omega} = (x,y,z) \in \R^3,$$

 * $$\widehat{\boldsymbol{\omega}} =\boldsymbol{\omega}\cdot \boldsymbol{L} = x \boldsymbol{L}_x + y \boldsymbol{L}_y + z \boldsymbol{L}_z =\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix} \in \mathfrak{so}(3).$$

This identification is sometimes called the hat-map. Under this identification, the $$\mathfrak{so}(3)$$ bracket corresponds in $$\R^3$$ to the cross product,

 * $$\left [\widehat{\boldsymbol{u}},\widehat{\boldsymbol{v}} \right ] = \widehat{\boldsymbol{u} \times \boldsymbol{v}}.$$

The matrix identified with a vector $$\boldsymbol{u}$$ has the property that

 * $$\widehat{\boldsymbol{u}}\boldsymbol{v} = \boldsymbol{u} \times \boldsymbol{v},$$

where the left-hand side we have ordinary matrix multiplication. This implies $$\boldsymbol{u}$$ is in the null space of the skew-symmetric matrix with which it is identified, because $$\boldsymbol{u} \times \boldsymbol{u} = \boldsymbol{0}.$$

A note on Lie algebras
In Lie algebra representations, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant is simply the sum of the squares of the generators, $$\boldsymbol{J}_x, \boldsymbol{J}_y, \boldsymbol{J}_z,$$ of the algebra

[\boldsymbol{J}_x, \boldsymbol{J}_y] = \boldsymbol{J}_z, \quad [\boldsymbol{J}_z, \boldsymbol{J}_x] = \boldsymbol{J}_y, \quad [\boldsymbol{J}_y, \boldsymbol{J}_z] = \boldsymbol{J}_x. $$ That is, the Casimir invariant is given by
 * $$\boldsymbol{J}^2\equiv \boldsymbol{J}\cdot \boldsymbol{J} =\boldsymbol{J}_x^2+\boldsymbol{J}_y^2+\boldsymbol{J}_z^2 \propto \boldsymbol{I}.$$

For unitary irreducible representations $ψ$, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality $$2j+1$$. That is, the eigenvalues of this Casimir operator are
 * $$\boldsymbol{J}^2=- j(j+1) \boldsymbol{I}_{2j+1},$$

where $L$ is integer or half-integer, and referred to as the spin or angular momentum.

So, the 3 × 3 generators L displayed above act on the triplet (spin 1) representation, while the 2 × 2 generators below, t, act on the doublet (spin-1/2) representation. By taking Kronecker products of $x$ with itself repeatedly, one may construct all higher irreducible representations $$. That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large $$, can be calculated using these spin operators and ladder operators.

For every unitary irreducible representations $$ there is an equivalent one, $Π(g_{α, β})$. All infinite-dimensional irreducible representations must be non-unitary, since the group is compact.

In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin $$ characterize bosonic representations, while half-integer values fermionic representations. The antihermitian matrices used above are utilized as spin operators, after they are multiplied by $$, so they are now hermitian (like the Pauli matrices). Thus, in this language,

[\boldsymbol{J}_x, \boldsymbol{J}_y] = i\boldsymbol{J}_z, \quad [\boldsymbol{J}_z, \boldsymbol{J}_x] = i\boldsymbol{J}_y, \quad [\boldsymbol{J}_y, \boldsymbol{J}_z] = i\boldsymbol{J}_x. $$ and hence
 * $$\boldsymbol{J}^2= j(j+1) \boldsymbol{I}_{2j+1}.$$

Explicit expressions for these $$ are,
 * $$\begin{align}

\left (\boldsymbol{J}_z^{(j)}\right )_{ba} &= (j+1-a)\delta_{b,a}\\ \left (\boldsymbol{J}_x^{(j)}\right )_{ba} &=\frac{1}{2} \left (\delta_{b,a+1}+\delta_{b+1,a} \right ) \sqrt{(j+1)(a+b-1)-ab}\\ \left (\boldsymbol{J}_y^{(j)}\right )_{ba} &=\frac{1}{2i} \left (\delta_{b,a+1}-\delta_{b+1,a} \right ) \sqrt{(j+1)(a+b-1)-ab}\\ \end{align}$$ where $D^{j}$ is arbitrary and $$1 \le a, b \le 2j+1$$.

For example, the resulting spin matrices for spin 1 ($$j = 1$$) are
 * $$\begin{align}

\boldsymbol{J}_x &= \frac{1}{\sqrt{2}} \begin{pmatrix} 0 &1 &0\\ 1 &0 &1\\ 0 &1 &0 \end{pmatrix} \\ \boldsymbol{J}_y &= \frac{1}{\sqrt{2}} \begin{pmatrix} 0 &-i &0\\ i &0 &-i\\ 0 &i &0 \end{pmatrix} \\ \boldsymbol{J}_z &= \begin{pmatrix} 1 &0 &0\\ 0 &0 &0\\ 0 &0 &-1 \end{pmatrix} \end{align}$$

Note, however, how these are in an equivalent, but different basis, the spherical basis, than the above $j$L  in the Cartesian basis.

For higher spins,  such as spin $D^{j}$ ($$j=\tfrac{3}{2}$$):
 * $$\begin{align}

\boldsymbol{J}_x &= \frac{1}{2} \begin{pmatrix} 0 &\sqrt{3} &0 &0\\ \sqrt{3} &0 &2 &0\\ 0 &2 &0 &\sqrt{3}\\ 0 &0 &\sqrt{3} &0 \end{pmatrix} \\ \boldsymbol{J}_y &= \frac{1}{2} \begin{pmatrix} 0 &-i\sqrt{3} &0 &0\\ i\sqrt{3} &0 &-2i &0\\ 0 &2i &0 &-i\sqrt{3}\\ 0 &0 &i\sqrt{3} &0 \end{pmatrix} \\ \boldsymbol{J}_z &=\frac{1}{2} \begin{pmatrix} 3 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &-1 &0\\ 0 &0 &0 &-3 \end{pmatrix}. \end{align}$$

For spin $j$ ($$j = \tfrac{5}{2}$$),
 * $$\begin{align}

\boldsymbol{J}_x &= \frac{1}{2} \begin{pmatrix} 0 &\sqrt{5} &0 &0 &0 &0 \\ \sqrt{5} &0 &2\sqrt{2} &0 &0 &0 \\ 0 &2\sqrt{2} &0 &3 &0 &0 \\ 0 &0 &3 &0 &2\sqrt{2} &0 \\ 0 &0 &0 &2\sqrt{2} &0 &\sqrt{5} \\ 0 &0 &0 &0 &\sqrt{5} &0 \end{pmatrix} \\ \boldsymbol{J}_y &= \frac{1}{2} \begin{pmatrix} 0 &-i\sqrt{5} &0 &0 &0 &0 \\ i\sqrt{5} &0 &-2i\sqrt{2} &0 &0 &0 \\ 0 &2i\sqrt{2} &0 &-3i &0 &0 \\ 0 &0 &3i &0 &-2i\sqrt{2} &0 \\ 0 &0 &0 &2i\sqrt{2} &0 &-i\sqrt{5} \\ 0 &0 &0 &0 &i\sqrt{5} &0 \end{pmatrix} \\ \boldsymbol{J}_z &= \frac{1}{2} \begin{pmatrix} 5 &0 &0 &0 &0 &0 \\ 0 &3 &0 &0 &0 &0 \\ 0 &0 &1 &0 &0 &0 \\ 0 &0 &0 &-1 &0 &0 \\ 0 &0 &0 &0 &-3 &0 \\ 0 &0 &0 &0 &0 &-5 \end{pmatrix}. \end{align}$$

Isomorphism with 𝖘𝖚(2)
The Lie algebras $$\mathfrak{so}(3)$$ and $$\mathfrak{su}(2)$$ are isomorphic. One basis for $$\mathfrak{su}(2)$$ is given by

 * $$\boldsymbol{t}_1 = \frac{1}{2}\begin{bmatrix}0 & -i\\ -i & 0\end{bmatrix}, \quad \boldsymbol{t}_2 = \frac{1}{2} \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}, \quad \boldsymbol{t}_3 = \frac{1}{2}\begin{bmatrix}-i & 0\\ 0 & i\end{bmatrix}.$$

These are related to the Pauli matrices by

 * $$\boldsymbol{t}_i \longleftrightarrow \frac{1}{2i} \sigma_i.$$

The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by $D^{j}$, the exponential map (below) is defined with an extra factor of $j$ in the exponent and the structure constants remain the same, but the definition of them acquires a factor of $i$. Likewise, commutation relations acquire a factor of $D^{j}$. The commutation relations for the $$\boldsymbol{t}_i$$ are

 * $$[\boldsymbol{t}_i, \boldsymbol{t}_j] = \varepsilon_{ijk}\boldsymbol{t}_k,$$

where $Π_{u}$ is the totally anti-symmetric symbol with $φ, θ, ψ$. The isomorphism between $$\mathfrak{so}(3)$$ and $$\mathfrak{su}(2)$$ can be set up in several ways. For later convenience, $$\mathfrak{so}(3)$$ and $$\mathfrak{su}(2)$$ are identified by mapping

 * $$\boldsymbol{L}_x \longleftrightarrow \boldsymbol{t}_1, \quad \boldsymbol{L}_y \longleftrightarrow \boldsymbol{t}_2, \quad \boldsymbol{L}_z \longleftrightarrow \boldsymbol{t}_3,$$

and extending by linearity.

Exponential map
The exponential map for $α, β$, is, since $α. β$ is a matrix Lie group, defined using the standard matrix exponential series,

 * $$\begin{cases}

\exp : \mathfrak{so}(3) \to \operatorname{SO}(3) \\ A \mapsto e^A = \sum_{k=0}^\infty \frac{1}{k!} A^k = I + A + \tfrac{1}{2} A^2 + \cdots. \end{cases}$$

For any skew-symmetric matrix $2:1$, $SO(3)$ is always in $SU(2)$. The proof uses the elementary properties of the matrix exponential

 * $$\left(e^A\right)^\textsf{T} e^A = e^{A^\textsf{T}} e^A = e^{A^\textsf{T} + A} = e^{-A + A} = e^{A - A} = e^A \left(e^A\right)^\textsf{T} = e^0 = I.$$

since the matrices $SO(3)$ and $3 × 3$ commute, this can be easily proven with the skew-symmetric matrix condition. This is not enough to show that $A^{T}A = I, A ∈ SO(3)$ is the corresponding Lie algebra for $[A_{1}, A_{2}] = A_{1}A_{2} − A_{2}A_{1}$, and shall be proven separately.

The level of difficulty of proof depends on how a matrix group Lie algebra is defined. defines the Lie algebra as the set of matrices

 * $$\left\{A \in \operatorname{M}(n, \R) \left| e^{tA} \in \operatorname{SO}(3) \forall t\right.\right\},$$

in which case it is trivial. uses for a definition derivatives of smooth curve segments in $3$ through the identity taken at the identity, in which case it is harder.

For a fixed $D^{1/2}$, $D^{−j−1}$ is a one-parameter subgroup along a geodesic in $ε_{ijk}$. That this gives a one-parameter subgroup follows directly from properties of the exponential map.

The exponential map provides a diffeomorphism between a neighborhood of the origin in the $ε_{123} = 1$ and a neighborhood of the identity in the $SO(3)$. For a proof, see Closed subgroup theorem.

The exponential map is surjective. This follows from the fact that every $SO(3)$, since every rotation leaves an axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form

 * $$D = \begin{pmatrix}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0 & 0 & 1\end{pmatrix} = e^{\theta L_z},$$

such that $A ∈ 𝖘𝖔(3)$, and that

 * $$Be^{\theta L_z}B^{-1} = e^{B\theta L_zB^{-1}},$$

together with the fact that $e^{A}$ is closed under the adjoint action of $SO(3)$, meaning that $A$.

Thus, e.g., it is easy to check the popular identity

 * $$e^{-\pi L_x/2} e^{\theta L_z} e^{\pi L_x/2} = e^{\theta L_y}.$$

As shown above, every element $AT$ is associated with a vector $𝖘𝖔(3)$, where $SO(3)$ is a unit magnitude vector. Since $SO(3)$ is in the null space of $j$, if one now rotates to a new basis, through some other orthogonal matrix $A ≠ 0$, with $e^{tA}, −∞ < t < ∞$ as the $i$ axis, the final column and row of the rotation matrix in the new basis will be zero.

Thus, we know in advance from the formula for the exponential that $SO(3)$ must leave $𝖘𝖔(3)$ fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of $SO(3)$, because its existence would violate the hairy ball theorem; but direct exponentiation is possible, and yields

 * $$\begin{align}

\exp(\tilde{\boldsymbol{\omega}}) &= \exp(\theta(\boldsymbol{u\cdot L})) = \exp\left(\theta \begin{bmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{bmatrix}\right)\\[4pt] &= \boldsymbol{I} + 2cs(\boldsymbol{u\cdot L}) + 2s^2 (\boldsymbol{u \cdot L})^2 \\[4pt] &= \begin{bmatrix} 2 \left(x^2 - 1\right) s^2 + 1 & 2 x y s^2 - 2 z c s & 2 x z s^2 + 2 y c s \\ 2 x y s^2 + 2 z c s & 2 \left(y^2 - 1\right) s^2 + 1 & 2 y z s^2 - 2 x c s \\ 2 x z s^2 - 2 y c s & 2 y z s^2 + 2 x c s & 2 \left(z^2 - 1\right) s^2 + 1 \end{bmatrix}, \end{align}$$

where $c = \cos\frac{\theta}{2}$ and $s = \sin\frac{\theta}{2}$. This is recognized as a matrix for a rotation around axis $R ∈ SO(3)$ by the angle $3⁄2$: cf. Rodrigues' rotation formula.

Logarithm map
Given $A = BDB^{−1}$, let $$A = \tfrac{1}{2} \left(R - R^\mathrm{T}\right)$$ denote the antisymmetric part and let $\|A\| = \sqrt{-\frac{1}{2}\operatorname{Tr}\left(A^2\right)}.$ Then, the logarithm of $5⁄2$ is given by

 * $$\log R = \frac{\sin^{-1}\|A\|}{\|A\|}A.$$

This is manifest by inspection of the mixed symmetry form of Rodrigues' formula,

 * $$e^X = I + \frac{\sin \theta}{\theta}X + 2\frac{\sin^2\frac{\theta}{2}}{\theta^2}X^2, \quad \theta = \|X\|,$$

where the first and last term on the right-hand side are symmetric.

Uniform random sampling
$$SO(3)$$ is doubly covered by the group of unit quaternions, which is isomorphic to the 3-sphere. Since the Haar measure on the unit quaternions is just the 3-area measure in 4 dimensions, the Haar measure on $$SO(3)$$ is just the pushforward of the 3-area measure.

Consequently, generating a uniformly random rotation in $$\R^3$$ is equivalent to generating a uniformly random point on the 3-sphere. This can be accomplished by the following$$(\sqrt{1-u_1}\sin(2\pi u_2), \sqrt{1-u_1}\cos(2\pi u_2), \sqrt{u_1}\sin(2\pi u_3), \sqrt{u_1}\cos(2\pi u_3))$$

where $$u_1, u_2, u_3$$ are uniformly random samples of $$[0, 1]$$.

Baker–Campbell–Hausdorff formula
Suppose $i$ and $i$ in the Lie algebra are given. Their exponentials, $𝖘𝖔(3)$ and $SO(3)$, are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some $i$ in the Lie algebra, $BθL_{z}B^{−1} ∈ 𝖘𝖔(3)$, and one may tentatively write

 * $$ Z = C(X, Y),$$

for $i$ some expression in $A ∈ 𝖘𝖔(3)$ and $ω = θ u$. When $u = (x,y,z)$ and $u$ commute, then $O$, mimicking the behavior of complex exponentiation.

The general case is given by the more elaborate BCH formula, a series expansion of nested Lie brackets. For matrices, the Lie bracket is the same operation as the commutator, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,

 * $$Z = C(X, Y) = X + Y + \frac{1}{2} [X, Y] + \tfrac{1}{12} [X, [X, Y]] - \frac{1}{12} [Y, [X, Y]] + \cdots.$$

The infinite expansion in the BCH formula for $u$ reduces to a compact form,

 * $$Z = \alpha X + \beta Y + \gamma[X, Y],$$

for suitable trigonometric function coefficients $exp(OAO^{T})$.

The $u$ are given by
 * $$\alpha = \phi \cot\left(\frac{\phi}{2}\right) \gamma, \qquad \beta = \theta \cot\left(\frac{\theta}{2}\right)\gamma, \qquad \gamma = \frac{\sin^{-1}d}{d}\frac{c}{\theta \phi},$$

where

 * $$\begin{align}

c &= \frac{1}{2}\sin\theta\sin\phi - 2\sin^2\frac{\theta}{2}\sin^2\frac{\phi}{2}\cos(\angle(u, v)),\quad a = c \cot\left(\frac{\phi}{2}\right), \quad b = c \cot\left(\frac{\theta}{2}\right), \\ d &= \sqrt{a^2 + b^2 + 2ab\cos(\angle(u, v)) + c^2 \sin^2(\angle(u, v))}, \end{align}$$

for

 * $$\theta = \|X\|,\quad \phi = \|Y\|,\quad \angle(u, v) = \cos^{-1}\frac{\langle X, Y\rangle}{\|X\|\|Y\|}.$$

The inner product is the Hilbert–Schmidt inner product and the norm is the associated norm. Under the hat-isomorphism,

 * $$\langle u, v\rangle = \frac{1}{2}\operatorname{Tr}X^\mathrm{T}Y,$$

which explains the factors for $A$ and $z$. This drops out in the expression for the angle.

It is worthwhile to write this composite rotation generator as

 * $$\alpha X + \beta Y + \gamma[X, Y]\underset{\mathfrak{so}(3)}{=} X + Y + \frac{1}{2} [X, Y] + \frac{1}{12} [X, [X, Y]] - \frac{1}{12} [Y, [X, Y]] + \cdots,$$

to emphasize that this is a Lie algebra identity.

The above identity holds for all faithful representations of $u$. The kernel of a Lie algebra homomorphism is an ideal, but $u$, being simple, has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the 2×2 derivation for SU(2).

The Pauli vector version of the same BCH formula is the somewhat simpler group composition law of SU(2),

e^{i a'\left(\hat{u} \cdot \vec{\sigma}\right)}e^{i b'\left(\hat{v} \cdot \vec{\sigma}\right)} = \exp\left(   \frac{c'}{\sin c'} \sin a' \sin b' \left(\left(i\cot b'\hat{u} + i \cot a' \hat{v}\right)\cdot\vec{\sigma} + \frac{1}{2} \left[i \hat{u} \cdot \vec{\sigma}, i \hat{v} \cdot \vec{\sigma}\right]\right) \right), $$

where

 * $$\cos c' = \cos a' \cos b' - \hat{u} \cdot\hat{v} \sin a' \sin b',$$

the spherical law of cosines. (Note $R ∈ SO(3)$ are angles, not the $exp(X)$ above.)

This is manifestly of the same format as above,
 * $$Z = \alpha' X + \beta' Y + \gamma' [X, Y],$$

with
 * $$X = i a'\hat{u} \cdot \mathbf{\sigma}, \quad Y = ib'\hat{v} \cdot \mathbf{\sigma} \in \mathfrak{su}(2),$$

so that

 * $$\begin{align}

\alpha' &= \frac{c'}{\sin c'}\frac{\sin a'}{a'}\cos b' \\ \beta' &= \frac{c'}{\sin c'}\frac{\sin b'}{b'}\cos a' \\ \gamma' &= \frac{1}{2}\frac{c'}{\sin c'}\frac{\sin a'}{a'}\frac{\sin b'}{b'}. \end{align}$$

For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of $θ$-matrices, $exp(Y)$, so that
 * $$a' \mapsto -\frac{\theta}{2}, \quad b' \mapsto - \frac{\phi}{2}.$$

To verify then these are the same coefficients as above, compute the ratios of the coefficients,
 * $$\begin{align}

\frac{\alpha'}{\gamma'} &= \theta\cot\frac{\theta}{2} &= \frac{\alpha}{\gamma}\\ \frac{\beta'}{\gamma'} &= \phi\cot\frac{\phi}{2} &= \frac{\beta}{\gamma}. \end{align}$$ Finally, $exp(Z) = exp(X) exp(Y)$ given the identity $X$.

For the general $Y$ case, one might use Ref.

The quaternion formulation of the composition of two rotations RB and RA also yields directly the rotation axis and angle of the composite rotation RC = RBRA.

Let the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,

 * $$S = \cos\frac{\phi}{2} + \sin\frac{\phi}{2} \mathbf{S}.$$

Then the composition of the rotation RR with RA is the rotation RC = RBRA with rotation axis and angle defined by the product of the quaternions

 * $$A = \cos\frac{\alpha}{2} + \sin\frac{\alpha}{2}\mathbf{A}\quad\text{ and }\quad B = \cos\frac{\beta}{2} + \sin\frac{\beta}{2}\mathbf{B},$$

that is

C = \cos\frac{\gamma}{2} + \sin\frac{\gamma}{2}\mathbf{C} = \left(\cos\frac{\beta}{2} + \sin\frac{\beta}{2}\mathbf{B}\right)\left(\cos\frac{\alpha}{2} + \sin\frac{\alpha}{2}\mathbf{A}\right).$$

Expand this product to obtain

\cos\frac{\gamma}{2} + \sin\frac{\gamma}{2} \mathbf{C} = \left(   \cos\frac{\beta}{2}\cos\frac{\alpha}{2} -    \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B}\cdot \mathbf{A}  \right) + \left(    \sin\frac{\beta}{2}\cos\frac{\alpha}{2} \mathbf{B} +    \sin\frac{\alpha}{2}\cos\frac{\beta}{2} \mathbf{A} +    \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B} \times \mathbf{A}  \right). $$

Divide both sides of this equation by the identity, which is the law of cosines on a sphere,

 * $$\cos\frac{\gamma}{2} = \cos\frac{\beta}{2}\cos\frac{\alpha}{2} - \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B}\cdot \mathbf{A},$$

and compute

 * $$\tan\frac{\gamma}{2} \mathbf{C} = \frac{\tan\frac{\beta}{2}\mathbf{B} + \tan\frac{\alpha}{2} \mathbf{A} + \tan\frac{\beta}{2}\tan\frac{\alpha}{2} \mathbf{B} \times \mathbf{A}}{1 - \tan\frac{\beta}{2}\tan\frac{\alpha}{2} \mathbf{B} \cdot \mathbf{A}}.$$

This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).

The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles.

Realizations of rotations
We have seen that there are a variety of ways to represent rotations:
 * as orthogonal matrices with determinant 1,
 * by axis and rotation angle
 * in quaternion algebra with versors and the map 3-sphere S3 → SO(3) (see quaternions and spatial rotations)
 * in geometric algebra as a rotor
 * as a sequence of three rotations about three fixed axes; see Euler angles.

Spherical harmonics
The group $exp(X)$ of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space

 * $$L^2\left(\mathbf{S}^2\right) = \operatorname{span} \left\{ Y^\ell_m, \ell \in \N^+, -\ell \leq m \leq \ell \right\}, $$

where $$Y^\ell_m$$ are spherical harmonics. Its elements are square integrable complex-valued functions on the sphere. The inner product on this space is given by

If $R$ is an arbitrary square integrable function defined on the unit sphere $exp(Y)$, then it can be expressed as

where the expansion coefficients are given by

The Lorentz group action restricts to that of $Z = X + Y$ and is expressed as

This action is unitary, meaning that

The $exp$ can be obtained from the $SO(3)$ of above using Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional $(α, β, γ)$-representation (the 3-dimensional one is exactly $(α, β, γ)$). In this case the space $𝖘𝖔(3)$ decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations $𝖘𝖔(3)$ according to

This is characteristic of infinite-dimensional unitary representations of $a', b', c'$. If $X$ is an infinite-dimensional unitary representation on a separable Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations. Such a representation is thus never irreducible. All irreducible finite-dimensional representations $a, b, c$ can be made unitary by an appropriate choice of inner product,

 * $$\langle f, g\rangle_U \equiv \int_{\operatorname{SO}(3)} \langle\Pi(R)f, \Pi(R)g\rangle \, dg = \frac{1}{8\pi^2} \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} \langle \Pi(R)f, \Pi(R)g\rangle \sin \theta \, d\phi \, d\theta \, d\psi, \quad f,g \in V,$$

where the integral is the unique invariant integral over $σ → 2i t$ normalized to $γ = γ'$, here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on $d = sin 2c'$.

Generalizations
The rotation group generalizes quite naturally to n-dimensional Euclidean space, $$\R^n$$ with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension n(n − 1)/2.

In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.

The rotation group SO(3) can be described as a subgroup of E+(3), the Euclidean group of direct isometries of Euclidean $$\R^3.$$ This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation, or put differently, a combination of an element of SO(3) and an arbitrary translation.

In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

title: 3D rotation group


text:
David Hartley (philosopher)

David Hartley (baptized 21 June 1705 Old Style; died 28 August 1757) was an English philosopher and founder of the Associationist school of psychology.

Early life and family history
David Hartley was born in 1705 in the vicinity of Halifax, Yorkshire. His mother died three months after his birth. His father, an Anglican clergyman, died when David was fifteen. Hartley was educated at Bradford Grammar School and in 1722 was admitted as a Sizar to Jesus College, Cambridge where he was a Rustat scholar. He received his BA in 1726 and MA in 1729. In April 1730 he became the first layperson to be Master of Magnus Grammar School (Magnus Church of England Academy), Newark, and it was there that he began to practise medicine. On 21 April 1730, Hartley married Alice Rowley (1705–31). The couple moved to Bury St Edmunds, and Alice died there giving birth to their son David Hartley (the Younger) (1731–1813). While in Bury, Hartley met his second wife, Elizabeth Packer (1713–78), the fifth child and only daughter of Robert Packer (died 1731) and Mary Winchcombe, a wealthy and influential family with estates in Gloucestershire, Oxfordshire, and Berkshire, including Donnington Castle House, Shellingford Manor and Bucklebury House, Berks. (Mary Winchcombe was the daughter of Sir Henry Winchcombe, Bart., and the sister of Frances, wife of Henry St John, 1st Viscount Bolingbroke.) Despite the opposition of Elizabeth's family, David and Elizabeth wed on 25 August 1735, after agreeing to a severe set of restrictions that kept the £5,000 Elizabeth received upon her marriage completely out of the hands of her husband. Their first child, Mary (1736–1803), was born eleven months later. In 1736 the family moved to London, and then in 1742 to Bath, Somerset. When Elizabeth's last surviving elder brother died without issue in 1746, their son Winchcombe Henry (1740–94) inherited the family estates, making the family (though not Hartley himself) the possessors of significant wealth. Hartley died in Bath on 28 August 1757. He was buried at St John the Baptist Church, Old Sodbury, Gloucestershire.

Education and professional career
At Cambridge, Hartley studied with Nicholas Saunderson, who, though blind since infancy, became the fourth Lucasian Professor of Mathematics. Hartley was later instrumental in raising the subscription for the posthumous publication of Saunderson's Elements of Algebra (1740). Upon graduation, Hartley declined to sign the Thirty-nine Articles, a requirement for ordination in the Church of England. Although one point at issue may have been the doctrine of the Trinity, Hartley's main dissent from Anglican orthodoxy was his assent to universal reconciliation. Writing to his friend Joseph Lister in 1736, Hartley stated he believed "[t]hat Universal Happiness is the Fundamental Doctrine both of Reason & Scripture", adding that "nothing is so irreconcilable [with] Reason as eternal Punishment, nothing so contrary to all the Intimations God has given us in his Works. Have you read Sr. Is. Newton’s Commt. upon Danl. & the Apocalypse?" For Hartley, on the gates of hell there could be no locks.

In the same letter to Lister, Hartley writes that "a Man who disregards himself, who entirely abandons Self-Interest & devotes his Labours to the Service of Mankind, or in that beautiful and expressive phrase of the Scriptures, who loves his neighbor as himself is sure to meet with private Happiness". This conviction became a guiding principle in Hartley's life, and it led him to devote himself to a various philanthropic projects. These include the publication of Saunderson's Elements of Algebra and the promotion of the shorthand system of his friend John Byrom (a system that Hartley believed could be a "universal character" and step toward the creation of a philosophical language).

Shortly after turning to medicine, Hartley became an advocate of variola inoculation for smallpox. Variolation confers personal immunity, and if widespread would be a "service to mankind" by furthering herd immunity. However, deliberate infection with the smallpox virus ran the risk of disfigurement or death. (Queen Caroline, wife of George II, was an advocate and had three of their children variolated, but Jonathan Edwards died from it in 1758.) The public good, then, could appear to be at odds with private interest. In his first publication, Some Reasons why the Practice of Inoculation ought to be Introduced into the Town of Bury at Present (1733), Hartley developed a statistical argument to show that the conflict is only apparent, that being inoculated furthers both the public good and a person's self-interest.

By the time of his move to London in 1736, Hartley was known by other campaigners of variolation, such as Hans Sloane and James Jurin, president of the Royal Society. He also had the support of important Whig families in Suffolk, notably of Charles Townshend, 2nd Viscount Townshend ('Turnip' Townshend). Hartley's daughter Mary wrote that "the old Lord Townshend (then Secretary of State) treated him with as much kindness as he had had been an additional son, and all the sons and daughters as an additional brother". He was inducted into the Royal Society, and he also became a physician to Thomas Pelham-Holles, 1st Duke of Newcastle, and his wife. In 1736 he offered to "recommend" John Byrom to George II.

By 1740, Hartley was known to every physician in London and to other medical men throughout Europe. He had thrown himself into a controversial attempt to harmonize private interest and public good. Hartley had started to experience symptoms of "the stone" (bladder stone) in early 1736. A bladder stone, sometimes as large as an egg, could function as a ball-cock on a toilet tank, causing an inability to urinate, excruciating pain, and sometimes death. (Benjamin Franklin, a sufferer, sometimes had to stand on his head to relieve himself.) Treatment by surgical removal (lithotomy) was a procedure many failed to survive.

Hartley thought a herbalist called Joanna Stephens had developed a lithontriptic, an oral medicine that would dissolve a stone in situ. He published Ten Cases of Persons who have Taken Mrs. Stephens’s Medicines for the Stone (1738), which includes an unsparing account of his own agonies. To make a proprietary medicine freely available to the public, Hartley convinced Parliament to pay Stephens £5,000 for her "secret".

With Stephens's recipe in hand, Hartley set to work with Stephen Hales, along with two colleagues in France, to locate the medicine's chemically active ingredients. These were slaked lime (calcium hydroxide) and Alicant soap (predominantly potassium oleate, like other soaps an alkaline salt of a fatty acid). Hales had shown that some bladder stones rapidly dissolved in boiled soap-lye (caustic potash, potassium hydroxide). What was needed, then, was a safely ingestible preparation that would turn a person's urine alkaline; and this, they concluded, is what the slaked lime and soap combination did. In 1739 Hales won the Copley Medal for his work, and the following year Hartley published their results in a Latin volume, De Lithontriptico, in Basel and Leiden, the latter being home at the time to the foremost medical school in Europe.

In 1742 Hartley and his family moved to Bath, Somerset. He continued to practise medicine, and he devoted himself to writing his major work, Observations on Man, His Frame, His Duty, and His Expectations, published in 1749 by Samuel Richardson.

He was a vegetarian.

Observations on Man: "A New and Most Extensive Science"
In the 18th century, "Observations" appears in the titles of scientific works – compare Benjamin Franklin's Experiments and Observations on Electricity (1751) and Joseph Priestley's Experiments and Observations of Different Kinds of Air (1774). Priestley, Hartley's champion, would declare that Hartley's work "contains a new and most extensive science. … [T]he study of it will be like entering upon a new world. …" The science is the science of "man", and the "new world" is the one embodied in the human "frame" itself.

The result, on the one hand, is a "vast haystack of a book". As one would expect from a physician with an inquiring mind and active medical practice, Hartley draws together a wide range of observations – to name a few, on phantom limbs, savant syndrome, and the experiences and mental development of the blind and the deaf (OM 1, props. 34, 69, 78, 80, and 93).

On the other hand, just as Newton's law of universal gravitation unified celestial and terrestrial mechanics, Hartley proposed a single "law" — "association" — to account for any and all observations of "man". Hartley's many observations are meant to be illustrations of the law.

Moreover, "association" has explanatory power. For example, in the section "The Affections by which we rejoice at the Misery of Others" (OM 1.1.4.97–98), Hartley presents a detailed analysis of the process by which an abused, bullied child becomes an abusive, bullying adult. He traces out how the child's automatic gesture of raising an arm to ward off a blow becomes, through a series of associative substitutions, the fist the adult raises to strike a child. An initial defensive gesture becomes a general aggressive stance, and thus the source of the insulting words and threatening actions by which the adult "goes on multiplying perpetually … the occasions of anger and the expressions of it".

Nonetheless, Hartley believed that it was no-one's destiny to be permanently trapped in such a hell. He was, rather, a religious visionary, and his fundamental belief breathtakingly optimistic: that association "has a tendency to reduce the state of those who have eaten of the tree of the knowledge of good and evil, back again to a paradisiacal one" (OM 1.1.2.14, Cor. 9). From this vantage point, Hartley's Observations on Man is a psychological epic, a story of "paradise regained" — but an epic describing, ultimately, the life of each human being.

Theories
Like John Locke, he asserted that, prior to sensation, the human mind is a blank slate. By a growth from simple sensations, those states of consciousness which appear most remote from sensation come into being. And the one law of growth of which Hartley took account was the law of contiguity, synchronous and successive. By this law he sought to explain, not only the phenomena of memory, which others had similarly explained before him, but also the phenomena of emotion, of reasoning, and of voluntary and involuntary action (see Association of Ideas). A friend, associate, and one of his chief advocates, was Joseph Priestley (1733–1804), the discoverer of oxygen. Priestley was one of the foremost scientists of his age.

Doctrine of vibrations
Hartley's physical theory gave birth to the modern study of the intimate connection of physiological and psychical facts. He believed that sensation is the result of a vibration of the minute particles of the medullary substance of the nerves, to account for which he postulated, with Newton, a subtle elastic ether, rare in the interstices of solid bodies and in their close neighbourhood, and denser as it recedes from them. Pleasure is the result of moderate vibrations, pain of vibrations so violent as to break the continuity of the nerves. These vibrations leave behind them in the brain a tendency to fainter vibrations or "vibratiuncles" of a similar kind, which correspond to "ideas of sensation." This accounts for memory.

Doctrine of associations
The course of reminiscence and of the thoughts generally, when not immediately dependent upon external sensation, is accounted for by the idea that there are always vibrations in the brain on account of its heat and the pulsation of its arteries. The nature of these vibrations is determined by each man's past experience, and by the circumstances of the moment, which causes one or another tendency to prevail over the rest. Sensations which are often associated together become each associated with the ideas corresponding to the others; and the ideas corresponding to the associated sensations become associated together, sometimes so intimately that they form what appears to be a new simple idea, not without careful analysis resolvable into its component parts.

Free will
Starting from a detailed account of the phenomena of the senses, Hartley tried to show how, by the above laws, all the emotions, which he analyses with considerable skill, may be explained. Locke's phrase "association of ideas" is employed throughout, "idea" being taken as including every mental state but sensation. He emphatically asserts the existence of pure disinterested sentiment, while declaring it to be a growth from the self-regarding feelings. Voluntary action is explained as the result of a firm connexion between a motion and a sensation or "idea," and, on the physical side, between an "ideal" and a motory vibration. Therefore, in the Freewill controversy Hartley took his place as a determinist. It was only with reluctance, and when his speculations were nearly complete, that he came to a conclusion on this subject in accordance with his theory.

List of major works

 * Conjecturae quaedam de sensu, motu, et idearum generatione, Appendix to De Lithontriptico a Joanna Stephens nuper invento Dissertatio Epistolaris (Bath, 1746); repr. in Samuel Parr (ed.), Metaphysical Tracts by English Philosophers (1837); trans. Robert E.A. Palmer
 * Various Conjectures on the Perception, Motion, and Generation of Ideas, with an Introduction and notes by Martin Kallich (Augustan Reprint Society, Publication no. 77–8, Los Angeles, 1959).
 * Observations on Man, His Frame, His Duty, and His Expectations. In Two Parts (1749; 2nd edn, trans. from the German, with A Sketch of the Life and Character of David Hartley by his son David Hartley, 1791; 1st edn repr. with an Introduction by Theodore L. Huguelet, Delmar, New York, 1976).
 * Prayers and Religious Meditations (Bath, 1810; R. Cruthwell, 1814).

David Hartley also published numerous medical works.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: David Hartley (philosopher)


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Thomas Holcroft

Thomas Holcroft (10 December 1745 – 23 March 1809) was an English dramatist, miscellanist, poet and translator. He was sympathetic to the early ideas of the French Revolution and helped Thomas Paine to publish the first part of The Rights of Man.

Early life
Holcroft was born in Orange Court, Leicester Fields, London. His father had a shoemaker's shop and kept riding horses for hire, but he fell into difficulties and was reduced to hawking as a pedlar. The son accompanied his parents on their travels. He obtained work as a stable boy at Newmarket, at the stables of Hon. Richard Vernon, where he spent his evenings chiefly on miscellaneous reading and the study of music. He gradually obtained a knowledge of French, German and Italian.

When Holcroft's job at the stables came to an end, he returned to assist his father, who had resumed his trade of shoemaker in London. Around 1765, he became a teacher in a small school in Liverpool. However, he failed in an attempt to set up a private school, and instead became the prompter in a Dublin theatre. He went on to act in various strolling companies until 1778, when he produced the play The Crisis; or, Love and Famine, at Drury Lane. Duplicity followed in 1781.

Literary and political career
Two years later Holcroft went to Paris as correspondent of the Morning Herald. Here he attended the performances of Beaumarchais's Mariage de Figaro until he had memorized the whole. His translation of it, with the title The Follies of the Day, was produced at Drury Lane in 1784. His comedy The Road to Ruin, his most successful play, was produced in 1792; a revival in 1873 ran for 118 nights.

His novels include Alwyn (1780), an account, largely autobiographical, of a strolling comedian, Anna St. Ives (the first British Jacobin novel, published in 1792), and The Adventures of Hugh Trevor (1794–1797). He also wrote Travels from Hamburg through Westphalia, Holland and the Netherlands to Paris, some volumes of verse, and translations from French and German. One of these was Letters Between Frederic II and M. De Voltaire (1789).

Sympathetic to the early ideals of the French Revolution, Holcroft assisted in publishing the first part of Thomas Paine's The Rights of Man in 1791. He joined the Society for Constitutional Information (SCI) in 1792 and was appointed a member of a liaison committee to work with the LCS in early 1794. As a result of his activism, Holcroft was indicted in the autumn of 1794 for high treason and held in Newgate Prison whilst three other treason trials proceeded. In early December 1794 Holcroft was discharged without trial after those cases, against London Corresponding Society secretary Thomas Hardy, and SCI figure John Horne Tooke, resulted in acquittals.

As one of what Secretary of War William Windham called "acquitted felons", Holcroft's post-arrest reputation meant that his plays achieved little success after 1795, although he was instrumental in bringing melodrama to Britain at the end of the decade with his Deaf and Dumb (1801) and A Tale of Mystery (1802, an unacknowledged translation of de Pixerécourt's Cœlina, ou, l'enfant du mystère). Despite a modicum of success with A Tale of Mystery, the remainder of the decade was marked by unsuccessful attempts to return to the public eye. He died in 1809, not long after a deathbed reconciliation with his closest friend from the 1790s (lately estranged), William Godwin. His Memoirs written by Himself and continued down to the Time of his Death, from his Diary, Notes and other Papers, by William Hazlitt, appeared in 1816, and was reprinted, in a slightly abridged form, in 1852.

Personal life

Thomas Holcroft married four times.

From his first wife, whom he married around 1765 and whose name is unknown, he had a daughter Ann (1766–1841), who in 1797 married Colonel William Tooke Harwood (1757–1824), a close associate of John Horne Tooke (1736–1812) and a fervent follower of Joanna Southcott (1750–1814).

In 1772 Holcroft married Matilda Tipler from Nottingham and had with her two children: a son William (1773–1789), who being only sixteen, committed suicide while attempting to escape to the West Indies after robbing his father of £40 (Memoirs, pp. 140–142), and a daughter Sophia (1775–1850), who in 1794 married William Cole, a merchant from Exeter. She resided later at Hamburg, and in 1805, after Cole's death, was married to Georges Danton's cousin Georges Nicholas Mergez (1772–1846), a general in the Napoleonic army.

In 1778, three years after the death of his second wife, Holcroft married Diana Robinson, who died in 1780 after giving birth to a daughter Fanny Margaretta (1780–1844). Fanny Holcroft was the author of the noted Romantic anti-slavery poem, "The Negro" (1797), as well as novels such as Fortitude and Frailty (1817) and The Wife and the Lover (1813–14). From 1805 to 1806, she also translated seven plays (from German, Italian, and Spanish) for her father's "Theatrical Recorder" and later wrote a melodrama of her own.

After nine years as a widower, Holcroft married his fourth wife, Louisa Mercier (1779–1853), in March 1799. She was the daughter of a longstanding friend, Charles-André Mercier, brother of the French dramatist Louis-Sébastien Mercier (1740–1814). From this marriage came four sons and two daughters. The daughter Louisa (1801–1869) became the wife of Carlyle's friend John Badams (Carlyle, Reminiscences, ed. C. E. Norton, 1887, i., pp. 93–95) in 1828; after Badam's death (1833), she in 1835 married Barham Cole Mergez, her half-sister's Sophia son from her second marriage who in 1846 inherited the title "baron" from his father. The son Thomas Holcroft Jr. (1803–1852) was a clerk in the House of Commons and spent several years in India, before becoming a journalist in 1822, who some time was Paris correspondent for the Morning Herald and secretary of the Asiatic Society. The widowed Louisa Mercier Holcroft remarried James Kenney (1780–1849), the dramatist, in 1812 and became the mother of three sons and three daughters.

Selected plays

 * Duplicity (1781)
 * Seduction (1787)
 * The German Hotel (1790)
 * The School for Arrogance (1791)
 * The Road to Ruin (1792)
 * Love's Frailties (1794)
 * The Deserted Daughter (1795)
 * The Force of Ridicule (1796)
 * The Man of Ten Thousand (1796)
 * The Inquisitor (1797)
 * He's Much to Blame (1798)
 * Knave or Not? (1798)
 * Hear Both Sides (1803)
 * The Vindictive Man (1806)

title: Thomas Holcroft


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Talk:Alt.zines

Untitled
Are we going to have entries for every newsgroup?

 * The entry for List of newsgroups specifically states: "It would be impossible for Wikipedia to list each and every newsgroup. However, several newsgroups are notable and popular, and some have had a considerable history. When adding newsgroups to this list, please try to include noteworthy newsgroups that may be worthy of their own Wikipedia entries." -- Modemac

finishing the merge
I moved the content here over to the sine article, per the RFD consensus. I'm not sure what the proper procedure for removing this page is though, so if someone does know, please go for it (after checking my work over there, naturally.) Thanks.Msalt (talk) 18:20, 28 January 2009 (UTC)
 * The proper procedure is to redirect the merged page to the target page, and tag it with R from merge. For more information, see Help:Merging. Jafeluv (talk) 11:45, 7 August 2009 (UTC)

Thanks! And obviously I meant Zine, not Sine. ;) Msalt (talk) 04:29, 4 September 2009 (UTC)

title: Talk:Alt.zines


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Candlestick Park

Candlestick Park was an outdoor stadium on the West Coast of the United States, located in San Francisco's Hunters Point area. The stadium was originally the home of Major League Baseball's San Francisco Giants, who played there from 1960 until 1999, after which the Giants moved into Pacific Bell Park (since renamed Oracle Park) in 2000. It was also the home field of the San Francisco 49ers of the National Football League from 1971 through 2013. The 49ers moved to Levi's Stadium in Santa Clara for the 2014 season. The last event held at Candlestick was a concert by Paul McCartney in August 2014, and the demolition of the stadium was completed in September 2015. As of 2019, the site is planned to be redeveloped into office space.

The stadium was situated at Candlestick Point on the western shore of San Francisco Bay. Candlestick Point was named for the "candlestick birds" (long-billed curlews) that populated the area for many years. Due to Candlestick Park's location next to the bay, strong winds often swirled down into the stadium, creating unusual playing conditions. At the time of its construction in the late 1950s, the stadium site was one of the few pieces of land available in the city that was suitable for a sports stadium, and had room for the 10,000 parking spaces that had been promised to the Giants.

The surface of the field for most of its existence was natural bluegrass, but for nine seasons, from 1970 to 1978, the stadium had artificial turf. A "sliding pit" configuration, with dirt cut-outs only around the bases, was installed in 1971, primarily to keep the dust down in the breezy conditions. Following the 1978 football season, the playing surface was restored to natural grass.

Park history
When the New York Giants arrived in San Francisco in 1958, they played their home games at the old Seals Stadium at 16th and Bryant Streets. As part of the agreement regarding the Giants' relocation to the West Coast, the city of San Francisco promised to build a new stadium for the team. Most of the land at Candlestick Point was purchased from Charles Harney, a local contractor. Harney purchased the land in 1952 for a quarry and industrial development. He made a profit of over $2 million when he sold the land for the stadium. Harney received a no-bid contract to build the stadium. The entire deal was the subject of a grand jury investigation in 1958.

Ground was broken in 1958 for the stadium and the Giants selected the name of Candlestick Park, after a name-the-park contest on March 3, 1959 (for the derivation of which, see below). Prior to the choice of the name, its construction site had been shown on maps as the generic Bay View Stadium. It was the first modern baseball stadium, as it was the first to be built entirely of reinforced concrete. Then-Vice President Richard Nixon threw out the ceremonial first pitch on the opening day of Candlestick Park on April 12, 1960, and the Oakland Raiders played the final three games of the 1960 season and their entire 1961 American Football League season at Candlestick Park. With only 77 home runs hit in 1960 (46 by Giants, 31 by visitors), the fences were moved in, from left-center to right-center, for the 1961 season.

Following the 1970 season, the first with AstroTurf, Candlestick Park was enclosed, with grandstands around the outfield. This was in preparation for the 49ers in 1971, who were moving from their long-time home of Kezar Stadium. The result was that the wind speed dropped marginally, but often swirled irregularly throughout the stadium, and the view of San Francisco Bay was lost.

Candlestick Park played host to two Major League Baseball All-Star Games in its life as home for the Giants. The stadium hosted the first of two games in 1961 and later hosted the 1984 All-Star Game. The Giants played a total of six postseason series at Candlestick Park; they played host to the NLCS in 1971, 1987, and 1989, the World Series in 1962 and 1989, and one NLDS in 1997.

The 49ers hosted eight NFC Championship games during their time at Candlestick Park. The first was in January 1982 when Dwight Clark caught a game-winning touchdown pass from Joe Montana to lead the 49ers to Super Bowl XVI by defeating the Dallas Cowboys. Clark's play went down as one of the more famous in football history, and was dubbed "The Catch". The last of these came in January 2012, when Lawrence Tynes kicked a field goal in overtime to defeat the 49ers and send the New York Giants to Super Bowl XLVI. The final postseason game hosted by the 49ers at Candlestick Park was the 2012 NFC Divisional Playoff matchup between the 49ers and the Green Bay Packers, won by the 49ers by a score of 45–31. The 49ers' record in NFC Championship games at Candlestick Park was 4-4; they defeated the Cowboys twice, in 1981 and 1994, the Chicago Bears in 1984, and the Los Angeles Rams in 1989. Their losses came against the Cowboys in 1992, the Giants in 1990 and 2011, and the Packers in 1997.

In addition to Clark's famous touchdown catch, two more plays referred to as "The Catch" took place during games at Candlestick Park. The play dubbed "The Catch II" came in the 1998 NFC Wild Card round, as Steve Young found Terrell Owens for a touchdown with eight seconds left to defeat the two-time defending NFC Champion Packers. The play called "The Catch III" came in the 2011 NFC Divisional Playoffs, when Alex Smith threw a touchdown pass to Vernon Davis with nine seconds remaining to provide the winning margin against the New Orleans Saints.

On October 17, 1989, the Loma Prieta earthquake (magnitude 6.9) struck San Francisco, minutes before Game 3 of the World Series was to begin at Candlestick Park. No one within the stadium was injured, although minor structural damage was incurred to the stadium. Al Michaels and Tim McCarver, who called the game for ABC, later credited the stadium's design for saving thousands of lives. An ESPN documentary about the earthquake revealed that the local stadium authority demanded that Candlestick Park undertake a major engineering project to shore up perceived safety red flags in the stadium. The authority pushed reluctant officials to get this done between the 1988 and 1989 baseball seasons, which prevented a "collapse wave" that could have killed thousands of fans and led to there being very few casualties of any kind in Candlestick Park after such a massive natural disaster. The World Series between the Giants and their Bay rivals the Oakland Athletics was subsequently delayed for 10 days, in part to give engineers time to check the stadium's overall structural soundness (and that of the Athletics' nearby home, the Oakland–Alameda County Coliseum). During this time, the 49ers moved their game against the New England Patriots on October 22 to Stanford Stadium, where they had defeated the Miami Dolphins 38–16 to win Super Bowl XIX on January 20, 1985.

The NFL awarded Super Bowl XXXIII to Candlestick Park on November 2, 1994. After planned renovations in preparation for the game were not made, the NFL owners instead awarded Super Bowl XXXIII to the Miami area during their meeting on October 31, 1996. The league promised to award Super Bowl XXXVII following construction of a new football stadium, which was approved by voters in 1997, but the forced sale of the team by owner Eddie DeBartolo Jr. caused plans to fall through.

In 2000, the Giants moved to the new Pacific Bell Park (now called Oracle Park) in the China Basin neighborhood, leaving the 49ers as the sole professional sports team to use Candlestick Park. The final baseball game was played on September 30, 1999, against their long-time rivals, the Los Angeles Dodgers, who won 9–4. In that game, all nine Dodgers starters had at least one base hit, while the stadium's final home run came from Dodgers' right fielder Raúl Mondesí in the 6th inning. The National League rivalry between the Giants and Dodgers, one of the oldest and most hotly contested in the Major Leagues, dated back to when both teams were based in New York City. When first the Dodgers, then the Giants, moved to California in 1958, the rivalry continued unabated.

For its last several years as home to just the 49ers, Candlestick Park was the only remaining NFL stadium to have begun as a baseball-only facility which later underwent an extensive redesign to accommodate football. That was evidenced by the stadium's curiously oblong and irregular shape, whereby views from a sizable section of lower-deck seating in the baseball configuration's right-field corner were so badly obstructed by the eastern grandstand of the football seating configuration that they were unusable for football games and would consequently sit empty. Since a football gridiron, including its end zones and benches along the sidelines, is much smaller than a baseball playing field and foul territory, this large grandstand, which provided thousands of prime seats along one whole sideline of the football field, was designed to be retractable. It would slide backwards for baseball games, under the upper deck, and provide a smaller section of baseball seating beyond the outfield wall in right. After the Giants played their 1999 season and moved away from Candlestick, this grandstand was left permanently in its football position, and the unusable seats were eventually removed.

On September 3, 2011, Candlestick Park hosted the first and only college football game in its history with a neutral site game between the California Golden Bears and Fresno State Bulldogs (Cal was designated the "home" team). This game was in San Francisco, because of the massive renovation and seismic retrofit at California's home stadium, California Memorial Stadium. The rest of the Golden Bears' home games in 2011 were played at AT&T Park. Cal won the game 36–21.

At approximately 5:19 p.m. local time on December 19, 2011, Candlestick Park experienced an unexpected power outage just before a Monday Night Football game between the 49ers and the Pittsburgh Steelers. An aerial shot shown live on ESPN showed a transformer sparking and then the stadium going completely dark. About 17 minutes later, however, the park's lights came back on in time for the game's kickoff. With 12:13 remaining in the second quarter, another power outage created yet another 30-minute delay before play resumed again. The 49ers 2011 season ended at Candlestick Park with a loss to the New York Giants in the NFC Championship Game.

The 49ers played their final game at Candlestick Park on Monday, December 23, 2013, against the Atlanta Falcons, winning 34–24 after a NaVorro Bowman interception that would be called The Pick at the Stick by some sports columnists. This game was the facility's 36th and final game on Monday Night Football, the most at any stadium used by the NFL.

Reputation
As a baseball field, the stadium was infamous for the windy conditions, damp air and dew from fog, and chilly temperatures. The wind often made it difficult for outfielders trying to catch fly balls, as well as for fans, while the damp grass further complicated play for outfielders who had to play in cold, wet shoes. Architect John Bolles designed the park with a boomerang-shaped concrete baffle in the upper tier in order to protect the park from wind. Unfortunately, it never worked properly. For Candlestick's first 10 seasons, the wind blew in from left-center and out toward right-center. When the park was expanded to accommodate the 49ers in 1971, it was thought that fully enclosing the park would cut down on the wind significantly. Instead, the wind swirled from all directions, and was as strong and cold as before. Giants Hall of Fame center fielder Willie Mays claimed the wind cost him over 100 home runs. (It may be noted that in the 12 years he played at Candlestick Park, from 1960 through 1971, Mays hit 396 home runs, 203 at Candlestick and 193 on the road. ) Nonetheless, he had less difficulty fielding balls in the windy conditions. Mays was used to playing in difficult and absurdly sized field conditions, beginning his career at the Polo Grounds in New York, which featured an enormous outfield where he made a famous World Series-saving catch.

During the first All Star Game of 1961 (one of two played in the park—the other was in 1984), Giants pitcher Stu Miller was blown off balance by a gust of wind and was charged with a balk. Two years later, wind picked up the entire batting cage and dropped it 60 feet (18 m) away on the pitcher's mound while the New York Mets were taking batting practice.

The stadium also had the reputation as the coldest park in Major League Baseball, with winds blowing directly off the Pacific Ocean. It was initially built with a radiant heating system of hot water pipes under the lower box seats in a space between the concrete and the ground. The pipes were not embedded in the concrete, however, and did not produce enough heat to offset the cold air. Both the city and the Giants balked at the cost of upgrading the system so it would work properly, which would have involved removing the seats and concrete, embedding larger pipes, and replacing the concrete and seats. As a result, the Giants played more day games than any Major League Baseball team except the Chicago Cubs, whose ballpark, Wrigley Field, did not have lights installed until 1988. Many locals, including Giants' broadcaster Lon Simmons, were surprised at the decision to build the park right on the bay, in one of the coldest areas of the city. Attorney Melvin Belli filed a claim against the Giants in 1960 because his six-seat box, which cost him almost $1,600, was unbearably cold. Belli won in court, claiming that the "radiant heating system" advertised was a failure.

The Giants eventually played on the reputation to bolster fan support with humorous promotions such as awarding the 'Croix de Candlestick' pin to fans who stayed for the duration of extra-inning night games. The pins featured the Giants' "SF" monogram capped with snow, along with the Latin slogan "Veni, vidi, vixi" ("I came, I saw, I survived"). Among many less-than-flattering fan nicknames for the park were "North Pole", "Cave of the Winds", "Windlestick", "The Quagmire", and "The Ashtray By The Bay". Older fans called it "The Dump" in honor of the former use of the land. Ironically, the Giants played their last night game at Candlestick (against the Los Angeles Dodgers) on September 29, 1999, under clear skies and a game time temperature of 74° as well as their last day game at Candlestick on September 30, 1999, under blue skies with no fog and a game time temperature of 82°, all of which was common for September games.

Giants owner Horace Stoneham visited the site as early as 1957 and was involved in the stadium's design from the outset. While he was aware of the weather conditions, he usually visited the park during the day, not knowing about the particularly cold, windy and foggy conditions that overtook it at night. Originally, Bolles' concrete baffle would have extended all the way to left field, which would have further reduced the prevailing winds. Nevertheless, the size of the structure was reduced for cost savings. In 1962, Stoneham commissioned a study of the wind conditions. The study revealed that had the windy conditions been known prior to construction, conditions would have been significantly improved by building the park 100 yards farther to the north and east. This would have meant building it on fill, however, which is less stable during earthquakes. The stadium's location on the bedrock of Bayview Hill provided more stability.

The winds were intense in the immediate area of the park. Studies showed they were no more frequent than other parts of San Francisco but are subject to higher gusts. This is because of a hill immediately adjacent to the park. This hill, in turn, is the first topographical obstacle met by the prevailing winds arriving from the Pacific Ocean 7 mi to the west. Arriving at Candlestick from the Pacific, these winds travel through what is known as the Alemany Gap before reaching the hill. The combination of ocean winds free-flowing to Candlestick, then swirling over the adjacent hill, created the cold and windy conditions that were the bane of the Giants' 40-year stay on Candlestick Point. It was indeed the wind and not the ambient air temperature that provided Candlestick's famed chill. The Giants' subsequent home, Oracle Park, is just one degree warmer, but is far less windy, creating a "warmer" (relatively speaking) effect. While the wind is a summer condition (hot inland, cool oceanside), winter weather is right in line with the rest of sea level Northern California (mild with occasional rain).

Other design flaws and irregularities
Candlestick was an object of scorn from baseball purists for reasons other than weather. Although originally built for baseball, foul territory was quite roomy. According to Simmons, nearly every seat was too far from the field even before the 1971 expansion. As with the radiant heating system in the grandstands, the heating systems in the dugouts were wholly inadequate. Players on other National League teams – especially if they had played for the Giants beforehand – complained that the visitors dugout was noticeably colder than the Giants' dugout. That was due to two factors. One was that the Giants' dugout included a tunnel to the clubhouse, so heat from the clubhouse flowed into the dugout. The other involved the placement of the dugouts. The Giants' dugout was located on the first base side, which was on the south side of the stadium. The visitors' dugout was located on the third base (west) side of the field.

The Beatles' final concert
The Beatles famously gave their last full public concert at Candlestick Park on August 29, 1966. Songs performed at the show were "Rock and Roll Music", "She's a Woman", "If I Needed Someone", "Day Tripper", "Baby's in Black", "I Feel Fine", "Yesterday", "I Wanna Be Your Man", "Nowhere Man", "Paperback Writer", and "Long Tall Sally".

An "official" bootleg recording of the 33-minute setlist was made by the Beatles' press officer, Tony Barrow, at the request of the band. As his cassette could only record 30 minutes per side, it ran out with a minute of the closing song, "Long Tall Sally", remaining. This recording has never been officially released, although it has been leaked on to the internet.

At the time, The Beatles had not announced that this was to be their final concert, and even if the foursome themselves knew, it was a closely guarded secret. Much of the existing color film footage of the concert was captured by a 15-year-old Beatles fan, Barry Hood. A relatively small amount of black-and-white footage was shot by local TV news in the San Francisco Bay Area and Sacramento. Hood released some of his film in a limited edition documentary titled The Beatles Live In San Francisco, but more of Hood's very rare footage remains in a vault, unseen by the public as of 2017. On August 14, 2014, former Beatle Paul McCartney returned to become the closing act of Candlestick Park's long musical history. To showcase the event, McCartney contacted Barry Hood and used a portion of his original 1966 Beatles film on a big screen at this last concert.

Papal Mass
Pope John Paul II celebrated a Papal Mass on September 18, 1987, at Candlestick Park during his tour of America. An estimated crowd of 70,000 attended the Mass.

In popular culture
Candlestick Park was home to dozens of commercial shoots as well as the location for the climactic scene in both the 1962 thriller Experiment in Terror and the 1974 Richard Rush comedy Freebie and the Bean. The stadium was featured in the 1976 Dirty Harry movie, The Enforcer. The stadium was featured in the 1994 family comedy Getting Even with Dad, where the father character (played by Ted Danson) takes his son (played by Macaulay Culkin) to a San Francisco Giants game. The stadium was featured briefly in the 1996 action film The Rock where a missile almost attacks the place before being diverted away by the last second. In February 2011, scenes for the film Contagion, starring Matt Damon, Kate Winslet and Jude Law, were filmed at the stadium. The Fan was also filmed there in 1996. In 2010, Candlestick Park was featured as the finishing point for the finale of The Amazing Race 16. And 1989 World Series, The series was known for one of the sites of the 1989 San Francisco earthquake.

Jackson, Mississippi landowner and developer Homer Lee Howie built a shopping center on Cooper Road in the city which opened in August 1963. Howie was originally going to name the shopping center Cooper Mart but one day watched the Giants on television at the stadium and thought the name Candlestick Park sounded catchy and would greatly help bring in customers and decided to name the center by that name without permission. The center was a popular shopping place and community center in southern Jackson for its first few years before being completely destroyed in the 1966 Candlestick Park tornado outbreak on March 3, 1966, which killed a total of 57 people, 12 of them in the shopping center. The center still exists today, but was never the same after the tornado.

Name changes

Some think that Candlestick Point was named for the indigenous "candlestick bird" (long-billed curlew), once common to the point. The book "California Geographic Names" lists Candlestick Point as being named for a pinnacle of rock first noted in 1781 by the De Anza Expedition. This pinnacle was also noted by the U.S. Geodetic Survey in 1869. The pinnacle disappeared around 1920.

The rights to the stadium name were licensed to 3Com Corporation from September 1995 until 2002, for $900,000 a year. During that time, the park became known as "3Com Park at Candlestick Point", or, simply, "3Com Park". In 2002, the naming rights deal expired, and the park then became officially known as "San Francisco Stadium at Candlestick Point". On September 28, 2004, a new naming rights deal was signed with Monster Cable, a maker of cables for electronic equipment, and the stadium was renamed "Monster Park". Just over a month later, however, a measure passed in the November 2 election stipulated that the stadium name revert to "Candlestick" permanently after the contract with Monster expired in 2008.

The City and County of San Francisco had trouble finding a new naming sponsor due in part to the downturn in the economy, but also because the stadium's tenure as 3Com Park was tenuous at best. Many local fans were annoyed with the change and continued referring to the park by its original name, regardless of the official name. The Giants reportedly continued to call the stadium "Candlestick Park" in media guides, because the naming rights were initiated by the 49ers. Some even mocked the 3Com sponsorship. Chris Berman, for instance, usually called it "Commercial-Stick Park". Local fans sometimes called it "Dot-com Park" (see Dot-com bubble). Freeway signs in the vicinity were changed to read "Monster Park" as part of an overall signage upgrade to national standards on California highways, but in 2008 those signs were changed back to "Candlestick Park".

The name change also ended up being confusing for the intended branding purposes, as without the "Cable" qualifier in the official name, many erroneously thought the stadium was named for the Monster.com employment website or Monster Energy Drink, not the cable vendor.

On August 10, 2007, San Francisco mayor Gavin Newsom announced that the playing field would be renamed "Bill Walsh Field" in honor of the former Stanford and 49ers coach, who died on July 30 that year, pending the approval of the city government. The stadium itself retained its name as was contractually obligated. Commentators still use this name occasionally, most recently when Jerry Rice's jersey was retired.

On September 18, 2009, Sports Illustrated's Peter King used the mock-combination name "Candle3Monsterstick" in reference to the many name changes the stadium has gone through.

Despite numerous official and unofficial name changes over the history of the stadium and surrounding park/facilities, the stadium was lovingly referred to as "the Stick" by many locals and die-hard fans from its original titling of "Candlestick Park" in 1960.

Replacement and demolition
By 1997, plans were underway to construct a new 68,000-seat stadium at Candlestick Point. On November 8, 2006, however, the 49ers announced that they would abandon their search for a location in San Francisco and begin to pursue the idea of building a stadium in Santa Clara. Because its centerpiece stadium was lost, San Francisco withdrew its bid for the 2016 Olympics on November 13, 2006. Ground-breaking for the Santa Clara stadium occurred on April 19, 2012. On May 8, 2013, the media announced that the name of the new stadium would be Levi's Stadium. The stadium opened on July 17, 2014, in time for the 2014 NFL season. The 49ers christened their new home a month after it opened.

A grassroots movement for the Giants to play another baseball game at Candlestick had existed since 2009. Many fans had hoped to see another game in 2010, the 50th anniversary of the Giants' first season at Candlestick Park, but the idea was dropped due to the cost. Although many fans wished for another Giants game at the Stick, the Giants never returned to their former stadium for a final game.

With the departure of the 49ers, Candlestick Park was left without any permanent tenants. Demolition was expected to occur soon after the 49ers played their final game of the 2013 season, but over time the date of demolition was moved back to late 2014, with several special events planned for the intervening period. In April 2014, Paul McCartney announced that he would perform a concert as the last scheduled event in the 54-year-old stadium on August 14, 2014. The Beatles had performed their last scheduled concert at Candlestick Park 48 years earlier.

Demolition began in November 2014 as workers tore out seats. In January 2015, the developer withdrew a request to implode the stadium, possibly to be broadcast as part of the Super Bowl halftime entertainment. Instead, mechanized structural demolition commenced, which was favored over implosion due to local dust pollution concerns. Demolition was expected to be complete by March 2015, but was not completed until September 24, 2015.

In 2014, 1,000 historic Candlestick Park Stadium seats were installed at Kezar Stadium for the public to enjoy. The renovation was funded by the city's Capital Planning General Fund. Mayor Edwin M. Lee helped re-open the stadium with a warm-up run.

In December 2016, 4,000 additional historic Candlestick seats were acquired and installed at Kezar. The seats were paid for by the San Francisco Deltas as a part of a $1-million improvement the team agreed upon to make use of the stadium.

In November 2014, Lennar and Macerich announced plans to build a dense "urban outlet" center incorporating retail and housing with underground parking on the Candlestick Park site. The proponents suggested that the new development would be completed in 2017. The project has not proceeded, and the plan was suspended by its proponents in April 2018.

Croix de Candlestick

The Croix de Candlestick is an award pin that was given out to baseball fans as they exited Candlestick Park at the conclusion of a night game that went extra innings. In reference to the ballpark's legendarily cold winds, the pin carried the motto, "Veni, Vidi, Vixi" ("I came, I saw, I survived").

In order to receive a pin, the fans would have to redeem their ticket stub for the pin at Patrick & Co. Stationery store in San Francisco. The pin, developed by team marketing director Patrick J. Gallagher, was first issued in 1983. In 1983 the San Francisco Giants played in five extra inning night games, with a total attendance of 70,933 and in 1984 they played in five extra inning night games with a total attendance of 44,031. The pin was given out for several years. On September 28–30, 1999, tens of thousands of fans received the pin for attending the Giants' final three-game home stand at Candlestick, against the team's archrival, the Los Angeles Dodgers. A San Francisco Chronicle columnist later called it "the smartest marketing promotional in Bay Area history".

The Croix de Candlestick received national exposure during the 1983 All-Star Game at Chicago's Comiskey Park. Giants pitcher Atlee Hammaker wore it on his cap and was mentioned by NBC announcer Joe Garagiola.

"Mayor Ed Lee...: I'm a real San Franciscan, because I've EARNED a Croix de Candlestick and whenever I hear the phrase "the catch" I have to take a moment..."

"They don't give out a Croix de Candlestick to fans who stay 'til the bitter end at Levi's, or even a Croix de Fiddlesticks, but this time the late birds got their reward."

title: Candlestick Park