text: User talk:217.169.53.163

title: User talk:217.169.53.163


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Talk:Zhu Ziqing

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there is a nice picture at: http://www.xys.org/pages2/Zhu-ZQ.html but i am not sure about copyrights.

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title: Talk:Zhu Ziqing


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Hicklin test

The Hicklin test is a legal test for obscenity established by the English case R. v Hicklin (1868). At issue was the statutory interpretation of the word "obscene" in the Obscene Publications Act 1857, which authorized the destruction of obscene books. The court held that all material tending "to deprave and corrupt those whose minds are open to such immoral influences" was obscene, regardless of its artistic or literary merit.

History
The modern English law of obscenity began with the Obscene Publications Act 1857, also known as Lord Campbell's Act. Lord Campbell, the Chief Justice of Queen's Bench, introduced the bill, which provided for the seizure and summary disposition of obscene and pornographic materials. The Act also granted authority to issue search warrants for premises suspected of housing such materials.

Regina v Hicklin involved one Henry Scott, who resold copies of an anti-Catholic pamphlet entitled "The Confessional Unmasked: shewing the depravity of the Romish priesthood, the iniquity of the Confessional, and the questions put to females in confession." When the pamphlets were ordered destroyed as obscene, Scott appealed the order to the court of quarter sessions. Benjamin Hicklin, the official in charge of such orders as Recorder, revoked the order of destruction. Hicklin held that Scott's purpose had not been to corrupt public morals but to expose problems within the Catholic Church; hence, Scott's intention was innocent. The authorities appealed Hicklin's reversal, bringing the case to the consideration of the Court of Queen's Bench.

Chief Justice Cockburn, on April 29, 1868, reinstated the order of the lower court, holding that Scott's intention was immaterial if the publication was obscene in fact. Justice Cockburn reasoned that the Obscene Publications Act allowed banning of a publication if it had a "tendency… to deprave and corrupt those whose minds are open to such immoral influences, and into whose hands a publication of this sort may fall." Hicklin therefore allowed portions of a suspect work to be judged independently of context. If any portion of a work was deemed obscene, the entire work could be outlawed.

In United States
Adoption of obscenity laws in the United States was largely by the efforts of Anthony Comstock, whose intense lobbying led to the passage in 1873 of an anti-obscenity statute, known as the Comstock Act. Comstock was appointed postal inspector to enforce the new law. Twenty-four states passed similar prohibitions on materials distributed within the states. The law criminalized not only sexually explicit material, but also material dealing with birth control and abortion. Although lower courts in the U.S. had used the Hicklin standard sporadically since 1868, it was not until 1879, when prominent federal judge Samuel Blatchford upheld the obscenity conviction of D. M. Bennett using Hicklin, that the constitutionality of the Comstock Law became firmly established. In 1896, the Supreme Court in Rosen v. United States,, adopted the Hicklin test as the appropriate test of obscenity.

However, in 1933, the Hicklin test ended on the federal level when, in United States v. One Book Called Ulysses, 72 F.2d 705 (2d Cir. 1933), Judge John Woolsey found Ulysses to not be obscene. Avoiding the Hicklin test, he said instead that in evaluating obscenity, a court must consider (1) the work as a whole, not just selected passages that could be interpreted out of context; (2) the effect on an average, rather than the most susceptible person; and (3) contemporary community standards. This ruling refuted those who argued against adult possession of material that could hypothetically corrupt a child.

Finally, in 1957, the Supreme Court ruled in Roth v. United States, that the Hicklin test was inappropriate. In Roth, Justice Brennan, writing for the majority, noted that some American courts had adopted the Hicklin standard, but that later decisions more commonly relied upon the question of "whether to the average person, applying contemporary community standards, the dominant theme of the material taken as a whole appeals to prurient interest." This Roth test became essentially the new definition of obscenity in the United States.

title: Hicklin test


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Triesenberg

Triesenberg is a municipality in Liechtenstein with a population of 2,636. Its area of 29.69 km2 makes it the largest municipality in Liechtenstein. The center of the municipality rests at an elevation of 884 –.

History
Triesenberg is noted for its distinct dialect, dating from the influence of Walser migrants in the Middle Ages, who arrived in the region early in the 14th century. This dialect is actively promoted by the municipality. The existence of this dialect is one evidence of remarkable linguistic diversity within the small Principality, as it is spoken alongside the Standard German and Alemannic dialect common to the country.

Geography
The municipality includes eight villages: Gaflei, Malbun, Masescha, Rotenboden, Silum, Steg, Sücka and Wangerberg. Malbun is the only ski-resort village in the country, located not far from the border with Austria (Vorarlberg).

Notable people

 * Maria Foser, Deputy Government Councillor for Social Affairs
 * Franz Schädler (1917–1941), alpine skier
 * Gustav Schädler (1883 Triesenberg – 1961 in Vaduz), politician and a former Prime Minister of Liechtenstein
 * Marco Schädler (born 1964 in Triesenberg) composer, he studied music at the conservatory in Feldkirch, Austria.

title: Triesenberg


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Rotations in 4-dimensional Euclidean space

In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.

In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment $[0, π]$ except where mentioned or clearly implied by the context otherwise.

A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.

Geometry of 4D rotations
Four-dimensional rotations are of two types: simple rotations and double rotations.

Simple rotations
A simple rotation $R$ about a rotation centre $O$ leaves an entire plane $A$ through $O$ (axis-plane) fixed. Every plane $B$ that is completely orthogonal to $A$ intersects $A$ in a certain point $P$. For each such point $P$ is the centre of the 2D rotation induced by $R$ in $B$. All these 2D rotations have the same rotation angle $α$.

Half-lines from $O$ in the axis-plane $A$ are not displaced; half-lines from $O$ orthogonal to $A$ are displaced through $α$; all other half-lines are displaced through an angle less than $α$.

Double rotations
For each rotation $R$ of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes $A$ and $B$ each of which is invariant and whose direct sum $A ⊕ B$ is all of 4-space. Hence $R$ operating on either of these planes produces an ordinary rotation of that plane. For almost all $R$ (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles $α$ in plane $A$ and $β$ in plane $B$ – both assumed to be nonzero – are different. The unequal rotation angles $α$ and $β$ satisfying $−π < α$, $β < π$ are almost uniquely determined by $A$. Assuming that 4-space is oriented, then the orientations of the 2-planes $B$ and $A$ can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (${α, β }$), $B$ is sometimes termed a "double rotation".

In that case of a double rotation, $R$ and $A$ are the only pair of invariant planes, and half-lines from the origin in $B$, $R$ are displaced through $A$ and $B$ respectively, and half-lines from the origin not in $A$ or $B$ are displaced through angles strictly between $α$ and $β$.

Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from $A$ are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through $B$ are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.

Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation $α$, and take an orientation-consistent ordered set ${−α, −β }$ of mutually perpendicular half-lines at $β$ (denoted as $O$) such that $O$ and $R$ span an invariant plane, and therefore $O$ and $OUXYZ$ also span an invariant plane. Now assume that only the rotation angle $OU$ is specified. Then there are in general four isoclinic rotations in planes $OX$ and $OY$ with rotation angle $OZ$, depending on the rotation senses in $α$ and $OUX$.

We make the convention that the rotation senses from $OYZ$ to $α$ and from $OUX$ to $OYZ$ are reckoned positive. Then we have the four rotations ${α, −β }$, ${−α, β }$, $α ≠ β$ and $OU, OX, OY, OZ$. $R_{1} = (+α, +α)$ and $R_{2} = (−α, −α)$ are each other's inverses; so are $R_{3} = (+α, −α)$ and $R_{4} = (−α, +α)$. As long as $OU$ lies between 0 and π, these four rotations will be distinct.

Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic. Left- and right-isoclinic rotations are represented respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below.

The four rotations are pairwise different except if $R_{1}$ or $R_{2}$. The angle $R_{3}$ corresponds to the identity rotation; $R_{4}$ corresponds to the central inversion, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic.

Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However, when another isoclinic rotation $OX$ with its own axes $OY$, $OZ$, $α$, $R′$ is selected, then one can always choose the order of $OU′$, $OX′$, $OY′$, $OZ′$ such that $U′$ can be transformed into $X′$ by a rotation rather than by a rotation-reflection (that is, so that the ordered basis $Y′$, $Z′$, $OUXYZ$, $OU′X′Y′Z′$ is also consistent with the same fixed choice of orientation as $OU′$, $OX′$, $OY′$, $OZ′$). Therefore, once one has selected an orientation (that is, a system $OU$ of axes that is universally denoted as right-handed), one can determine the left or right character of a specific isoclinic rotation.

Group structure of SO(4)
SO(4) is a noncommutative compact 6-dimensional Lie group.

Each plane through the rotation centre $OX$ is the axis-plane of a commutative subgroup isomorphic to SO(2). All these subgroups are mutually conjugate in SO(4).

Each pair of completely orthogonal planes through $OY$ is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to SO(2) × SO(2).

These groups are maximal tori of SO(4), which are all mutually conjugate in SO(4). See also Clifford torus.

All left-isoclinic rotations form a noncommutative subgroup $α = 0$ of SO(4), which is isomorphic to the multiplicative group $α = π$ of unit quaternions. All right-isoclinic rotations likewise form a subgroup $α = 0$ of SO(4) isomorphic to $α = π$. Both $S^{3}_{L}$ and $S^{3}$ are maximal subgroups of SO(4).

Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product $S^{3}_{R}$ with normal subgroups $S^{3}$ and $S^{3}_{L}$; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to $S^{3}_{R}$. (This is not SO(4) or a subgroup of it, because $S^{3}_{L} × S^{3}_{R}$ and $S^{3}_{L}$ are not disjoint: the identity $OZ$ and the central inversion $S^{3}_{R}$ each belong to both $S^{3}$ and $S^{3}_{L}$.)

Each 4D rotation $OUXYZ$ is in two ways the product of left- and right-isoclinic rotations $S^{3}_{R}$ and $−I$. $S^{3}_{L}$ and $S^{3}_{R}$ are together determined up to the central inversion, i.e. when both $A_{L}$ and $A_{R}$ are multiplied by the central inversion their product is $O$ again.

This implies that $A_{L}$ is the universal covering group of SO(4) — its unique double cover — and that $A_{R}$ and $A_{L}$ are normal subgroups of SO(4). The identity rotation $O$ and the central inversion $A_{R}$ form a group $S^{3}_{L} × S^{3}_{R}$ of order 2, which is the centre of SO(4) and of both $S^{3}_{L}$ and $S^{3}_{R}$. The centre of a group is a normal subgroup of that group. The factor group of C2 in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of $−I$3L by C2 and of $C_{2}$3R by C2 are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by $S^{3}_{L}$3L and of SO(4) by $S^{3}_{R}$3R are each isomorphic to SO(3).

The topology of SO(4) is the same as that of the Lie group SO(3) × Spin(3) = SO(3) × SU(2), namely the space $$\mathbb{P}^3 \times \mathbb{S}^3$$ where $$\mathbb{P}^3$$ is the real projective space of dimension 3 and $$\mathbb{S}^3$$ is the 3-sphere. However, it is noteworthy that, as a Lie group, SO(4) is not a direct product of Lie groups, and so it is not isomorphic to SO(3) × Spin(3) = SO(3) × SU(2).

Special property of SO(4) among rotation groups in general
The odd-dimensional rotation groups do not contain the central inversion and are simple groups.

The even-dimensional rotation groups do contain the central inversion $S$ and have the group C2 = { $S$, $S$ } as their centre. For even n ≥ 6, SO(n) is almost simple in that the factor group SO(n)/C2 of SO(n) by its centre is a simple group.

SO(4) is different: there is no conjugation by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. Reflections transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of all isometries with fixed point $I$ the distinct subgroups $S$ and $−I$ are conjugate to each other, and so cannot be normal subgroups of O(4). The 5D rotation group SO(5) and all higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any two isoclinic rotations through the same angle are conjugate. The set of all isoclinic rotations is not even a subgroup of SO(2$I$), let alone a normal subgroup.

Algebra of 4D rotations
SO(4) is commonly identified with the group of orientation-preserving isometric linear mappings of a 4D vector space with inner product over the real numbers onto itself.

With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order orthogonal matrices with determinant +1.

Isoclinic decomposition
A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows:

Let
 * $$A=

\begin{pmatrix} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \\ \end{pmatrix} $$ be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called associate matrix
 * $$M=

\frac{1}{4} \begin{pmatrix} a_{00}+a_{11}+a_{22}+a_{33} & +a_{10}-a_{01}-a_{32}+a_{23} & +a_{20}+a_{31}-a_{02}-a_{13} & +a_{30}-a_{21}+a_{12}-a_{03} \\ a_{10}-a_{01}+a_{32}-a_{23} & -a_{00}-a_{11}+a_{22}+a_{33} & +a_{30}-a_{21}-a_{12}+a_{03} & -a_{20}-a_{31}-a_{02}-a_{13} \\ a_{20}-a_{31}-a_{02}+a_{13} & -a_{30}-a_{21}-a_{12}-a_{03} & -a_{00}+a_{11}-a_{22}+a_{33} & +a_{10}+a_{01}-a_{32}-a_{23} \\ a_{30}+a_{21}-a_{12}-a_{03} & +a_{20}-a_{31}+a_{02}-a_{13} & -a_{10}-a_{01}-a_{32}-a_{23} & -a_{00}+a_{11}+a_{22}-a_{33} \end{pmatrix} $$

$A$ has rank one and is of unit Euclidean norm as a 16D vector if and only if $A$ is indeed a 4D rotation matrix. In this case there exist real numbers $−I$ and $S^{3}_{L}$ such that

 * $$M=

\begin{pmatrix} ap &   aq  &   ar  &   as  \\ bp &   bq  &   br  &   bs  \\ cp &   cq  &   cr  &   cs  \\ dp &   dq  &   dr  &   ds \end{pmatrix} $$

and
 * $$(ap)^2 + \cdots + (ds)^2 = \left(a^2 + b^2 + c^2 + d^2\right)\left(p^2 + q^2 + r^2 + s^2\right) = 1.$$

There are exactly two sets of $S^{3}_{R}$ and $N$ such that $a, b, c, d$ and $p, q, r, s$. They are each other's opposites.

The rotation matrix then equals
 * $$\begin{align}A&=

\begin{pmatrix} ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\ bp+aq-dr+cs&-bq+ap+ds+cr&-br+as-dp-cq&-bs-ar-dq+cp\\ cp+dq+ar-bs&-cq+dp-as-br&-cr+ds+ap+bq&-cs-dr+aq-bp\\ dp-cq+br+as&-dq-cp-bs+ar&-dr-cs+bp-aq&-ds+cr+bq+ap\end{pmatrix} \\&= \begin{pmatrix} a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end{pmatrix} \begin{pmatrix} p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end{pmatrix} .\end{align} $$

This formula is due to Van Elfrinkhof (1897).

The first factor in this decomposition represents a left-isoclinic rotation, the second factor a right-isoclinic rotation. The factors are determined up to the negative 4th-order identity matrix, i.e. the central inversion.

Relation to quaternions
A point in 4-dimensional space with Cartesian coordinates $a, b, c, d$ may be represented by a quaternion $p, q, r, s$.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion $a^{2} + b^{2} + c^{2} + d^{2} = 1$. In matrix-vector language this is

\begin{pmatrix} u'\\x'\\y'\\z' \end{pmatrix} = \begin{pmatrix} a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end{pmatrix} \begin{pmatrix} u\\x\\y\\z \end{pmatrix}. $$

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion $p^{2} + q^{2} + r^{2} + s^{2} = 1$, which is in matrix-vector form

\begin{pmatrix} u'\\x'\\y'\\z' \end{pmatrix} = \begin{pmatrix} u\\x\\y\\z \end{pmatrix} \begin{pmatrix} p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end{pmatrix}. $$

In the preceding section it is shown how a general 4D rotation is split into left- and right-isoclinic factors.

In quaternion language Van Elfrinkhof's formula reads
 * $$u' + x'i + y'j + z'k = (a + bi + cj + dk)(u + xi + yj + zk)(p + qi + rj + sk), $$

or, in symbolic form,
 * $$P' = Q_\mathrm{L} P Q_\mathrm{R}.\, $$

According to the German mathematician Felix Klein this formula was already known to Cayley in 1854.

Quaternion multiplication is associative. Therefore,
 * $$P' = \left(Q_\mathrm{L} P\right) Q_\mathrm{R} = Q_\mathrm{L} \left(P Q_\mathrm{R}\right),\,$$

which shows that left-isoclinic and right-isoclinic rotations commute.

The eigenvalues of 4D rotation matrices
The four eigenvalues of a 4D rotation matrix generally occur as two conjugate pairs of complex numbers of unit magnitude. If an eigenvalue is real, it must be ±1, since a rotation leaves the magnitude of a vector unchanged. The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions $(u, x, y, z)$ and $P = u + xi + yj + zk$ are equal in magnitude and have the same sign. If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of $Q_{L} = a + bi + cj + dk$ and $Q_{R} = p + qi + rj + sk$ are not equal then all eigenvalues are complex, and the rotation is a double rotation.

The Euler–Rodrigues formula for 3D rotations
Our ordinary 3D space is conveniently treated as the subspace with coordinate system 0XYZ of the 4D space with coordinate system UXYZ. Its rotation group SO(3) is identified with the subgroup of SO(4) consisting of the matrices

\begin{pmatrix} 1 & \,\, 0 & \,\, 0 & \,\, 0 \\ 0 & a_{11} & a_{12} & a_{13} \\ 0 & a_{21} & a_{22} & a_{23} \\ 0 & a_{31} & a_{32} & a_{33} \end{pmatrix}. $$

In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to $Q_{L}$, $Q_{R}$, $Q_{L}$, $Q_{R}$, or in quaternion representation: $p = a$. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations

\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} = \begin{pmatrix} a^2 + b^2 - c^2 - d^2 & 2(bc - ad)& 2(bd + ac) \\ 2(bc + ad) & a^2 - b^2 + c^2 -d^2 & 2(cd - ab) \\ 2(bd - ac) & 2(cd + ab) & a^2 - b^2 - c^2 + d^2 \end{pmatrix}, $$

which is the representation of the 3D rotation by its Euler–Rodrigues parameters: $q = −b$.

The corresponding quaternion formula $r = −c$, where $s = −d$, or, in expanded form:
 * $$x'i + y'j + z'k = (a + bi + cj + dk)(xi + yj + zk)(a - bi - cj - dk)$$

is known as the Hamilton–Cayley formula.

Hopf coordinates
Rotations in 3D space are made mathematically much more tractable by the use of spherical coordinates. Any rotation in 3D can be characterized by a fixed axis of rotation and an invariant plane perpendicular to that axis. Without loss of generality, we can take the $I$-plane as the invariant plane and the $O$-axis as the fixed axis. Since radial distances are not affected by rotation, we can characterize a rotation by its effect on the unit sphere (2-sphere) by spherical coordinates referred to the fixed axis and invariant plane:
 * $$\begin{align}

x &= \sin\theta \cos \phi \\ y &= \sin\theta \sin \phi \\ z &= \cos\theta \end{align}$$

Because $Q_{R} = Q_{L}′ = Q_{L}^{−1}$, the points (x,y,z) lie on the unit 2-sphere. A point with angles $a, b, c, d$, rotated by an angle $M$ about the $A$-axis, becomes the point with angles $P′ = QPQ^{−1}$. While hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by Hopf coordinates $Q = Q_{L}$, which are a set of three angular coordinates specifying a position on the 3-sphere. For example:
 * $$\begin{align}

u &= \cos\xi_1 \sin\eta \\ z &= \sin\xi_1 \sin\eta \\ x &= \cos\xi_2 \cos\eta \\ y &= \sin\xi_2 \cos\eta \end{align}$$

Because $x^{2} + y^{2} + z^{2} = 1$, the points lie on the 3-sphere.

In 4D space, every rotation about the origin has two invariant planes which are completely orthogonal to each other and intersect at the origin, and are rotated by two independent angles ${θ_{0}, φ_{0} }$ and ${θ_{0}, φ_{0} + φ }$. Without loss of generality, we can choose, respectively, the $xy$- and $z$-planes as these invariant planes. A rotation in 4D of a point ${ξ_{1}, η, ξ_{2} }$ through angles $u^{2} + x^{2} + y^{2} + z^{2} = 1$ and $ξ_{1}$ is then simply expressed in Hopf coordinates as $ξ_{2}$.

Visualization of 4D rotations

Every rotation in 3D space has a fixed axis unchanged by rotation. The rotation is completely specified by specifying the axis of rotation and the angle of rotation about that axis. Without loss of generality, this axis may be chosen as the $φ$-axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 3D space, the spherical coordinates ${ξ_{10}, η_{0}, ξ_{20} }$ may be seen as a parametric expression of the 2-sphere. For fixed $z$ they describe circles on the 2-sphere which are perpendicular to the $uz$-axis and these circles may be viewed as trajectories of a point on the sphere. A point $ξ_{1}$ on the sphere, under a rotation about the $xy$-axis, will follow a trajectory $ξ_{2}$ as the angle $z$ varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: ${ξ_{10} + ξ_{1}, η_{0}, ξ_{20} + ξ_{2} }$, with $θ$ being an "angular velocity".

Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a point). The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the $z$- and $z$-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 4D space, the Hopf angles ${θ, φ }$ parameterize the 3-sphere. For fixed $φ$ they describe a torus parameterized by ${θ_{0}, φ_{0} }$ and ${θ_{0}, φ_{0} + φ }$, with $φ = ωt$ being the special case of the Clifford torus in the $ω$- and $uz$-planes. These tori are not the usual tori found in 3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be stereographically projected onto the whole Euclidean 3D-space, and these tori are then seen as the usual tori of revolution. It can be seen that a point specified by ${ξ_{1}, η, ξ_{2} }$ undergoing a rotation with the $xy$- and $η$-planes invariant will remain on the torus specified by $ξ_{1}$. The trajectory of a point can be written as a function of time as $ξ_{2}$ and stereographically projected onto its associated torus, as in the figures below. In these figures, the initial point is taken to be $η = π⁄4$, i.e. on the Clifford torus. In Fig. 1, two simple rotation trajectories are shown in black, while a left and a right isoclinic trajectory is shown in red and blue respectively. In Fig. 2, a general rotation in which ${ξ_{10}, η_{0}, ξ_{20} }$ and $η_{0}$ is shown, while in Fig. 3, a general rotation in which ${ξ_{10} + ω_{1}t, η_{0}, ξ_{20} + ω_{2}t }$ and ${0, π⁄4, 0 }$ is shown.

Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus described above is depicted in its rectangular (wrapping) form.

Generating 4D rotation matrices
Four-dimensional rotations can be derived from Rodrigues' rotation formula and the Cayley formula. Let $xy$ be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix $uz$ can be uniquely decomposed as
 * $$A =\theta_1 A_1+\theta_2 A_2$$

into two skew-symmetric matrices $ω_{1} = 1$ and $ω_{2} = 5$ satisfying the properties $ω_{1} = 5$, $ω_{2} = 1$ and $A_{1}$, where $A_{2}$ and $A_{1}A_{2} = 0$ are the eigenvalues of $uz$. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices $A_{1}^{3} = −A_{1}$ and $A_{2}^{3} = −A_{2}$ by Rodrigues' rotation formula and the Cayley formula.

Let $xy$ be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues
 * $$\left\{\theta_1 i,-\theta_1 i,\theta_2 i,-\theta_2 i : {\theta_1}^2 + {\theta_2}^2 > 0\right\}.$$

Then $A$ can be decomposed as
 * $$A=\theta_1 A_1+\theta_2 A_2$$

where $∓θ_{1}i$ and $∓θ_{2}i$ are skew-symmetric matrices satisfying the properties
 * $$A_1 A_2=A_2 A_1=0, \qquad {A_1}^3=-A_1, \quad \text{and} \quad {A_2}^3=-A_2.$$

Moreover, the skew-symmetric matrices $A_{1}$ and $A_{2}$ are uniquely obtained as
 * $$A_1 = \frac{{\theta_2}^2 A + A^3}{\theta_1 \left({\theta_2}^2 - {\theta_1}^2\right)}$$

and
 * $$A_2 = \frac{{\theta_1}^2 A + A^3}{\theta_2 \left({\theta_1}^2 - {\theta_2}^2\right)}.$$

Then,
 * $$R = e^A = I + \sin\theta_1 A_1 + \left(1-\cos\theta_1\right) {A_1}^2 + \sin\theta_2 A_2 + \left(1-\cos\theta_2\right) {A_2}^2$$

is a rotation matrix in $A_{1}$, which is generated by Rodrigues' rotation formula, with the set of eigenvalues
 * $$\left\{e^{\theta_1 i}, e^{-\theta_1 i}, e^{\theta_2 i}, e^{-\theta_2 i}\right\}.$$

Also,
 * $$R = (I+A)(I-A)^{-1} = I+\frac{2\theta_1}{1+{\theta_1}^2}A_1+\frac{2{\theta_1}^2}{1+{\theta_1}^2}{A_1}^2+\frac{2\theta_2}{1+{\theta_2}^2}A_2+\frac{2{\theta_2}^2}{1+{\theta_2}^2}{A_2}^2$$

is a rotation matrix in $A_{2}$, which is generated by Cayley's rotation formula, such that the set of eigenvalues of $A$ is,
 * $$\left\{\frac{\left(1+\theta_1 i\right)^2}{1+{\theta_1}^2},\frac{\left(1-\theta_1 i\right)^2}{1+{\theta_1}^2},\frac{\left(1+\theta_2 i\right)^2}{1+{\theta_2}^2},\frac{\left(1-\theta_2 i\right)^2}{1+{\theta_2}^2}\right\}.$$

The generating rotation matrix can be classified with respect to the values $A_{1}$ and $A_{2}$ as follows:
 * 1) If $E^{4}$ and $E^{4}$ or vice versa, then the formulae generate simple rotations;
 * 2) If $θ_{1}$ and $θ_{2}$ are nonzero and $θ_{1} = 0$, then the formulae generate double rotations;
 * 3) If $θ_{2} ≠ 0$ and $θ_{1}$ are nonzero and $θ_{2}$, then the formulae generate isoclinic rotations.


text: Category:Guatemalan sportspeople

title: Category:Guatemalan sportspeople


text: Category:Heavy metal singers

title: Category:Heavy metal singers


text:
Fernando Gabeira

Fernando Paulo Nagle Gabeira (born 17 February 1941) is a Brazilian politician, author and journalist. He was a federal representative for the state of Rio de Janeiro from 1995 to 2011.

He is known for his 1979 book O Que É Isso, Companheiro? (loosely translated, What Was That, Man?). The book tells of the armed resistance to the military dictatorship in Brazil, and describes the 1969 kidnapping of American ambassador Charles Burke Elbrick, in which Gabeira took part as a member of MR8, an armed group fighting the military dictatorship then ruling Brazil.

The book was made into a movie in 1997, titled Four Days in September. The movie was nominated for many awards, including an Oscar as Best Foreign Language Film at the 70th Academy Awards. (Dutch film Karakter won the category.)

Because of his role in kidnapping its ambassador, the United States considered Gabeira to be a terrorist and refused him a visa to visit the United States. Over the years, Gabeira requested and was denied a visa three times. In 1998, he declared a visa denial would speak not of him, but as a U.S. act against Brazilian sovereignty. Folha de São Paulo reported that Gabeira's lack of contrition may have contributed to the continuing refusals. Years later, in a 2009 Ragga interview, Gabeira called the kidnapping a mistake, and acknowledged positive views of the United States, saying Brazil had much it could learn from and admire about the U.S.

Biography
Fernando Paulo Nagle Gabeira was born February 17, 1941, in Juiz de Fora, Minas Gerais, to Lebanese immigrant parents.

Gabeira started his journalism career young, contributing to magazines and newspapers in Juiz de Fora while still in high school. His interest in politics was evident from the start.

After a short stay in Belo Horizonte in the 1960s, Gabeira moved to Rio de Janeiro, where he worked for the Jornal do Brasil, a prominent carioca newspaper. With the military coup in April 1964, Gabeira got involved in the armed resistance to the new regime.

In 1969, the Ação Libertadora Nacional, the main armed leftwing organization in Brazil, and the MR8, to which Gabeira belonged, orchestrated the kidnapping of U.S. Ambassador Charles Burke Elbrick in Rio de Janeiro. Gabeira helped plan and execute the attack. To begin negotiations, the kidnappers demanded their manifesto be printed in local media, which was done. They also threatened to murder Elbrick if their demands were not met within 48 hours. Elbrick was released after 78 hours in exchange for the release of 15 political prisoners imprisoned by the military dictatorship, who were exiled to Mexico and then Cuba.

In 1970, Gabeira was jailed in São Paulo. He tried to escape but was shot in the attempt, striking his back and perforating a kidney as well as his stomach and liver. In June of that year, he and 39 others were released in exchange for the release of German ambassador Ehrenfried von Holleben, who had been kidnapped by terrorists just as Elbrick had been. The 40 were banned from Brazil and sent into exile.

During his exile in the 1970s, Gabeira lived in several countries including Chile, Sweden, and Italy. In Stockholm, where he spent most of his exile, he earned a degree in anthropology from Stockholm University and worked as a journalist as well as a train conductor.

After Brazil passed a general amnesty for those convicted of political crimes in 1979, Gabeira returned to Brazil. He worked as a journalist and writer, arguing for an end to the military dictatorship. In this period he wrote O Que É Isso, Companheiro?, describing the 1969 kidnapping of Ambassador Elbrick, as well as his role in it.

After 1985, Gabeira shifted his attention to minority rights and the environment. He was one of the founding members of the Green Party of Brazil, and repeatedly voiced his ideological support for progressive causes like the legalization of marijuana, marriage equality, and the legalization of abortion.

Gabeira left the Green Party in 2001 to join the Workers' Party. He remained affiliated with the Workers' Party for only two years, however, and later rejoined the Green Party in 2005. He ran for mayor of the city of Rio de Janeiro in 2008, but was narrowly defeated by Eduardo Paes in a runoff (49.3% – 50.7%). He lost a bid to become governor of the state of Rio de Janeiro in 2010.

The knitted swimsuit affair
Just after Gabeira's return from exile, a photo of him wearing a very small knitted swimsuit on Ipanema beach became a national scandal. Years later, Gabeira revealed that the suit was actually the bottom of one of his cousin Leda Nagle's bikinis.

Literary career
In 1979, Gabeira wrote the book O Que É Isso, Companheiro?, about his participation in the armed struggle against the military regime in Brazil (1964–1985) and his subsequent exile in Europe. The book won the Jabuti Literature Prize in the biography/memoirs category in 1980 and was made into the film Four Days in September by filmmaker Bruno Barreto in 1997.

In 1980, he released O Crepúsculo do Macho, a continuation of O Que É Isso, Companheiro?.

In 1981, he launched Entradas e Bandeiras, a book in which he chronicles his return to Brazil and his abandonment of Marxist ideology, shifting to fight for issues such as ecology, pleasure, and sexual freedom. In the same year, he launched Hôte da utopia, in which he deepens his new ideological positioning.

In 1982, he launched Sinais de Vida on Planeta Minas, in which he chronicles feminist struggles against the conservative society of Brazil's Minas Gerais state, through the biographies of five mineiras, including Dona Beja and Ângela Diniz.

In 1987, he published Goiânia, Rua 57 — o Nuclear na Terra do Sol, in which he reported on a radiological accident that occurred in Goiânia.

In 2000, he launched A Maconha, in which he discussed the decriminalization of marijuana, its therapeutic functions, the societal role it plays, etc.

In 2006, he launched Navegação na Neblina, under a Creative Commons license, addressing the 2005 "Bloodsuckers Scandal," in which elected and governing officials stole money meant to buy ambulances.

In 2012, he released Onde Está Tudo Aquilo Agora.

In 2017, he released Democracia Tropical: Caderno de um Aprendiz, in which he recounted Dilma Rousseff's impeachment, as well as an overview of the prior thirty years of Brazilian democracy.

Personal life
Gabeira is the son of Paulo Gabeira and Isabel Nagle, both Lebanese immigrants. His surname was aportuguesado (Portuguese language equivalent to anglicization) from the transliterations Jabara or Gebara (in Arabic: جبارة).

During his exile, Gabeira married his fellow militant Vera Sílvia Magalhães. They ended their relationship while living in Sweden.

After his return to Brazil, Gabeira married Yamê Reis, a fashion designer. He and Reis have two daughters. The couple divorced in 1999.

Currently, Gabeira is married to businesswoman Neila Tavares.

title: Fernando Gabeira


text:
File:Need for Speed Underground 2.PNG

Summary
Screenshot from the video game Need for Speed: Underground 2.

Licensing: Fair use for Need for Speed: Underground 2
The author asserts the image qualifies for fair use for the following reasons:
 * 1) EA Games and its affiliates have not released screenshots into public domain.
 * 2) The image is needed to critically illustrate a computer or video game and improve the quality of the articles in question, per Article development.
 * 3) The image is not intended to hinder the copyright holders' ability to market and sell its games.
 * 4) This image is only to be used on the articles related to the Need for Speed: Underground 2.

title: File:Need for Speed Underground 2.PNG


text:
Wikipedia:Collaboration of the week/Heraldry

Heraldry

 * Nominated on 21:16, 25 July 2005 (UTC); needs 5 votes by August 1st.

I am nominating this article (which I have worked a bit on) because I feel that it can be improved through collaboration. For example, heraldic terms could be clairified for those not familiar with heraldry, and more images of armorial bearings could be added.

Support:
 * 1) Mb1000 21:16, 25 July 2005 (UTC)

Comments:
 * Should be moved to the Article Improvement Drive section. KNewman 12:34, July 26, 2005 (UTC)


text:
The King Beyond the Gate

The King Beyond The Gate is a fantasy novel by British writer David Gemmell. It was published in 1985. It was the second book published by Gemmell, after Legend, published a year earlier. The book is set in the same fictional world as Legend, that of the Drenai, but is not a sequel in the usual sense as the events of the two books take place around a century apart. Thus the main protagonists of Legend are long since dead and play little part in The King Beyond the Gate, other than passing mentions. This set a precedent for the entire Drenai series, in which very few characters appear in more than one novel, the gaps between novels sometimes running to centuries, giving a more epic, historical flavour to the series.

Synopsis
The King Beyond the Gate takes place a century after Legend: Druss The Legend is gone, the Earl of bronze is a legend himself. The Drenai are now under the rule of Ceska, an usurper of the throne who over time has become a mad power-hungry emperor. Ceska rules with an iron fist, with the Joinings, and the Dark Templars as his tools of terror. The Joinings are horrifying werebeasts, made by the merging of man and animal, an ancient machine of the old times making the merging of two species possible, the result, a towering beast of such savagery that one alone becomes an army itself. The Dark Templars are priests of darkness, with power beyond that of normal men. Thus it falls to men of courage and those that choose the path of light to overthrow a tyrant--- Tenaka Khan and his companions must forge alliances and unlikely friendships to overcome a time of horror and turmoil.

Issues
Following on from other Gemmell works where the main characters are powerful men past their prime, the three characters of Tenaka, Ananaïs and Decado are all three less able to rely on physical endurance and dumb luck, and instead use experience and bluff to survive. This story also offers a much grittier view of the Drenai world compared to Legend. The “Joinings” are new to the saga, melding of animals and men (and women).

title: The King Beyond the Gate


text: User talk:213.84.230.131

title: User talk:213.84.230.131


text: User talk:195.93.60.106

title: User talk:195.93.60.106


text: User talk:86.134.15.218

title: User talk:86.134.15.218


text:
User talk:IDude 101

Welcome from Redwolf24
Welcome!

Hello, and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. We as a community are glad to have you and thank you for creating a user account! Here are a few good links for newcomers:
 * The Five Pillars of Wikipedia
 * How to edit a page
 * Editing, policy, conduct, and structure tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style
 * Merging, redirecting, and renaming pages
 * If you're ready for the complete list of Wikipedia documentation, there's also Topical index.

Yes some of the links appear a bit boring at first, but they are VERY helpful if you ever take the time to read them.

Remember to place any articles you create into a category so we don't get orphans.

I hope you enjoy editing here and being a Wikipedian! By the way, please be sure to sign your name on Talk and vote pages using four tildes (~~~~) to produce your name and the current date, or three tildes (~~~) for just your name. If you have any questions, see the help pages, add a question to the village pump or ask me on my Talk page. Again, welcome.

Redwolf24 (Talk) 21:18, 25 July 2005 (UTC)

P.S. I like messages :-P

Featured articles
Hi IDude. Welcome to Wikipedia.

You recently added Saint Lucia Jazz Festival to Featured articles. If you read the top of that page, you'll notice it says that you can't just add them there, but you have to go through a nomination/peer review process at http://en.wikipedia.org/wiki/Wikipedia:Featured_article_candidates. I've removed it, but feel free to nominate it. I will caution you, though, it doesn't meet several of the criteria for what a feature article should be. →Raul654 04:48, July 26, 2005 (UTC)

St Lucia Jazz Festival
Hello Ian,

Thanks for the information that you originally contributed to start the St Lucia Jazz Festival page which was factually correct. I hope you don't mind that I edited it. The purpose of my edit was to correct spelling and grammatical errors rather than the substance of what you wrote. Also, I changed the order of the text so that it reads better.

Let me know if you agree with the changes or if you have any further comments. After all, Wikipedia is a collaborative effort - by reviewing and contributing to each other's work we will eventually produce an informative, accurate and entertaining information resource.

I'm glad that you've joined the list of St Lucian Wikipedians that I set up. Hopefully before long, many other Lucians will join us.

All the best.

James Vivenot 15:30, 28 July 2005 (UTC)

Ian,

I've noticed that Wikipedia are in the process of re-implementing the userlists of Wikipedians into a set of Categories, so I've set up Category:Saint Lucian Wikipedians.

To add yourself to this list, insert the following on your user page:

Cheers!

Vivenot 16:02, 29 July 2005 (UTC)

The Transporter, too?
I'm sorry I didn't look more closesly for your TT2 article, its a shame the two overlapped. I do think however, that the article should be at The Transporter 2 because that is the actual title of the movie. Alternatively, both of our articles could point to The Transporter, too? which is utterly random and nondescript for thos trying to find it, but funny nonetheless.

re: List of Cancelled Shows on FOX
It's not obligatory policy to inform the original editor of an article that you're nominating said article for deletion (especially being that Wikipedia is an open source encyclopedia, and any number of editors could have co-authored a single article). It wasn't done out of spite or to upset anyone, but it was just a superfluous article (all shows are cancelled, and every show on the list was already listed at List of Cancelled Shows on FOX is already at List of programs broadcast by Fox, where it properly belongs (practically all shows are cancelled at some point and time, with the exception of programs such as the news). --FuriousFreddy 23:10, 18 August 2005 (UTC)

Your message on User talk:209.68.71.39
Hi there, I found your User page when on the lookout for Wikipedians with "high stress". I'm not sure if you're currently feeling stressed by goings on on Wikipedia, but if you are, and you want to talk about it, feel free to drop me a note. I did notice that you recently left a message for User:209.68.71.39 that was a little hostile. I think this was because you were concerned about vandalism. This is of course quite understandable. Do note however, that IPs are often shared by many unconnected people, and many perfectly innocent users may read the messages you leave on that talk page and assume it was directed at them. I wonder if you might consider not leaving such messages in future, but instead use the standard ones available for warning vandals (see WP:VIP). Kind regards enceph  alon  15:47, 9 October 2005 (UTC)

Image copyright problem with Image:Piece Of Shit Car.gif
Thanks for uploading Image:Piece Of Shit Car.gif. However, the image may soon be deleted unless we can determine the copyright holder and copyright status. The Wikimedia Foundation is very careful about the images included in Wikipedia because of copyright law (see Wikipedia's Copyright policy).

The copyright holder is usually the creator, the creator's employer, or the last person who was transferred ownership rights. Copyright information on images is signified using copyright templates. The three basic license types on Wikipedia are open content, public domain, and fair use. Find the appropriate template in Image copyright tags and place it on the image page like this:.

Please signify the copyright information on any other images you have uploaded or will upload. Remember that images without this important information can be deleted by an administrator. If you have any questions, feel free to contact me. Thank you. Longhair 11:51, 3 February 2006 (UTC)

Your Userpage
A bit rude, eh? Some people might be very offended. The Neokid - Wikihalo Guiding Director talk 17:07, 8 February 2006 (UTC)

Infobox
There is a consensus discussion on Wikipedia talk:WikiProject Aircraft on adopting a non-specifications summary infobox for aircraft articles. Your comments would be appreciated. Thanks! - Emt147 Burninate!  18:34, 24 April 2006 (UTC)

Aviation Newsletter delivery
The March 2007 issue of the Aviation WikiProject newsletter has been published. You may read the newsletter, change the format in which future issues will be delivered to you, or unsubscribe from this notification by following the link. Thank you. Trevor MacInnis (Contribs) 16:12, 9 March 2007 (UTC)

Meetup
Meetup/Tampa -- You're invited! Hires an editor (talk) 15:11, 1 August 2008 (UTC)

An exciting opportunity to get involved!
As a member of the Aviation WikiProject or one of its subprojects, you may be interested in testing your skills in the Aviation Contest! I created this contest, not to pit editor against editor, but to promote article improvement and project participation and camraderie. Hopefully you will agree with its usefulness. Sign up here, read up on the rules here, and discuss the contest here. The first round of the contest may not start until September 1st-unless a large number of editors signup and are ready to compete immediately! Since this contest is just beginning, please give feedback here, or let me know what you think on my talkpage. -  Trevor  MacInnis   contribs  00:41, 22 August 2009 (UTC)

Speedy deletion of MetroPCS
Hello IDude 101, this is a message from an automated bot to inform you that the page you created on September 11 2005, MetroPCS, has been marked for speedy deletion by User:AOC25 (page has mainspace links, and more than 49 edits). This has been done because the page is a blatant advert that would require a substantial rewrite in order to become an encyclopedia article (see CSD). If you think the tag was placed in error, please add " " to the page text, and edit the to explain why the page should not be deleted. If you have a question about this bot, please ask it at User talk:SDPatrolBot II. If you have a question for the user who tagged the article, see User talk:AOC25. Thanks, - SDPatrolBot II (talk) on behalf of 08:45, 4 December 2010 (UTC)

Nomination of Mad TV recurring characters for deletion
A discussion is taking place as to whether the article Mad TV recurring characters is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The article will be discussed at Articles for deletion/Mad TV recurring characters until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion notice from the top of the article. StewdioMACK (talk) 14:19, 21 January 2015 (UTC)

title: User talk:IDude 101


text:
User talk:Freeq

Welcome from Redwolf24
Welcome!

Hello, and welcome to Wikipedia. Thank you for your contributions. I hope you like the place and decide to stay. We as a community are glad to have you and thank you for creating a user account! Here are a few good links for newcomers:
 * The Five Pillars of Wikipedia
 * How to edit a page
 * Editing, policy, conduct, and structure tutorial
 * Picture tutorial
 * How to write a great article
 * Naming conventions
 * Manual of Style
 * Merging, redirecting, and renaming pages
 * If you're ready for the complete list of Wikipedia documentation, there's also Topical index.

Yes some of the links appear a bit boring at first, but they are VERY helpful if you ever take the time to read them.

Remember to place any articles you create into a category so we don't get orphans.

I hope you enjoy editing here and being a Wikipedian! By the way, please be sure to sign your name on Talk and vote pages using four tildes (~~~~) to produce your name and the current date, or three tildes (~~~) for just your name. If you have any questions, see the help pages, add a question to the village pump or ask me on my Talk page. Again, welcome.

Redwolf24 (Talk) 21:19, 25 July 2005 (UTC)

P.S. I like messages :-P

Image:May31 26.JPG
Please stop adding nonsense to Wikipedia. It is considered vandalism. If you want to experiment, please use the sandbox. Thank you. jni 09:23, 26 July 2005 (UTC)

title: User talk:Freeq