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Special unitary group

In mathematics, the special unitary group of degree $n$, denoted $SU(n)$, is the Lie group of $n × n$ unitary matrices with determinant 1.

The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case.

The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group $U(n)$, consisting of all $n×n$ unitary matrices. As a compact classical group, $U(n)$ is the group that preserves the standard inner product on $$\mathbb{C}^n$$. It is itself a subgroup of the general linear group, $$\operatorname{SU}(n) \subset \operatorname{U}(n) \subset \operatorname{GL}(n, \mathbb{C} ).$$

The $U(n)$ groups find wide application in the Standard Model of particle physics, especially $SU(n)$ in the electroweak interaction and $SU(n)$ in quantum chromodynamics.

The simplest case, $SU(2)$, is the trivial group, having only a single element. The group $SU(3)$ is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from $SU(1)$ to the rotation group $SU(2)$ whose kernel is $SU(2)$. Since the quaternions can be identified as the even subalgebra of the Clifford Algebra $SO(3)$, ${+I, −I}$ is in fact identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Properties
The special unitary group $SU(2) → SO(3)$ is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is $Cl(3)$. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).

The center of $SU(2)$ is isomorphic to the cyclic group $$\mathbb{Z}/n\mathbb{Z}$$, and is composed of the diagonal matrices $SU(n)$ for $n^{2} − 1$ an $SU(n)$th root of unity and $ζ I$ the $ζ$ identity matrix.

Its outer automorphism group for $n$ is $$\mathbb{Z}/2\mathbb{Z},$$ while the outer automorphism group of $I$ is the trivial group.

A maximal torus of rank $n × n$ is given by the set of diagonal matrices with determinant $n ≥ 3$. The Weyl group of $SU(2)$ is the symmetric group $n − 1$, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is $1$).

The Lie algebra of $SU(n)$, denoted by $$\mathfrak{su}(n)$$, can be identified with the set of traceless anti‑Hermitian $S_{n}$ complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian $1$ complex matrices with Lie bracket given by $SU(n)$ times the commutator.

Lie algebra
The Lie algebra $$\mathfrak{su}(n)$$ of $$\operatorname{SU}(n)$$ consists of $n × n$ skew-Hermitian matrices with trace zero. This (real) Lie algebra has dimension $n × n$. More information about the structure of this Lie algebra can be found below in .

Fundamental representation
In the physics literature, it is common to identify the Lie algebra with the space of trace-zero Hermitian (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of $$i$$ from the mathematicians'. With this convention, one can then choose generators $−i$ that are traceless Hermitian complex $n × n$ matrices, where:
 * $$ T_a \, T_b = \tfrac{1}{\, 2n \,}\,\delta_{ab}\,I_n + \tfrac{1}{2}\,\sum_{c=1}^{n^2 -1}\left(if_{abc} + d_{abc} \right) \, T_c $$

where the $n^{2} − 1$ are the structure constants and are antisymmetric in all indices, while the $T_{a}$-coefficients are symmetric in all indices.

As a consequence, the commutator is:
 * $$ ~ \left[T_a, \, T_b\right] ~ = ~ i \sum_{c=1}^{n^2 -1} \, f_{abc} \, T_c \;,$$

and the corresponding anticommutator is:
 * $$ \left\{T_a, \, T_b\right\} ~ = ~ \tfrac{1}{n} \, \delta_{ab} \, I_n + \sum_{c=1}^{n^2 -1}{d_{abc} \, T_c} ~.$$

The factor of $n × n$ in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention.

The conventional normalization condition is

 * $$\sum_{c,e=1}^{n^2 - 1} d_{ace}\,d_{bce} = \frac{\, n^2 - 4 \,}{n} \, \delta_{ab}~ .$$

Adjoint representation
In the $f$-dimensional adjoint representation, the generators are represented by $d$ matrices, whose elements are defined by the structure constants themselves:
 * $$\left(T_a\right)_{jk} = -if_{ajk}.$$

The group SU(2)
Using matrix multiplication for the binary operation, $i$ forms a group,
 * $$\operatorname{SU}(2) = \left\{ \begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C}, |\alpha|^2 + |\beta|^2 = 1 \right\}~,$$

where the overline denotes complex conjugation.

Diffeomorphism with the 3-sphere S3
If we consider $$\alpha,\beta$$ as a pair in $$\mathbb{C}^2$$ where $$\alpha=a+bi$$ and $$\beta=c+di$$, then the equation $$|\alpha|^2 + |\beta|^2 = 1$$ becomes
 * $$ a^2 + b^2 + c^2 + d^2 = 1 $$

This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map
 * $$\begin{align}

\varphi \colon \mathbb{C}^2 \to{} &\operatorname{M}(2, \mathbb{C}) \\[5pt] \varphi(\alpha, \beta) ={} &\begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix}, \end{align}$$ where $$\operatorname{M}(2,\mathbb{C})$$ denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering $$\mathbb{C}^2$$ diffeomorphic to $$\mathbb{R}^4$$ and $$\operatorname{M}(2,\mathbb{C})$$ diffeomorphic to $$\mathbb{R}^8$$). Hence, the restriction of $(n^{2} − 1)$ to the 3-sphere (since modulus is 1), denoted $(n^{2} − 1) × (n^{2} − 1)$, is an embedding of the 3-sphere onto a compact submanifold of $$\operatorname{M}(2,\mathbb{C})$$, namely $SU(2)$.

Therefore, as a manifold, $φ$ is diffeomorphic to $S^{3}$, which shows that $φ(S^{3}) = SU(2)$ is simply connected and that $S^{3}$ can be endowed with the structure of a compact, connected Lie group.

Isomorphism with group of versors
Quaternions of norm 1 are called versors since they generate the rotation group SO(3): The $SU(2)$ matrix:
 * $$ \begin{pmatrix}

a + bi & c + di \\ -c + di & a - bi  \end{pmatrix} \quad (a, b, c, d \in \mathbb{R}) $$ can be mapped to the quaternion
 * $$a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k}$$

This map is in fact a group isomorphism. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix in $SU(2)$ is of this form and, since it has determinant $S^{3}$, the corresponding quaternion has norm $SU(2)$. Thus $SU(2)$ is isomorphic to the group of versors.

Relation to spatial rotations
Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from $1$ to $1$; consequently $SU(2)$ is isomorphic to the quotient group $SU(2)$, the manifold underlying $SO(3)$ is obtained by identifying antipodal points of the 3-sphere $SO(3)$, and $SU(2)/\{±I\}$ is the universal cover of $SO(3)$.

Lie algebra
The Lie algebra of $S^{3}$ consists of $SU(2)$ skew-Hermitian matrices with trace zero. Explicitly, this means
 * $$\mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~.$$

The Lie algebra is then generated by the following matrices,
 * $$u_1 = \begin{pmatrix}

0 & i \\ i & 0 \end{pmatrix}, \quad u_2 = \begin{pmatrix} 0 & -1 \\   1 &  0  \end{pmatrix}, \quad u_3 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}~, $$ which have the form of the general element specified above.

This can also be written as $$\mathfrak{s u}(2)=\operatorname{span}\left\{i \sigma_{1}, i \sigma_{2}, i \sigma_{3}\right\}$$ using the Pauli matrices.

These satisfy the quaternion relationships $$u_2\ u_3 = -u_3\ u_2 = u_1~,$$  $$u_3\ u_1 = -u_1\ u_3 = u_2~,$$  and  $$u_1 u_2 = -u_2\ u_1 = u_3~.$$ The commutator bracket is therefore specified by
 * $$\left[u_3, u_1\right] = 2\ u_2, \quad \left[u_1, u_2\right] = 2\ u_3, \quad \left[u_2, u_3\right] = 2\ u_1~.$$

The above generators are related to the Pauli matrices by $$u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2$$ and  $$u_3 = +i\ \sigma_3~.$$ This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity. They also correspond to the Pauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3d rotations about the axes of the Bloch sphere.

The Lie algebra serves to work out the representations of $SO(3)$.

SU(3)
The group $SU(2)$ is an 8-dimensional simple Lie group consisting of all $2 × 2$ unitary matrices with determinant 1.

Topology
The group $SU(2)$ is a simply-connected, compact Lie group. Its topological structure can be understood by noting that $SU(3)$ acts transitively on the unit sphere $$S^5$$ in $$\mathbb{C}^3 \cong \mathbb{R}^6$$. The stabilizer of an arbitrary point in the sphere is isomorphic to $3 × 3$, which topologically is a 3-sphere. It then follows that $SU(3)$ is a fiber bundle over the base $SU(3)$ with fiber $SU(2)$. Since the fibers and the base are simply connected, the simple connectedness of $SU(3)$ then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).

The $S^{5}$-bundles over $S^{3}$ are classified by $$\pi_4\mathord\left(S^3\right) = \mathbb{Z}_2$$ since any such bundle can be constructed by looking at trivial bundles on the two hemispheres $$S^5_\text{N}, S^5_\text{S}$$ and looking at the transition function on their intersection, which is a copy of $SU(3)$, so
 * $$S^5_\text{N} \cap S^5_\text{S} \simeq S^4$$

Then, all such transition functions are classified by homotopy classes of maps
 * $$\left[S^4, \mathrm{SU}(2)\right] \cong \left[S^4, S^3\right] = \pi_4\mathord\left(S^3\right) \cong \mathbb{Z}/2$$

and as $$\pi_4(\mathrm{SU}(3)) = \{0\}$$ rather than $$\mathbb{Z}/2$$, $SU(2)$ cannot be the trivial bundle $S^{5}$, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.

Representation theory
The representation theory of $S^{4}$ is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra $$\mathfrak{sl}(3; \mathbb{C})$$, may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for $SU(3)$.

Lie algebra
The generators, $T$, of the Lie algebra $$\mathfrak{su}(3)$$ of $SU(2) × S^{5} ≅ S^{3} × S^{5}$ in the defining (particle physics, Hermitian) representation, are
 * $$T_a = \frac{\lambda_a}{2}~, $$

where $SU(3)$, the Gell-Mann matrices, are the $SU(3)$ analog of the Pauli matrices for $SU(3)$:
 * $$\begin{align}

\lambda_1 ={} &\begin{pmatrix} 0 & 1 &  0 \\ 1 &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & \lambda_2 ={} &\begin{pmatrix} 0 & -i & 0 \\ i &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & \lambda_3 ={} &\begin{pmatrix} 1 & 0 &  0 \\ 0 & -1 &  0 \\ 0 & 0 &  0 \end{pmatrix}, \\[6pt] \lambda_4 ={} &\begin{pmatrix} 0 & 0 &  1 \\ 0 &  0 &  0 \\ 1 & 0 &  0 \end{pmatrix}, & \lambda_5 ={} &\begin{pmatrix} 0 & 0 & -i \\ 0 &  0 &  0 \\ i & 0 &  0 \end{pmatrix}, \\[6pt] \lambda_6 ={} &\begin{pmatrix} 0 & 0 &  0 \\ 0 &  0 &  1 \\ 0 & 1 &  0 \end{pmatrix}, & \lambda_7 ={} &\begin{pmatrix} 0 & 0 &  0 \\ 0 &  0 & -i \\ 0 & i &  0 \end{pmatrix}, & \lambda_8 = \frac{1}{\sqrt{3}} &\begin{pmatrix} 1 & 0 &  0 \\ 0 &  1 &  0 \\ 0 & 0 & -2 \end{pmatrix}. \end{align}$$

These $λ_{a}$ span all traceless Hermitian matrices $H$ of the Lie algebra, as required. Note that $SU(3)$ are antisymmetric.

They obey the relations
 * $$\begin{align}

\left[T_a, T_b\right] &= i \sum_{c=1}^8 f_{abc} T_c, \\ \left\{T_a, T_b\right\} &= \frac{1}{3} \delta_{ab} I_3 + \sum_{c=1}^8 d_{abc} T_c, \end{align}$$ or, equivalently,
 * $$\begin{align}

\left[\lambda_a, \lambda_b\right] &= 2i \sum_{c=1}^8 f_{abc} \lambda_c, \\ \{\lambda_a, \lambda_b\} &= \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c}. \end{align}$$

The $f$ are the structure constants of the Lie algebra, given by
 * $$\begin{align}

f_{123} &= 1, \\ f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} &= \frac{1}{2}, \\ f_{458} = f_{678} &= \frac{\sqrt{3}}{2}, \end{align}$$ while all other $SU(2)$ not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set $λ_{a}$.

The symmetric coefficients $λ_{2}, λ_{5}, λ_{7}$ take the values
 * $$\begin{align}

d_{118} = d_{228} = d_{338} = -d_{888} &= \frac{1}{\sqrt{3}} \\ d_{448} = d_{558} = d_{668} = d_{778} &= -\frac{1}{2\sqrt{3}} \\ d_{344} = d_{355} = -d_{366} = -d_{377} = -d_{247} = d_{146} = d_{157} = d_{256} &=  \frac{1}{2} ~. \end{align}$$

They vanish if the number of indices from the set $f_{abc}$ is odd.

A generic $\{2, 5, 7\}$ group element generated by a traceless 3×3 Hermitian matrix $1/6$, normalized as $f_{abc}$, can be expressed as a second order matrix polynomial in $H$:
 * $$\begin{align}

\exp(i\theta H) ={} &\left[-\frac{1}{3} I\sin\left(\varphi + \frac{2\pi}{3}\right) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin(\varphi) - \frac{1}{4}~H^2\right] \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta\sin(\varphi)\right)} {\cos\left(\varphi + \frac{2\pi}{3}\right) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt] & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{4}~H^{2}\right] \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi + \frac{2\pi}{3}\right)\right)} {\cos(\varphi) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt] & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{4}~H^2\right] \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi - \frac{2\pi}{3}\right)\right)} {\cos(\varphi)\cos\left(\varphi + \frac{2\pi}{3}\right)} \end{align}$$ where
 * $$\varphi \equiv \frac{1}{3}\left[\arccos\left(\frac{3\sqrt{3}}{2}\det H\right) - \frac{\pi}{2}\right].$$

Lie algebra structure
As noted above, the Lie algebra $$\mathfrak{su}(n)$$ of $d$ consists of $\{2, 5, 7\}$ skew-Hermitian matrices with trace zero.

The complexification of the Lie algebra $$\mathfrak{su}(n)$$ is $$\mathfrak{sl}(n; \mathbb{C})$$, the space of all $SU(3)$ complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in $$\mathbb C^n$$ whose entries sum to zero. The roots then consist of all the $tr(H^{2}) = 2$ permutations of $SU(n)$.

A choice of simple roots is
 * $$\begin{align}

(&1, -1, 0, \dots, 0,  0), \\ (&0, 1, -1, \dots, 0,  0), \\ &\vdots                   \\ (&0, 0,  0, \dots, 1, -1). \end{align}$$

So, $n × n$ is of rank $n × n$ and its Dynkin diagram is given by $n(n − 1)$, a chain of $(1, −1, 0, ..., 0)$ nodes: .... Its Cartan matrix is
 * $$\begin{pmatrix}

2 & -1 & 0 & \dots & 0 \\ -1 & 2 & -1 & \dots & 0 \\ 0 & -1 & 2 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 2 \end{pmatrix}. $$

Its Weyl group or Coxeter group is the symmetric group $SU(n)$, the symmetry group of the $n − 1$-simplex.

Generalized special unitary group
For a field $A_{n−1}$, the generalized special unitary group over F, $n − 1$, is the group of all linear transformations of determinant 1 of a vector space of rank $S_{n}$ over $(n − 1)$ which leave invariant a nondegenerate, Hermitian form of signature $F$. This group is often referred to as the special unitary group of signature $SU(p, q; F)$ over $n = p + q$. The field $F$ can be replaced by a commutative ring, in which case the vector space is replaced by a free module.

Specifically, fix a Hermitian matrix $(p, q)$ of signature $p q$ in $$\operatorname{GL}(n, \mathbb{R})$$, then all
 * $$M \in \operatorname{SU}(p, q, \mathbb{R})$$

satisfy
 * $$\begin{align}

M^{*} A M &= A \\ \det M &= 1. \end{align}$$

Often one will see the notation $F$ without reference to a ring or field; in this case, the ring or field being referred to is $$\mathbb C$$ and this gives one of the classical Lie groups. The standard choice for $F$ when $$\operatorname{F} = \mathbb{C}$$ is
 * $$A = \begin{bmatrix}

0 & 0      & i \\ 0 & I_{n-2} & 0 \\ -i & 0      & 0 \end{bmatrix}. $$

However, there may be better choices for $A$ for certain dimensions which exhibit more behaviour under restriction to subrings of $$\mathbb C$$.

Example
An important example of this type of group is the Picard modular group $$\operatorname{SU}(2, 1; \mathbb{Z}[i])$$ which acts (projectively) on complex hyperbolic space of degree two, in the same way that $$\operatorname{SL}(2,9;\mathbb{Z})$$ acts (projectively) on real hyperbolic space of dimension two. In 2005 Gábor Francsics and Peter Lax computed an explicit fundamental domain for the action of this group on $p q$.

A further example is $$\operatorname{SU}(1, 1; \mathbb{C})$$, which is isomorphic to $$\operatorname{SL}(2, \mathbb{R})$$.

Important subgroups
In physics the special unitary group is used to represent fermionic symmetries. In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups of $SU(p, q)$ that are important in GUT physics are, for $A$,
 * $$\operatorname{SU}(n) \supset \operatorname{SU}(p) \times \operatorname{SU}(n - p) \times \operatorname{U}(1),$$

where × denotes the direct product and $A$, known as the circle group, is the multiplicative group of all complex numbers with absolute value 1.

For completeness, there are also the orthogonal and symplectic subgroups,
 * $$\begin{align}

\operatorname{SU}(n) &\supset \operatorname{SO}(n), \\ \operatorname{SU}(2n) &\supset \operatorname{Sp}(n). \end{align}$$

Since the rank of $HC^{2}$ is $SU(n)$ and of $p > 1, n − p > 1$ is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. $U(1)$ is a subgroup of various other Lie groups,
 * $$\begin{align}

\operatorname{SO}(2n) &\supset \operatorname{SU}(n) \\ \operatorname{Sp}(n) &\supset \operatorname{SU}(n) \\ \operatorname{Spin}(4) &= \operatorname{SU}(2) \times \operatorname{SU}(2) \\ \operatorname{E}_6 &\supset \operatorname{SU}(6) \\ \operatorname{E}_7 &\supset \operatorname{SU}(8) \\ \operatorname{G}_2 &\supset \operatorname{SU}(3) \end{align}$$ See Spin group and Simple Lie group for $SU(n)$, $n − 1$, and $U(1)$.

There are also the accidental isomorphisms: $SU(n)$, $E_{6}$, and $E_{7}$.

One may finally mention that $G_{2}$ is the double covering group of $SU(4) = Spin(6)$, a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.

SU(1, 1)
$$\mathrm{SU}(1,1) = \left \{ \begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \in M(2,\mathbb{C}): u u^* - v v^* = 1 \right \},$$ where $$~u^*~$$ denotes the complex conjugate of the complex number $H$.

This group is isomorphic to $SU(2) = Spin(3) = Sp(1)$ and $Sp(n)$ where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression $$~u u^* - v v^*~$$ in the definition of $USp(2n)$ is an Hermitian form which becomes an isotropic quadratic form when $u$ and $Sp(n)$ are expanded with their real components.

An early appearance of this group was as the "unit sphere" of coquaternions, introduced by James Cockle in 1852. Let

j = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\,, \quad k = \begin{bmatrix} 1 & \;~0 \\ 0 & -1 \end{bmatrix}\,, \quad i = \begin{bmatrix} \;~0 & 1 \\ -1 & 0 \end{bmatrix}~. $$

Then $$~j\,k = \begin{bmatrix} 0 & -1 \\ 1 & \;~0 \end{bmatrix} = -i ~,~$$ $$~ i\,j\,k = I_2 \equiv \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}~,~$$ the 2×2 identity matrix, $$~k\,i = j ~,$$ and $$\;i\,j = k \;,$$ and the elements $u$ and $i, j,$ all anticommute, as in quaternions. Also $$i$$ is still a square root of $2n × 2n$ (negative of the identity matrix), whereas $$~j^2 = k^2 = +I_2~$$ are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of $k$$I$ and notated as $U(1) = Spin(2) = SO(2)$.

The coquaternion $$~q = w + x\,i + y\,j + z\,k~$$ with scalar $2$, has conjugate $$~q = w - x\,i - y\,j - z\,k~$$ similar to Hamilton's quaternions. The quadratic form is $$~q\,q^* = w^2 + x^2 - y^2 - z^2.$$

Note that the 2-sheet hyperboloid $$\left\{ x i + y j + z k : x^2 - y^2 - z^2 = 1 \right\}$$ corresponds to the imaginary units in the algebra so that any point $w$ on this hyperboloid can be used as a pole of a sinusoidal wave according to Euler's formula.

The hyperboloid is stable under $SU(2)$, illustrating the isomorphism with $SO(3)$. The variability of the pole of a wave, as noted in studies of polarization, might view elliptical polarization as an exhibit of the elliptical shape of a wave with pole $~p \ne \pm i~$. The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model, and the practice of $SL(2,ℝ)$ interferometry has been introduced.

When an element of $Spin(2,1)$ is interpreted as a Möbius transformation, it leaves the unit disk stable, so this group represents the motions of the Poincaré disk model of hyperbolic plane geometry. Indeed, for a point $SU(1,1)$ in the complex projective line, the action of $v$ is given by
 * $$\bigl[\;z,\;1\;\bigr]\,\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} = [\;u\,z + v^*, \, v\,z +u^*\;] \, = \, \left[\;\frac{uz + v^*}{vz +u^*}, \, 1 \;\right]$$

since in projective coordinates $$(\;u\,z + v^*, \; v\,z +u^*\;) \thicksim \left(\;\frac{\,u\,z + v^*\,}{v\,z +u^*}, \; 1 \;\right)~.$$

Writing $$\;suv + \overline{suv} = 2\,\Re\mathord\bigl(\,suv\,\bigr)\;,$$ complex number arithmetic shows
 * $$\bigl|u\,z + v^*\bigr|^2 = S + z\,z^* \quad \text{ and } \quad \bigl|v\,z + u^*\bigr|^2 = S + 1~,$$

where $$S = v\,v^* \left(z\,z^* + 1\right) + 2\,\Re\mathord\bigl(\,uvz\,\bigr).$$ Therefore, $$z\,z^* < 1 \implies \bigl|uz + v^*\bigr| < \bigl|\,v\,z + u^*\,\bigr|$$ so that their ratio lies in the open disk.

title: Special unitary group


text:
Talk:Ferdinand Philippe, Duke of Orléans

Protestant princess
Why is Princess Victoria of Saxe-Coburg and Gotha listed as a Protestant princess? Her branch of the family were Catholics not Protestants.--The Emperor's New Spy (talk) 02:44, 21 April 2013 (UTC)

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Penelope Wilton

Dame Penelope Alice Wilton, Lady Holm (born 3 June 1946) is an English actress.

Wilton is known for starring opposite Richard Briers in the BBC sitcom Ever Decreasing Circles (1984–1989); playing Homily in The Borrowers (1992) and The Return of the Borrowers (1993); and for her role as the widowed Isobel Crawley in the ITV drama Downton Abbey (2010–2015). She also played the recurring role of Harriet Jones in Doctor Who (2005–2008) and Anne in Ricky Gervais' Netflix dark comedy After Life.

Wilton has had an extensive career on stage, receiving six Olivier Award nominations. She was nominated for Man and Superman (1981), The Secret Rapture (1988), The Deep Blue Sea (1994), John Gabriel Borkman (2008) and The Chalk Garden (2009), before winning the 2015 Olivier Award for Best Actress for Taken at Midnight. Her film appearances include Clockwise (1986), Cry Freedom (1987), Blame It on the Bellboy (1992), Calendar Girls (2003), Shaun of the Dead (2004), Match Point (2005), Pride & Prejudice (2005), The Best Exotic Marigold Hotel (2012), The Girl (2012), The BFG (2016) and The Unlikely Pilgrimage of Harold Fry (2023).

Early life and background
Wilton was born in Scarborough, North Riding of Yorkshire, the second of three daughters of Cliff Wilton, a Cambridge-educated businessman and barrister who had played rugby union on the amateur and provincial level, going on to be an administrator in the sport, and Alice Linda Travers, a tap dancer and former actress.

She is a niece of actors Bill Travers and Linden Travers. Her cousins include actors Angela and Richard Morant. Her maternal grandparents owned theatres.

She attended the Drama Centre London from 1965 to 1968.

Career
Wilton began her career on stage in 1969 at the Nottingham Playhouse. Her early roles included Cordelia in King Lear, both in Nottingham and at The Old Vic.

She made her Broadway debut in March 1971 when she played Araminta in the original Broadway production of The Philanthropist, and made her West End debut in August 1971 opposite Sir Ralph Richardson in the John Osborne play West of Suez at the Cambridge Theatre. She had previously appeared in both plays at the Royal Court Theatre. She played Ruth in the original 1974 London stage production of Alan Ayckbourn's Norman Conquests trilogy, initially as understudy for Bridget Turner.

Her television acting career began in 1972, playing Vivie Warren in the BBC2's adaptation of Mrs. Warren's Profession opposite Coral Browne in the title role and Robert Powell. The production was repeated as part of the Play of the Month series in 1974 on BBC1. In 1994, Wilton portrayed Browne in a radio adaptation of An Englishman Abroad for the BBC World Service and repeated on various BBC radio formats since.

Following the broadcast of Mrs. Warren's Profession, Wilton then had several major TV roles, including two of the BBC Television Shakespeare productions (as Desdemona in Othello and Regan in King Lear ).

Wilton's film career includes roles in The French Lieutenant's Woman (1981), Cry Freedom (1987), Iris (2001), Calendar Girls (2003) and Shaun of the Dead (2004), Jane Austen's Pride & Prejudice (2005), Woody Allen's Match Point (2005), and in The History Boys (2006).

She did not garner fame until she appeared with Richard Briers in the 1984 BBC situation comedy, Ever Decreasing Circles, which ran for five years. She played Ann, long suffering wife of Martin (Briers), an obsessive and pedantic "do-gooder". In 2005, Wilton guest starred as Harriet Jones for two episodes in the BBC's revival of the popular TV science-fiction series Doctor Who. This guest role was written especially for her by the programme's chief writer and executive producer Russell T. Davies, with whom she had worked on Bob and Rose (ITV, 2001). The character of Jones returned as Prime Minister in the Doctor Who 2005 Christmas special "The Christmas Invasion". In the first part of the 2008 series finale, "The Stolen Earth", she made a final appearance, now as the former Prime Minister who sacrifices herself by extermination by the Daleks so that the Doctor's companions can contact him. Wilton appeared on television as Barbara Poole, the mother of a missing woman, in the BBC television drama series Five Days in 2005; and in ITV's drama Half Broken Things (October 2007) and the BBC production of The Passion (Easter 2008). Beginning in 2010, she appeared as Isobel Crawley in all six seasons of the hit period drama Downton Abbey. She was the castaway on BBC Radio 4's Desert Island Discs in April 2008. In December 2012 and February 2013, she was the narrator in Lin Coghlan's dramatisation of Elizabeth Jane Howard's The Cazalets, broadcast on BBC Radio.

Personal life
Between 1975 and 1984, Wilton was married to actor Daniel Massey. They had a daughter, Alice, born in 1977. Before that, they had a stillborn son.

In 1991, Wilton married actor Ian Holm. In 1992, they appeared together as Pod and Homily in the BBC's adaptation of The Borrowers. A year later, they appeared together in a follow-up The Return of the Borrowers. In 1998, Ian Holm was knighted and Wilton became Lady Holm. They divorced in 2001.

Honours
Wilton was appointed an Officer of the Order of the British Empire (OBE) in the 2004 New Year Honours and was elevated to become a Dame Commander of the Order of the British Empire (DBE) in the 2016 Birthday Honours, both for services to drama.

Awards and recognition
In 2012, Wilton received an honorary doctorate from the University of Hull Scarborough Campus.

title: Penelope Wilton


text:
Tom Ford

Thomas Carlyle Ford (born August 27, 1961) is an American fashion designer and filmmaker. He launched his eponymous brand in 2005, having previously been the creative director at Gucci and Yves Saint Laurent. Ford wrote and directed the films A Single Man (2009) and Nocturnal Animals (2016). From 2019 to 2022 he was chairman of the Council of Fashion Designers of America.

Early life
Thomas Carlyle Ford was born on August 27, 1961, in Austin, Texas, the son of realtors Shirley Burton (née Shirley Ann Thrasher) and Thomas David Ford (1932–2020). He spent his early life in the suburbs of Houston, Texas, and in San Marcos, outside Austin. He rearranged furniture in the house at 6, and gave his mother advice on her hair and shoes. His family moved to Santa Fe, New Mexico, when he was 11. In Santa Fe, he entered St. Michael's High School and later moved to Santa Fe Preparatory School, from which he graduated in 1979.

At age 16, he enrolled at Bard College at Simon's Rock, but quickly dropped out. He moved to New York City to study art history at New York University. There he met Ian Falconer, who took him to Studio 54 for the first time. Ford dropped out after a year, focusing on acting in television commercials.

Ford began studying interior architecture at The New School's art and design college, Parsons The New School for Design, in New York City. He continued to frequent Studio 54, where he realized he was gay. The club's disco-era glamor would be a major influence on his later designs. Before his last year at New School, Ford spent a year and a half in Paris, where he worked as an intern in Chloé's press office, inspiring his interest in fashion. He spent his final year at The New School studying fashion, but graduated with a degree in architecture.

Early career
When interviewing for jobs after graduation, Ford said that he had attended The New School's Parsons division, but concealed that he graduated in architecture, and that his work at Chloé was a low-level public relations position. Despite his lack of experience in fashion, Ford called American designer Cathy Hardwick every day for a month in hopes of securing a job at her sportswear company. Hardwick eventually agreed to interview him. She later recalled the incident: "I had every intention of giving him no hope. I asked him who his favorite European designers were. He said, 'Armani and Chanel.' Months later I asked him why he said that, and he said, 'Because you were wearing something Armani.' Is it any wonder he got the job?" Ford worked as a design assistant for Hardwick for two years.

In 1988, Ford moved to Perry Ellis, where he knew both Robert McDonald, the company's president, and Marc Jacobs, its designer, socially. He worked at the company for two years, but grew tired of working in American fashion. In a later interview with The New York Times, he commented, "If I was ever going to become a good designer, I had to leave America. My own culture was inhibiting me. Too much style in America is tacky. It's looked down upon to be too stylish. Europeans, however, appreciate style."

At the time, Italian fashion house Gucci was struggling financially and was seeking to strengthen its women's ready-to-wear presence as a part of a brand overhaul. The company's creative director, Dawn Mello said, "no one would dream of wearing Gucci". In 1990, Mello hired Ford as the brand's chief women's ready-to-wear designer and Ford moved to Milan. "I was talking to a lot of people, and most didn't want the job," Mello said. "For an American designer to move to Italy to join a company that was far from being a brand would have been pretty risky." Ford and his longtime partner, fashion journalist Richard Buckley, relocated to Milan that September.

Ford's role at Gucci rapidly expanded; he was designing menswear within six months, and shoes soon after that. When Richard Lambertson left as design director in 1992, Ford took over his position, heading the brand's ready-to-wear, fragrances, image, advertising, and store design. In 1993, when he was in charge of designing eleven product lines, Ford worked eighteen-hour days. During these years, there were creative tensions between Ford and Maurizio Gucci, the company's chairman and 50% owner. According to Mello, "Maurizio always wanted everything to be round and brown, and Tom wanted to make it square and black." Though Maurizio Gucci wanted to fire Ford, Domenico De Sole insisted that he remain. Nonetheless, Ford's work during the early 1990s was primarily behind the scenes; his contributions to Gucci were overshadowed by those of Mello, who was the company's public face.

Creative Director of Gucci and Saint Laurent

In 1994, Ford was promoted to Creative Director of Gucci. In his first year at the helm, he introduced Halston-style velvet hipsters, skinny satin shirts and car-finish metallic patent boots. In 1995, he brought in French stylist Carine Roitfeld and photographer Mario Testino to create a series of new ad campaigns for the company. Between 1995 and 1996, sales at Gucci increased by 90%. At one point, Ford was the largest individual shareholder of Gucci stock and options. By 1999, the house, which had been almost insolvent when Ford joined, was valued at more than $4 billion.

When Gucci acquired the house of Yves Saint Laurent (YSL) in 1999, Ford was named Creative Director of that label as well. Saint Laurent did not hide his displeasure with Ford's designs, stating "The poor man does what he can." During his time as creative director for YSL, Ford nonetheless won numerous Council of Fashion Designers of America Awards. Ford was able to pull the classic fashion house back into the mainstream. His advertising campaigns for the YSL fragrances Opium (with a red-haired Sophie Dahl naked wearing only a necklace and stiletto heels in a sexually suggestive pose) and YSL M7 (with martial arts champion Samuel de Cubber in complete full-frontal nudity) were controversial and provocative.

In April 2004, Ford parted ways with the Gucci group after he and CEO Domenico de Sole, who is credited as Ford's partner in Gucci's success, failed to agree with Pinault Printemps Redoute's boss over control of the Group. He has since referred to this experience as "devastating" and as a "midlife crisis" because he had "put everything into that for fifteen years". When Ford left in 2004, Gucci Group was valued at $10 billion. Four people were hired to split the work Ford had done.

Tom Ford label
After leaving Gucci, Ford launched a line of menswear, beauty, eyewear, and accessories in 2006, named after himself. De Sole became chairman of the label. Ford has described "the Tom Ford customer" as international, cultured, well-traveled, and possessing disposable income. For women, he added "strong women, … intelligent women who know their own style".

First Lady Michelle Obama wore an ivory floor-length evening gown designed by Ford to Buckingham Palace in 2011. He has also dressed Beyoncé, Jennifer Lopez, Gwyneth Paltrow, Anne Hathaway, Daniel Craig, Tom Hanks, Johnny Depp, Ryan Gosling, Will Smith, Julianne Moore, Hugh Jackman, Jon Hamm, and Henry Cavill. Ford designed Daniel Craig's suits for his final four James Bond films: Quantum of Solace (2008), Skyfall (2012), Spectre (2015), and No Time to Die (2021).

In 2013, Ford was mentioned in Justin Timberlake's song "Suit & Tie", which was a collaboration with Jay-Z. Ford created the suits, shirts, and accessories for the Grammy winning "Suit & Tie" music video. He went on to dress Timberlake's 20/20 Experience World Tour, designing over 600 pieces for the tour. The same year, Jay-Z released a song titled "Tom Ford" with "Tom Ford" rapped numerous times within the song. Ford responded that he was flattered and "it means that one has really penetrated and made an impact on popular culture." Following the song's release, Ford received a huge spike in online search engine queries. The song would go on to sell over a million copies and become certified platinum.

In November 2022, the Tom Ford brand was purchased by Estée Lauder for $2.8 billion. Forbes estimated that Ford would earn $1.1 billion from the deal. The following April, Ford stepped down as the brand's creative director and was succeeded by designer Peter Hawkings.

Controversies
Ford has been criticized for using naked women in various ad campaigns. Various journalists asserted that the ads were vulgar, sexist, or objectified women. One ad featured a nude woman holding a bottle of the perfume between her legs. Another featured a naked woman ironing a man's pants while he read a newspaper. A separate ad was banned in Italy.

Responding to criticism that he objectified women, Ford stated he is an "equal opportunity objectifier" and is "just as happy to objectify men". He argued "you can't show male nudity in our culture in the way you can show female nudity" and pointed out that he did a male nude ad while at Yves Saint Laurent which got pulled.

In 2014, Ford released a new product, called the "Penis Pendant Necklace". The product caused some controversy, with Christians calling it offensive due to the pendant being shaped similar to a Christian cross or crucifix. Ford replied that "it was not meant to be a cross, it was a phallus" and "people read into things what they want to".

In 2022, Ford criticized the Met Gala, stating the event had "turned into a costume party."

A Single Man
In March 2005, Ford announced the launch of his film production company, Fade to Black. In 2009 he made his directorial debut with A Single Man, based on the novel of the same name by Christopher Isherwood. The drama stars Colin Firth as an LA-based college professor who is gay, alongside Julianne Moore, Nicholas Hoult and Matthew Goode. The novel was adapted by David Scearce and Ford; Ford was also one of the producers.

A Single Man premiered on September 11, 2009, at the 66th Venice International Film Festival, where it was nominated for top award the Golden Lion. Colin Firth was awarded the Volpi Cup as Best Actor for his performance. He won a BAFTA Award for Best Actor in a Leading Role, and was nominated for an Academy Award, Golden Globe, Independent Spirit Award and Screen Actors Guild Award. The film won AFI Film of the Year and the GLAAD Media Award for Outstanding Film – Wide Release.

Other nominations for the film included two further Golden Globe categories: Julianne Moore for Best Supporting Actress, and Abel Korzeniowski for Best Original Score. At the Independent Spirit Awards, the film was nominated for Best First Feature and Best First Screenplay. Ford and Scearce also received a nomination for Best Adapted Screenplay at the Broadcast Film Critics Association Awards.

Nocturnal Animals
In 2015, Ford became attached to direct Nocturnal Animals, an adaptation of the Austin Wright novel Tony and Susan. The film was released in 2016. Jake Gyllenhaal and Amy Adams played the lead roles of Tony and Susan, and Michael Shannon, Armie Hammer, Aaron Taylor-Johnson, and Isla Fisher co-starred.

The film received praise from critics, as well as winning the Grand Jury Prize at the Venice Film Festival. The film has an approval rating of 72% on Rotten Tomatoes, based on 143 reviews, with an average rating of 7.1/10, and the site's critical consensus reading: "Well-acted and lovely to look at, Nocturnal Animals further underscores writer-director Tom Ford's distinctive visual and narrative skill."

Personal life
Ford married Richard Buckley in 2014, a journalist and former editor in chief of Vogue Hommes International; they had been in a relationship since meeting in 1986. The couple have a son who was born in September 2012 via gestational surrogate. The family lived in Italy, where Ford moved from New York in 1990, and in London for 17 years. They lived in his residences in New York, Los Angeles, Santa Fe, and London. Ford and Buckley have owned smooth fox terriers, which have appeared on the runway and in his film A Single Man. Ford also has two nephews and a niece, the children of his sister Jennifer.

Ford constructed a 24,000-acre private tract designed by Japanese architect Tadao Ando, in Santa Fe. It is called Cerro Pelon Ranch and has additional structures that were designed by Marmol Radziner. The property also has a fictional town known as Silverado that is used as a filming location for Western movies.

He told Vogue he had adopted a vegan diet after viewing the Netflix documentary called What the Health. As of 2019, he has been a teetotaler and has been open about using fillers and Botox.

He has said in multiple interviews that his first lover was the late artist Ian Falconer, who went on to write and illustrate the very popular Olivia the Pig children's book series. Ford maintains that he and Falconer remained good friends until Falconer's death in 2023; decades after their breakup, Ford lent Falconer's last name to the title character of his first movie, A Single Man (in the source novel the character originally only had a first name).

Richard Buckley died on September 19, 2021, at the age of 72 after a long illness.

Politics
Ford is a Democrat. He opposed America's invasion of Iraq in 2003, stating that it made him feel "ashamed to be an American". His comment drew public criticism within America. He hosted a fundraiser for Barack Obama. He voted for Hillary Clinton in the 2016 US general election.

Ford has advocated for federal recognition of same-sex marriage in the United States. In a 2009 interview, he said he preferred the term "civil partnership" for both opposite-sex and same-sex partnerships, and to leave "marriage" to religion to decide.

Ford maintains a policy of not dressing politicians regardless of party.

"I think that whoever is the President, or the First Lady, should be wearing clothes at a price point that are accessible to most Americans, and wearing clothes made in America. My clothes are made in Italy, they're very, very expensive. I don't think most women or men in our country can relate to that, and I think the First Lady or the President should represent all people."

Ford considers "obsession with political correctness" as something which negatively impacts modern fashion designers. He has stated that "Cancel culture inhibits design because rather than feeling free, the tendency is to start locked into a set of rules. Everything is now considered appropriation. We used to be able to celebrate other cultures. Now you can't do that."

Public image and legacy
Ford has been included in several best-dressed lists, such as International Best Dressed List, The Guardians "The 50 best-dressed over-50s", and British GQs "50 Best Dressed Men in Britain 2015". He was featured on the cover of the 2011 spring/summer issue of Another Man, giving his opinion on what makes the modern day gentleman. He has been called a "fashion icon" and a "style icon",  and he was included in "All-TIME 100 Fashion Icons" list. He won many awards including several VH1/Vogue Fashion Awards and Council of Fashion Designers of America (CFDA) Awards. In 2014, the CFDA awarded him the Geoffrey Beene Lifetime Achievement Award.

While Ford was in a monogamous relationship for many years, he "continue[d] to promote himself with a youthful and sexually charged image". He is known for sexy clothes, making provocative statements, and creating racy advertisements. Ford's designs convey a "sophisticated sex appeal" and he has been credited for "bringing sexy back". His advertisement campaigns have drawn controversy for use of nudity and "provocative sexual imagery". Ford is also known to pose with celebrities and models in his ad campaigns. He has been called the "King of Sex" and "the straightest gay man in the world".

Ford saved Gucci from near bankruptcy and transformed it into a fashion powerhouse. His decade as the creative director was hailed as a "golden era" for Gucci. He turned the brand around, replacing the "grunge look" with "sexy, yet sophisticated, clothes". He is known for his bold designs. The New York Magazine wrote "Every season, Ford created an 'It' piece, a must-have, a season-defining trend, photographed to death, knocked off ad nauseam." Ford says it is important for designers to be contemporary and current with the changing standard of beauty.

In 2004, Ford published an eponymous monograph, detailing his early career and his design work for Gucci and Yves Saint Laurent from 1990 to 2004. In 2021, seventeen years later, Ford published a follow-up volume entitled Tom Ford 002, which described his career from 2005 including the creation of his own fashion label and the production of his two films. Both books are published by Rizzoli International Publications and co-authored by fashion journalist Bridget Foley, with forewords by Anna Wintour.

In popular culture
In September 2013, hip-hop artist Jay-Z released the song "Tom Ford" as a single from his album Magna Carta Holy Grail. Ford responded favorably to the song saying, "Who would not be flattered to have an entire Jay-Z track named after them? I mean, [...] it's pretty rare that something like that happens. It's a kind of validation of one's work, as it means that one has really penetrated and made an impact on popular culture." The song was nominated for 56th Grammy Award for Best Rap Performance in 2014.

Tom Ford was referenced by Swedish rapper Bladee on his 2018 mixtape Icedancer.

In the 2021 film House of Gucci directed by Ridley Scott, Ford was portrayed by actor Reeve Carney.

Awards and nominations
Ford has been recognized by important design and cultural councils worldwide, including the Cooper Hewitt Design Museum and Time.
 * 1995: International Award – Council of Fashion Designers of America (CFDA)
 * 1997: People Magazine's 50 Most Beautiful People
 * 1999: Style Icon Award – Elle Style Awards UK
 * 2000: Best International Designer – VH1/Vogue Awards
 * 2000: Fashion Editors Club of Japan Award
 * 2000: British GQ International Man of the Year Award
 * 2000: Superstar Award – Fashion Group International
 * 2001: Womenswear Designer of the Year – CFDA
 * 2001: Best Fashion Designer – Time Magazine
 * 2001: Designer of the Year – GQ USA
 * 2002: Accessory Designer of the Year Award for Yves Saint-Laurent – CFDA
 * 2003: Fashion Design Achievement Award – Cooper-Hewitt Design Museum's National Design Awards
 * 2004: Board of Directors Special Tribute – CFDA
 * 2004: Rodeo Drive Walk of Style Award
 * 2004: International Best Dressed List Hall of Fame
 * 2005: André Leon Talley Lifetime Achievement Award – Savannah College of Art & Design
 * 2006: Accessory Brand Launch – Accessories Council Excellence Awards
 * 2007: GLAAD Media Awards – Vito Russo Award
 * 2007: DNR's Person of the Year
 * 2008: Menswear Designer of the Year – CFDA
 * 2009: Venice Film Festival – Golden Lion for A Single Man (nominee)
 * 2009: Venice Film Festival – Queer Lion for A Single Man
 * 2009: Critics' Choice Awards – Best Adapted Screenplay for A Single Man (nominee)
 * 2009: Independent Spirit Awards – Best First Screenplay for A Single Man (nominee)
 * 2009: Independent Spirit Awards – Best First Feature for A Single Man (nominee)
 * 2009: Honored as one of GQ USA's Men of the Year
 * 2009: GQ Germany Man of the Year
 * 2010: GLAAD Media Awards – Outstanding Film Wide Release for A Single Man
 * 2010: Menswear Designer of the Year – CFDA (nominee)
 * 2012: All-TIME 100 Fashion Icons – Member
 * 2013: Named one of the 50 best-dressed over 50s by The Guardian.
 * 2014: Geoffrey Beene Lifetime Achievement Award – CFDA
 * 2015: Menswear Designer of the Year – CFDA
 * 2015: Named in British GQs 50 best-dressed men in Britain
 * 2016: Venice Film Festival – Grand Jury Prize for Nocturnal Animals
 * 2016: Satellite Auteur Award

title: Tom Ford


text:
My Three Sons

My Three Sons is an American television sitcom that aired from September 29, 1960, to April 13, 1972. The series was broadcast on ABC during its first five seasons, before moving to CBS for the remaining seven seasons. My Three Sons chronicles the life of widower and aeronautical engineer Steven Douglas (Fred MacMurray) as he raises his three sons.

The series originally featured William Frawley (who had first co-starred with Fred MacMurray 25 years earlier in the film Car 99) as the boys' maternal grandfather and live-in housekeeper, William Michael Francis "Bub" O'Casey. William Demarest, playing Bub's brother, "Uncle Charley", replaced Frawley in 1965 because of Frawley's declining health. In September 1965 (when the show moved from ABC to CBS and began to be filmed in color), eldest son Mike (Tim Considine) married fiancée Sally Ann Morrison (Meredith MacRae), and his character was written out of the show. To keep the emphasis on "three sons", original youngest son Chip's (Stanley Livingston) friend Ernie (Barry Livingston, Stanley's real-life brother) was adopted. In the program's later years, Steven Douglas married Barbara Harper (Beverly Garland) and adopted her young daughter Dorothy Anne ("Dodie") (Dawn Lyn).

The series was a cornerstone of the ABC and CBS lineups in the 1960s. Disney producer Bill Walsh often mused on whether the concept of the show was inspired by the film The Shaggy Dog, as in his view they shared "the same dog, the same kids, and Fred MacMurray".

A Lawrence Welk cover version of the theme song reached #12 in Canada, April 24, 1961.

ABC years
The show began on ABC in black-and-white. The first season, consisting of 36 episodes, was directed in its entirety by Peter Tewksbury, the Emmy-winning director of Father Knows Best, who produced and occasionally scripted the programs.

During the 1964 fall season, William Frawley, who played Bub, was declared too ill to work by Desilu Studios, as the company was informed that insuring the actor would be too costly. Frawley continued in the role until a suitable replacement could be found at midseason. He was replaced by William Demarest, who played his hard-nosed brother (great) Uncle Charley, introduced partway through the 1964–1965 season (the last on ABC). According to the storyline, Bub returns to Ireland to help his Aunt Kate celebrate her 104th birthday. Soon after, brother Charley visits and stays on. Charley, a cello-playing merchant sailor, was a soft-hearted curmudgeon, who proved to be a responsible caregiver. Frawley left the series before the end of the 1964–1965 season. He died March 3, 1966.

The address of the fictional home in Bryant Park was 837 Mill Street. The actual address of the home used for the Bryant Park episodes was 837 5th Avenue in Los Angeles California. Many of the 1960s My Three Sons external shoots were actually done on location on 5th Avenue.

Directors
Peter Tewksbury directed the first season. The succeeding director, Richard Whorf, took over the reins for one season and was in turn followed by former actor-turned-director Gene Reynolds from 1962 to 1964. James V. Kern, an experienced Hollywood television director who had previously helmed the "Hollywood" and "Europe" episodes of I Love Lucy, continued in this role for two years until his untimely death at 57 in late 1966. Director James Sheldon was also contracted to finish episodes that had been partly completed by Kern to complete that season. Fred De Cordova was the show's longest and most consistent director of the series (108 episodes) until he left in 1971 to produce The Tonight Show Starring Johnny Carson. Earl Bellamy rounded out the series as director of the show's final year.

CBS years
My Three Sons moved to the CBS television network for the 1965–1966 season after ABC declined to underwrite the expense of producing the program in color. Along with the change in networks and the transition to color, Tim Considine (who had earlier worked with Fred MacMurray on The Shaggy Dog), playing eldest son Mike, had chosen not to renew his contract due to a clash with executive producer Don Fedderson over Considine's wish to direct but not co-star in the series. (Considine did, however, direct one of the last black-and-white episodes for ABC.) In an August 1989 interview on the Pat Sajak Show, he explained that he was also devoted to car racing, which his contract forbade. His character was written out, along with Meredith MacRae, who had played his fiancée Sally, in a wedding episode that was the premiere of the 1965–1966 season on CBS. After this episode, Mike is mentioned briefly in only four succeeding episodes (including the one in which the family adopts Ernie) and is never seen again, even at Robbie's or Steve's weddings. (Steve explains briefly in one of these episodes that he has another son who "lives away from home".) In the episode "Steve and the Huntress" (first aired January 27, 1966), Mike is specifically mentioned as teaching at a college. MacRae joined Petticoat Junction the following year, the last of three actresses to play Billie Jo Bradley.

To keep the show's title plausible, the show's head writer, George Tibbles, fashioned a three-part story arc in which an orphaned friend of youngest brother Richard (Chip, played by Stanley Livingston), Ernie Thompson (played by his real-life brother, Barry Livingston), awaits adoption when his current foster parents are transferred to the Orient. Steve offers to adopt Ernie, but faces antagonism from Uncle Charley, who finds Ernie a bit grating and forecasts major headaches over both the boy and his dog. It also transpires that a law requires a woman to live in the home of an adoptive family. A likable female social worker supervises the case and the Douglases speculate that Steve might marry the woman to make the adoption possible, but they agree that this is not reason enough for them to be married. Also, the family has no need for a housekeeper; Uncle Charley already has things running smoothly. The family soon appears before a judge who researches the law and determines that its intent is to ensure that a full-time caregiver is in the household. As Charley meets that role and has had a change of heart about Ernie, he assents to a legal fiction that declares him the Douglas family's "housemother".

While the three sons were always central to the storyline, several major changes took place by the late 1960s. In the spring of 1967, the series' ratings began to sag, and it finished its seventh season in 31st place in the Nielsen ratings. It was decided that the 1967–1968 season would bring the program not only a new time slot but also new storylines to spice up the ratings. In the fall of 1967, CBS moved My Three Sons to Saturday at 8:30 pm ET. In the season premiere episode, "Moving Day", the Douglas family and Uncle Charley relocate from the fictional Midwestern town of Bryant Park to Los Angeles, California. Robbie (Don Grady) marries his girlfriend Katie Miller. Katie is played by Tina Cole, who had appeared in different roles on three previous episodes: "House For Sale" from the fourth season (February 13, 1964), "The Coffee House Set" from the fifth season (November 19, 1964), and "Robbie and the Little Stranger" from the sixth season (February 17, 1966). At the end of the 1967–1968 season, the ratings had improved from the previous year with the series placing at 24th in the Nielsens. The following season, the newlyweds discover that Katie is pregnant and she gives birth to triplets named Robert, Steven, and Charles. Originally played by sets of uncredited twins, the boys were later played uncredited by Guy, Gunnar, and Garth Swanson; and in the last two seasons by Michael, Daniel, and Joseph Todd.

The following year in the 10th season, 1969–1970, Steve remarries. His new bride, widowed teacher Barbara Harper (Beverly Garland), brings with her a five-year-old daughter, Dorothy "Dodie" (Dawn Lyn), whom Steve adopts. Dodie is wary of Steve at first, believing that he wants her to just forget her late father, but he explains that he wants her to always remember and love him, but since he's no longer alive, Steve wants to raise her in his place and he hopes she'll come to love him, too. The series' last year and a half feature fewer appearances from both Don Grady and Stanley Livingston. Grady's character was written out at the end of the 11th season, which allowed for his wife Katie and their triplet sons to remain in the Douglas household for the following season (as a structural engineer Robbie was working on a bridge construction in Peru). Meanwhile, Chip and his teenage wife Polly (Ronne Troup), who had eloped after Polly's disciplinarian father refused to sanction their marriage, move into their own apartment.

At the end of the 1970–1971 season, the show's 11th year, My Three Sons was still garnering healthy ratings. By the spring of 1971, it had finished in 19th place. A 1971 television pilot with Don Grady and Tina Cole called Three of a Kind, then retitled Robbie—about Robbie, Katie, and the triplets moving to San Francisco—was filmed but not picked up as a series. The final episode of the 1970–1971 season, "After the Honeymoon", actually set up the premise for this pilot. Richard X. Slattery and Pat Carroll guest-starred as the landlords of the apartment block into which Robbie and Katie move. However, Don Grady had informed the producers of his intention to leave the series and pursue a new full-time career as a composer, which he ultimately did.

For the series' 12th and, ultimately, final season, CBS moved the show to Mondays at 10:00 pm ET. In addition to the time changes for the 12th season, a new four-part story arc is introduced with MacMurray in a second role, that of his cousin, the Laird (Lord) Fergus McBain Douglas of Sithian Bridge; English actor Alan Caillou's voice was dubbed over MacMurray's. The plot centers on Lord Douglas's arrival in Los Angeles from the family's native Scotland, in search of a bride to take back to Scotland with him.

He finds Terri Dowling (Anne Francis), a waitress at the Blue Berry Bowling Alley. While initially reluctant to give up her life in America and return to Scotland as nobility, she finally accepts. This storyline continues a plot idea that originally began in the fourth season, when the Douglases visited Scotland on the pretense of having been told they had inherited a castle in the Highlands.

With a later time slot, the show finished the season outside the top 30. To save the series, CBS moved it in midseason back to Thursdays at 8:30 pm ET, its old time slot. Nevertheless, My Three Sons ended its primetime run in the spring of 1972 after 12 years on the air. CBS also aired daytime reruns starting in September 1971 (only the CBS color shows), for about one season.

Main

 * Fred MacMurray - Steve Douglas (1960–1972)
 * William Frawley - William Michael Francis Aloysius "Bub" O'Casey, Steve's father-in-law (1960–1965)
 * William Demarest - Charles Leslie "Uncle Charley" O'Casey, Bub's younger brother (1965–1972)
 * Tim Considine - Mike Douglas, Steve's son (1960–1965)
 * Don Grady - Robbie Douglas, Steve's son (1960–1971)
 * Stanley Livingston - Richard "Chip" Douglas, Steve's son (1960–1972)
 * Barry Livingston - Ernie Thompson-Douglas, Steve's adoptive son (1963–1972)
 * Meredith MacRae - Sally Ann Morrison-Douglas, Mike's wife (1963–1965)
 * Tina Cole - Katie Miller-Douglas, Robbie's wife (1967–1972)
 * Beverly Garland - Barbara Harper-Douglas, Steve's 2nd wife (1969–1972)
 * Dawn Lyn - Dorothy Anne "Dodie" Harper-Douglas, Barbara's daughter (1969–1972)
 * Ronne Troup - Polly Williams-Douglas, Chip's wife (1970–1972)
 * Michael, Daniel and Joseph Todd playing - Robbie Jr., Steve II and Charley Douglas, Robbie & Katie's triplet sons respectively (1970–1972)

Recurring

 * Cynthia Pepper - Jean Pearson (1960–1961)
 * Peter Brooks - Hank Ferguson (1960–1963)
 * Cheryl Holdridge - Judy Doucette (1960–1961)
 * Ricky Allen - Hubert 'Sudsy' Pfeiffer (1961–1963)
 * Hank Jones - Pete (1964–1966)
 * Bill Erwin - Joe Walters (1962–1964)
 * Doris Singleton - Helen Morrison, Sally's mother (1964–65) and Margaret Williams, Polly's mother (1970)
 * John Howard - Dave Welch (1965–1967)
 * Joan Tompkins - Lorraine Miller, Katie's mother (1967–1970)
 * Eleanor Audley - Mrs. Beatrice Vincent (1969-1970)
 * Norman Alden - Tom Williams, Polly's father (1970)

Episodes
My Three Sons had 36 episodes each in the first two seasons. The series had at least thirty episodes in each of the first eight seasons; the episode output then decreased by two episodes until the eleventh season, which had twenty-four episodes, along with the twelfth season. The first five seasons were filmed in black & white, then after the move to CBS, it was filmed in color for the remainder of its run.

Production
The series was initially filmed at Desilu Studios in Hollywood, but at the start of the 1967–68 season, the cast and crew began filming the series at the CBS Studio Center in Studio City, California. The reasons behind this move concerned the sale of actress-comedian Lucille Ball's studios to the Gulf + Western conglomerate, which owned Paramount Pictures. Don Fedderson Productions, which produced My Three Sons (along with Family Affair starring Brian Keith), had to quickly make other arrangements for filming.

Distribution
Although Don Fedderson gets the credit, My Three Sons was created by George Tibbles and produced by Don Fedderson Productions throughout the show's run, with MCA Television co-distributing the series during its 1960–65 ABC airing. When the series moved to CBS in 1965, the latter network assumed full production responsibilities (in association with Fedderson Productions) until the end of the series in 1972. CBS now holds the series' copyright. CBS Media Ventures presently owns distribution rights to the entire series (including the more widely seen and aforementioned 1965–72 CBS episodes). The show did not get syndicated until September 1976, when Viacom Enterprises assumed off-network distribution (although CBS did air reruns of the show in its daytime lineup from December 1971 until the fall of 1972), and even then, only the CBS color episodes aired, while the black and white ABC episodes did not air on broadcast TV at all. At the time, the first half of season 11 was in the same syndication package as seasons 6-10, while season 12 and the second half of season 11 were distributed with seasons 1–5.

Nick at Nite aired My Three Sons from November 3, 1985, to October 28, 1991, with episodes from Seasons 1–5, the second half of season 11, and season 12. The Family Channel also aired only the black and white episodes from September 7, 1992, to July 30, 1993. The Seasons 1–3 episodes had the original Chevrolet closing credits. The Seasons 6–10 (and the first half of season 11) episodes were later aired on TV Land in the late 1990s. Odyssey ran all of the color episodes in the early 2000s. They also briefly aired the black and white episodes. In 2000, TV Land briefly aired the black & white episodes again, using the same syndication episode rights that were on Nick at Nite during the 1980s. In 2006 the Retro Television Network broadcast the show. Only seasons 6-10 and the first half of season 11 were aired.

In 2009, FamilyNet began airing the program as a lead-in for its Happy Days and Family Ties program block, which ended in February 2010. From 2012 to 2014, and also from October 2015 to February 2016, MeTV aired the Season 6-10 episodes in heavy rotation, and most closing credits included the original sponsor tags, such as those for Kellogg's. In 2016, Decades began airing the CBS (color) episodes in its "binge" blocks on some weekends.

MeTV began airing the black and white episodes on May 29, 2017, the first time the black and white episodes had aired on broadcast television since their original ABC airings. MeTV aired all episodes in order until August 3, 2018, when the series's last episode aired. This marked the first time a U.S. television station aired the entire series in full. MeTV started reairing the entire series again on August 6, 2018 and has continued to do so until May 27, 2024 when The Flintstones replaced it. Additionally, Catchy Comedy aired a Catchy Binge of the series from September 9 to 11, 2023.

Reunion special
MacMurray and most of the cast took part in Thanksgiving Reunion with The Partridge Family and My Three Sons, which aired on ABC on November 25, 1977. The retrospective special looked back at the history of My Three Sons and The Partridge Family (other than featuring single parents with a large family, the two series had no narrative, or even a studio, link). The special was notable for featuring footage from early black and white episodes of My Three Sons that, at this point in time, were not in syndication. While most of the collected casts gathered in a studio to reminisce, Demarest appeared in a brief pre-taped segment.

Home media
CBS DVD (distributed by Paramount) has released the first five seasons of My Three Sons on DVD in Region 1.

In most episodes, the soundtrack was edited to remove the background musical score, which were originally stock music from the Capitol Records library; the licensing agreement with Capitol only covered broadcast rights, not home video rights, and clearing the music for home video release with the individual composers who worked on the Capitol recordings was deemed cost-prohibitive. The Capitol scores were replaced instead with more modern, synthesized music. The original theme tune by Frank DeVol has been left unaltered; his musical scores in later seasons of the show, written specifically for the series, would less-likely be affected by licensing problems if the later seasons were released on DVD.

On May 13, 2019, the third season was released, in two volume sets, available exclusively through Amazon's made-on-demand (MOD) service. Season 4 was released on September 10, 2019 and season 5 on December 17, 2019.

title: My Three Sons


text:
Phar Lap

Phar Lap (4 October 1926 – 5 April 1932) was a New Zealand-born champion Australian Thoroughbred racehorse. Achieving incredible success during his distinguished career, his initial underdog status gave people hope during the early years of the Great Depression. He won the Melbourne Cup, two Cox Plates, the Australian Derby, and 19 other weight-for-age races.

One of his greatest performances was winning the Agua Caliente Handicap in Mexico in track-record time in his final race. He won in a different country, after a bad start many lengths behind the leaders, with no training before the race, and he split his hoof during the race.

After a sudden and mysterious illness, Phar Lap died in 1932 in Menlo Park, California. At the time, he was the third-highest stakes-winner in the world. His mounted hide is displayed at the Melbourne Museum, his skeleton at the Museum of New Zealand, and his heart at the National Museum of Australia.

Name
The name Phar Lap derives from the common Zhuang and Thai word for lightning: ฟ้าแลบ, literally 'sky flash'.

Phar Lap was called "The Wonder Horse," "The Red Terror," and "Big Red" (the latter nickname was also given to two of the greatest United States racehorses, Man o' War and Secretariat). He was affectionately known as "Bobby" to his strapper Tommy Woodcock He was also sometimes referred to as "Australia's Wonder Horse."

According to the Museum of Victoria, Aubrey Ping, a medical student at the University of Sydney, suggested "Farlap" as the horse's name. Ping knew the word from his father, a Zhuang-speaking Chinese immigrant. Phar Lap's trainer Harry Telford liked the name, but changed the F to PH to create a seven letter word, which was split in two in keeping with the dominant naming pattern of Melbourne Cup winners.

Early life
A chestnut gelding, Phar Lap was foaled on 4 October 1926 in Seadown near Timaru in the South Island of New Zealand. He was sired by Night Raid from Entreaty by Winkie. He was by the same sire as the Melbourne Cup winner Nightmarch. Phar Lap was a brother to seven other horses, Fortune's Wheel, Nea Lap (won 5 races), Nightguard, All Clear, Friday Night, Te Uira and Raphis, none of which won a principal (stakes) race. He was a half-brother to another four horses, only two of which were able to win any races at all.

Sydney trainer Harry Telford persuaded American businessman David J. Davis to buy the colt at auction, based on his pedigree. Telford's brother Hugh, who lived in New Zealand, was asked to bid up to 190 guineas at the 1928 Trentham Yearling Sales. When the horse was obtained for a mere 160 guineas, he thought it was a great bargain until the colt arrived in Australia. The horse was gangly, his face was covered with warts, and he had an awkward gait. Davis was furious when he saw the colt as well, and refused to pay to train the horse. Telford had not been particularly successful as a trainer, and Davis was one of his few remaining owners. To placate Davis, he agreed to train the horse for nothing, in exchange for a two-thirds share of any winnings. Telford leased the horse for three years and was eventually sold joint ownership by Davis.

Although standing a winning racehorse at stud could be quite lucrative, Telford gelded Phar Lap anyway, hoping the colt would concentrate on racing.

Racing career
Phar Lap finished last in the first race and did not place in his next three races. He won his first race on 27 April 1929, the Maiden Juvenile Handicap at Rosehill, ridden by Jack Baker of Armidale, a 17-year-old apprentice. He didn't race for several months but was then entered in a series of races, in which he moved up in class. Phar Lap took second in the Chelmsford Stakes at Randwick on 14 September 1929, and the racing community started treating him with respect. He won the Rosehill Guineas by three lengths on 21 September 1929, ridden by James L. Munro.

As his achievements grew, there were some who tried to halt his progress. Criminals tried to shoot Phar Lap on the morning of Saturday 1 November 1930 after he had finished track work. They missed, and later that day he won the Melbourne Stakes, and three days later the Melbourne Cup as odds-on favourite at 8 to 11.

In the four years of his racing career, Phar Lap won 37 of 51 races he entered, including the Melbourne Cup, being ridden by Jim Pike, in 1930 with 9 st 12 lb (138 lb). In that year and 1931, he won 14 races in a row. From his win as a three-year-old in the VRC St. Leger Stakes until his final race in Mexico, Phar Lap won 32 of 35 races. In the three races that he did not win, he ran second on two occasions, beaten by a short head and a neck, and in the 1931 Melbourne Cup he finished eighth when carrying 10 st 10 lb (150 lb).

Phar Lap at the time was owned by American businessman David J. Davis and leased to Telford. After their three-year lease agreement ended, Telford had enough money to become joint owner of the horse. Davis then had Phar Lap shipped to North America to race. Telford did not agree with this decision and refused to go, so Davis, who along with his wife traveled to Mexico with him, brought Phar Lap's strapper Tommy Woodcock as his new trainer. Phar Lap was shipped by boat to Agua Caliente Racetrack near Tijuana, Mexico, to compete in the Agua Caliente Handicap, which was offering the largest prize money ever offered in North America racing. Phar Lap won in track-record time while carrying 129 pounds (58.5 kg). The horse was ridden by Australian jockey Billy Elliot for his seventh win from seven rides. From there, the horse was sent to a private ranch near Menlo Park, California, while his owner negotiated with racetrack officials for special race appearances.

Death
Early on 5 April 1932, the horse's strapper for the North American visit, Tommy Woodcock, found him in severe pain and with a high temperature. Within a few hours, Phar Lap haemorrhaged to death. An autopsy revealed that the horse's stomach and intestines were inflamed, leading many to believe the horse had been deliberately poisoned. There have been alternative theories, including accidental poisoning from lead insecticide and a stomach condition. It was not until the 1980s that the infection could be formally identified.

In 2000, equine specialists studying the two necropsies concluded that Phar Lap probably died of duodenitis-proximal jejunitis, an acute bacterial gastroenteritis. In 2006, Australian Synchrotron Research scientists said it was almost certain Phar Lap was poisoned with a large single dose of arsenic in the hours before he died, perhaps supporting the theory that Phar Lap was killed on the orders of US gangsters, who feared the Melbourne Cup-winning champion would inflict big losses on their illegal bookmakers. No real evidence of involvement by a criminal element exists, however.

Sydney veterinarian Percy Sykes believes deliberate poisoning did not cause the death. He said "In those days, arsenic was quite a common tonic, usually given in the form of a solution (Fowler's Solution)", and suggests this was the cause of the high levels. "It was so common that I'd reckon 90 percent of the horses had arsenic in their system."

In December 2007, Phar Lap's mane was tested for multiple doses of arsenic which, if found, would point to accidental poisoning.

In April 2008, an 82-page handwritten notebook belonging to Telford and containing recipes for tonics given to Phar Lap in the days before swabbing was sold by a Melbourne auction house. It showed that Phar Lap was given tonics designed to boost his performance that included arsenic, strychnine, cocaine and caffeine. The find gave credence to Woodcock's deathbed admission in 1985 that Phar Lap may have been given an overdose of a tonic before the horse died in 1932. The notebook was sold to the Melbourne Museum for $37,000.

On 19 June 2008, the Melbourne Museum released the findings of the forensic investigation conducted by Ivan Kempson, University of South Australia, and Dermot Henry, Natural Science Collections at Museum Victoria. Kempson analysed six hairs from Phar Lap's mane at the Advanced Photon Source at Argonne National Laboratory near Chicago. These high resolution X-rays detect arsenic in hair samples, showing the specific difference "between arsenic, which had entered the hair cells via the blood and arsenic which had infused the hair cells by the taxidermy process when he was stuffed and mounted at the museum".

Kempson and Henry discovered that in the 30 to 40 hours before Phar Lap's death, the horse ingested a massive dose of arsenic. "We can't speculate where the arsenic came from, but it was easily accessible at the time", Henry said.

In October 2011 the Sydney Morning Herald published an article in which a New Zealand physicist and information from Phar Lap's strapper state that the great horse was never given any tonic with arsenic and that he died of an infection. Said Putt, "Unless we are prepared to say that Tommy Woodcock was a downright liar, which even today, decades after the loveable and respected horseman's death, would ostracise us with the Australian racing public, we must accept him on his word. The ineluctable conclusion we are left with, whether we like it or not, is that Phar Lap's impeccable achievements here and overseas were utterly tonic, stimulant, and drug-free."

Contradicting this is the tonic book of Harry Telford, Phar Lap's owner and trainer, on display in Museum Victoria, Melbourne. One recipe for a "general tonic" has a main ingredient of arsenic and has written below it: "A great tonic for all horses". Several theories have been proposed for the large amount of arsenic in Phar Lap's body.

Legacy
Following his death, Phar Lap's heart was donated to the Institute of Anatomy in Canberra and his skeleton to the New Zealand's National Museum in Wellington. After preparations of the hide by New York City taxidermist Louis Paul Jonas, Phar Lap's stuffed body was placed in the Australia Gallery at Melbourne Museum. The hide and the skeleton were put on exhibition together when Museum of New Zealand Te Papa Tongarewa lent the skeleton to the Melbourne Museum in September 2010 as part of celebrations for the 150th running of the 2010 Melbourne Cup.

Phar Lap's heart was remarkable for its size, weighing 6.2 kg, compared with a normal horse's heart at 3.2 kg. Now held at the National Museum of Australia in Canberra, it is the object visitors most often request to see. The author and film maker Peter Luck was convinced the heart is a fake. In Luck's 1979 television series This Fabulous Century, the daughter of Walker Neilson, the government veterinarian who performed the first post-mortem on Phar Lap, says her father told her the heart was necessarily cut to pieces during the autopsy, and the heart on display is that of a draughthorse. However the expression "a heart as big as Phar Lap" to describe a very generous or courageous person became a popular idiom.

Several books and films have featured Phar Lap, including the 1983 film Phar Lap, and the song "Phar Lap—Farewell To You".

Phar Lap was one of five inaugural inductees into both the Australian Racing Hall of Fame and New Zealand Racing Hall of Fame. In the Blood-Horse magazine ranking of the Top 100 U.S. Thoroughbred champions of the 20th century, Phar Lap was ranked No. 22.

The horse is considered to be a national icon in both Australia and New Zealand. In 1978 he was honoured on a postage stamp issued by Australia Post and features in the Australian citizenship test.

Phar Lap has been honoured with a $500,000 life-sized bronze memorial near his birthplace in Timaru, New Zealand, that was unveiled on 25 November 2009. The statue is located at the entrance to Phar Lap Raceway in Washdyke. There is also a life-sized bronze statue at Flemington Racecourse in Melbourne.

Phar Lap has several residential streets named after him in Australia, New Zealand, and the United States. (In many cases, the name is merged into a single word "Pharlap".)

In 1931, Gilbert Percy Whitley, an ichthyologist at the Australian Museum, proposed a new genus of seahorse, Farlapiscis, named after Phar Lap. Farlapiscis was subsequently categorized as a junior synonym of the genus Hippocampus.

1931/1932: Five-year-old season
Total: 51 starts – 37 wins, 3 seconds, 2 thirds, 2 fourths, 7 unplaced

title: Phar Lap


text:
Étienne Bonnot de Condillac

Étienne Bonnot de Condillac (, ; 30 September 1714 – 2 August or 3 August 1780) was a French philosopher, epistemologist, and Catholic priest, who studied in such areas as psychology and the philosophy of the mind.

Biography
He was born at Grenoble into a legal family, the youngest of three brothers. His two older brothers Jean and Gabriel took names associated with one of the family's properties at Mably, Loire, and were each known as "Bonnot de Mably". Étienne identified with another property at Condillac, Drôme, was known as "Bonnot de Condillac". Like his brother Gabriel, Condillac took holy orders (1733–1740) at Saint-Sulpice church in Paris. He was appointed as Abbot of Mureau.

Condillac devoted his whole life, with the exception of an interval as a court-appointed tutor to the court of Parma, to speculative thought. His works are:

 * Essai sur l'origine des connaissances humaines (1746);
 * Traité des systèmes (1749);
 * Traité des sensations (1754);
 * Traité des animaux (1755);
 * a comprehensive Cours d'études (1767–1773) in 13 vols., written for the young Duke Ferdinand of Parma, a grandson of Louis XV;
 * Le Commerce et le gouvernement, considérés relativement l'un a l'autre (1776); and two posthumous works,
 * Logique (1781) and the unfinished Langue des calculs (1798).

In Paris, Condillac was involved with the circle of Denis Diderot, the philosopher who was co-contributor to the Encyclopédie. He developed a friendship with Jean-Jacques Rousseau, which lasted in some measure to the end of his life. It likely started when Rousseau was a tutor to two of his brother Jean's sons in Lyon—Jean Bonnot de Mably was then provost of the police and known as Monsieur de Mably. Together with his brother Gabriel, who became the well-known political writer known as Abbé de Mably, Condillac introduced Rousseau to an intellectual circle.

Condillac's relations with unorthodox philosophers did not injure his career. He had already published several works when the French court sent him to Parma to educate the orphan duke, then a child of seven years.

On his return from Italy, Condillac was elected to the Académie française in 1768. Contrary to the popular idea that he attended only one meeting, he was a frequent attendee until two years before his death. He spent his later years in retirement at Flux, a small property which he had purchased near Beaugency on the river Loire. He died there on 2 or 3 August 1780.

Work
Condillac is important both as a psychologist and as having established systematically in France the principles of John Locke. Voltaire had made the English philosopher fashionable. Condillac developed his concept of empirical sensationism, and demonstrated "lucidity, brevity, moderation, and an earnest striving after logical method."

His first book, the Essai sur l'origine des connaissances humaines, keeps close to his English master. He accepts with some reluctance Locke's deduction of our knowledge from two sources, sensation and reflection. He uses as his main principle of explanation the association of ideas.

His next book, the Traité des systèmes, is a vigorous criticism of those modern systems which are based upon abstract principles or upon unsound hypotheses. His polemic, which is inspired throughout by Locke, is directed against the innate ideas of the Cartesians, Malebranche's faculty-psychology, Leibniz's monadism and pre-established harmony, and, above all, against the conception of substance set forth in the first part of the Ethics of Baruch Spinoza.

By far the most important of his works is the Traité des sensations, in which Condillac treats psychology in his own characteristic way. He questioned Locke's doctrine that the senses give us intuitive knowledge of objects, that the eye, for example, naturally judges shapes, sizes, positions, and distances. He believed it was necessary to study the senses separately, to distinguish precisely what ideas are owed to each sense, to observe how the senses are trained, and how one sense aids another. He believed that the conclusion has to be that all human faculty and knowledge are transformed sensation only, to the exclusion of any other principle, such as reflection.

The author imagines a statue organized inwardly like a man, animated by a soul which has never received an idea, into which no sense-impression has ever penetrated. He unlocks its senses one by one, beginning with smell, as the sense that contributes least to human knowledge. At its first experience of smell, the consciousness of the statue is entirely occupied by it; and this occupancy of consciousness is attention. The statue's smell-experience will produce pleasure or pain; and pleasure and pain will thenceforward be the master-principle which, determining all the operations of its mind, will raise it by degrees to all the knowledge of which it is capable. The next stage is memory, which is the lingering impression of the smell experience upon the attention: "memory is nothing more than a mode of feeling." From memory springs comparison: the statue experiences the smell, say, of a rose, while remembering that of a carnation; and "comparison is nothing more than giving one's attention to two things simultaneously." And "as soon as the statue has comparison it has judgment." Comparisons and judgments become habitual, are stored in the mind and formed into series, and thus arises the powerful principle of the association of ideas. From comparison of past and present experiences in respect of their pleasure-giving quality arises desire; it is desire that determines the operation of our faculties, stimulates the memory and imagination, and gives rise to the passions. The passions, also, are nothing but sensation transformed.

These indications will suffice to show the general course of the argument in the first section of the Traité des sensations. He thoroughly developed this idea through the subsequent chapters: "Of the Ideas of a Man limited to the Sense of Smell," "Of a Man limited to the Sense of Hearing," "Of Smell and Hearing combined," "Of Taste by itself, and of Taste combined with Smell and Hearing," "Of a Man limited to the Sense of Sight."

In the second section of the treatise, Condillac invests his statue with the sense of touch, which first informs it of the existence of external objects. In a very careful and elaborate analysis, he distinguishes the various elements in our tactile experiences-the touching of one's own body, the touching of objects other than one's own body, the experience of movement, the exploration of surfaces by the hands: he traces the growth of the statue's perceptions of extension, distance and shape. The third section deals with the combination of touch with the other senses. The fourth section deals with the desires, activities and ideas of an isolated man who enjoys possession of all the senses; and ends with observations on a "wild boy" who was found living among bears in the forests of Lithuania.

The conclusion of the whole work is that in the natural order of things, everything has its source in sensation, and yet that this source is not equally abundant in all men; men differ greatly in the degree of vividness with which they feel. Finally, he says that man is nothing but what he has acquired; all innate faculties and ideas are to be swept away. Modern theories of evolution and heredity have differed from this.

Condillac's work on politics and history, in his Cours d'études, is considered of less interest. In logic, on which he wrote extensively, he is far less successful than in psychology. He enlarges with much iteration on the supremacy of the analytic method; argues that reasoning consists in the substitution of one proposition for another which is identical with it; and lays it down that science is the same thing as a well-constructed language, a proposition which in his Langue des calculs, he tries to prove by the example of arithmetic. His logic is limited by his study of sensations and lack of knowledge of science other than mathematics. He rejects the medieval apparatus of the syllogism; but is precluded by his standpoint from understanding the active, spiritual character of thought; nor had he that interest in natural science and appreciation of inductive reasoning which form the chief merit of J. S. Mill. Some might claim that Condillac's anti-spiritual psychology, with its explanation of personality as an aggregate of sensations, leads straight to atheism and determinism. However, he denies both these consequences. What he says upon religion is always in harmony with his profession; and he vindicated the freedom of the will in a dissertation that has very little in common with the Traité des sensations to which it is appended. The common reproach of materialism should certainly not be made against him. He always asserts the substantive reality of the soul; and in the opening words of his Essai, "Whether we rise to heaven, or descend to the abyss, we never get outside ourselves—it is always our own thoughts that we perceive," we have the subjectivist principle that forms the starting-point of Berkeley.

Language
Condillac considered language as the vehicle by which senses and emotions were transformed into higher mental faculties. He believed that the structure of language reflects the structure of thought, and compared ideas to the sounds of a harpsichord. His theories had a major effect on the development of linguistics.

Condillac promoted "sensationalism," a theory that says all knowledge comes from the senses and there are no innate ideas. Condillac promoted an expressionist theory of linguistic creation that anticipates the prime features of later thoughts about language by German theorist Johann Gottfried Herder (1744–1803).

Economics
Condillac's 'Le Commerce et le Gouvernement' (published in 1776, the same year as Adam Smith's Wealth of Nations) attempted to place economics in a coherent logical framework. He was a friend of François Quesnay – leader of the Physiocrats. Much of Condillac's work reflected mainstream Physiocrats, particularly his analysis of the structure of taxation and proposals for the revival of the economy, but he also proposed another line of argument, claiming that producers work to obtain utility. Most physiocrats rejected utility and the idea was ignored until his 'rediscovery' by Stanley Jevons and Carl Menger in 1871.

In his theory of "vrai prix" [true price], Condillac proposed a theory of human history divided into two phases: progress and decline. Progress is marked by a rational development and use of resources; decline is precipitated by bad behavior from the upper classes that then trickles down to the workers, encouraging excess, luxury, and false prices that harm the masses. Condillac saw the remedy to this as "vrai prix," a true price created by the unimpeded interaction of supply and demand, to be achieved by complete deregulation. People would be taught to work toward their best interest in an open market through a reshaping of their perceptions. By advocating of a free market economy in contrast to the prevailing contemporary policy of state control in France, Condillac influenced classical liberal economics.

History
Condillac's Histoire ancienne and Histoire moderne (1758–1767) demonstrated how the experience and observation of the past aided man. History was not a mere retelling of the past, but a source of information and inspiration as well. Reason and critical thinking can improve man's lot and destroy superstition and fanaticism. History thus served as a moral, political, and philosophical textbook which taught man to live better. Thus the two histories present the basic program of the Enlightenment in crystallized form.

Legacy
As was fitting to a disciple of Locke, Condillac's ideas have had most importance in their effect upon English thought. In matters connected with the association of ideas, the supremacy of pleasure and pain, and the general explanation of all mental contents as sensations or transformed sensations, his influence can be traced upon the Mills and upon Bain and Herbert Spencer. And, apart from any definite propositions, Condillac did a notable work in the direction of making psychology a science; it is a great step from the desultory, genial observation of Locke to the rigorous analysis of Condillac, short-sighted and defective as that analysis may seem to us in the light of fuller knowledge.

His method, however, of imaginative reconstruction was by no means suited to English ways of thinking. In spite of his protests against abstraction, hypothesis and synthesis, his allegory of the statue is in the highest degree abstract, hypothetical and synthetic. James Mill, who stood more by the study of concrete realities, put Condillac into the hands of his youthful son with the warning that here was an example of what to avoid in the method of psychology. A modern historian has compared Condillac with Scottish Enlightenment philosopher and pre-evolutionary thinker Lord Monboddo, who had a similar fascination with abstraction and ideas. In France Condillac's doctrine, so congenial to the tone of 18th century philosophism, reigned in the schools for over fifty years, challenged only by a few who, like Maine de Biran, saw that it gave no sufficient account of volitional experience. Early in the 19th century, the romantic awakening of Germany had spread to France, and sensationism was displaced by the eclectic spiritualism of Victor Cousin.

Condillac's collected works were published in 1798 (23 vols.) and two or three times subsequently; the last edition (1822) has an introductory dissertation by A. F. Théry. The Encyclopédie méthodique has a very long article on Condillac by Naigeon. Biographical details and criticism of the Traité des systèmes in J. P. Damiron's Mémoires pour servir a l'histoire de to philosophie au dixhuitieme siècle, tome iii.; a full criticism in V Cousin's Cours de l'histoire de la philosophie moderne, ser. i. tome iii. Consult also F Rethoré, Condillac ou l'empirisme et le rationalisme (1864); L Dewaule, Condillac et la psychologie anglaise contemporaine (1891); histories of philosophy.

In Condillac's statue, a chapter in A Mind So Rare: The evolution of human consciousness, psychologist and cognitive neuroscientist Merlin Donald argues that Condillac was the first constructivist.

In the short story "Condillac's Statue, or Wrens in his Head", science fiction writer R. A. Lafferty brings the allegory of Condillac's statue to life, having Condillac build the statue in a park in the French countryside, and then slowly turning the statue's senses on one at a time.

title: Étienne Bonnot de Condillac