text:
Metatron

Metatron (Mishnaic Hebrew: Meṭāṭrōn),   or Matatron,  is an angel in Judaism, Christianity, and Islam. Metatron is mentioned three times in the Talmud, in a few brief passages in the Aggadah, the Targum, and in mystical Kabbalistic texts within Rabbinic literature. The figure forms one of the traces for the presence of dualist proclivities in the otherwise monotheistic visions of both the Tanakh and later Christian doctrine. In Rabbinic literature, he is sometimes portrayed as serving as the celestial scribe. The name Metatron is not mentioned in the Torah or the Bible, and how the name originated is a matter of debate. In Islamic tradition, he is also known as Mīṭaṭrūn (ميططرون), the angel of the veil.

In Jewish apocrypha, early Kabbalah, and rabbinic literature, Metatron is the name that Enoch received after his transformation into an angel.

Etymology
Numerous etymologies have been proposed to account for the name Metatron, but there is no consensus, and its precise origin is unknown. Some scholars, such as Philip Alexander, believe that if the name Metatron originated in Hekhalot literature and Merkabah texts such as 3 Enoch, then it may have been a magical word like and.

Hugo Odeberg Adolf Jellinek and Marcus Jastrow suggest the name may have originated from either (מטרא, ) or the verb  (ממטר, 'to guard' or 'to protect'). An early derivation of this can be seen in, where Enoch is clothed in light and is the guardian of the souls ascending to heaven. Odeberg also suggested that the name Metatron might have been adopted from the Old Persian name Mithra. Citing Wiesner, he drew up several parallels that appeared to link Mithra and Metatron based on their positions in heaven and duties.

Another hypothesis would derive Metatron from a combination of two Greek words, μετά (meta, meaning 'after') and θρóνος (thronos, meaning 'throne'), which, taken together, would suggest the idea of 'one who serves behind the throne' or 'one who occupies the throne next to the throne of glory'. The primary arguments against this etymology are that Metatron's function as a servant of the celestial throne emerges only later in the traditions regarding him, and θρóνος itself is not attested as a word in Talmudic literature.

A connection with the word σύνθρονος used as 'co-occupant of the divine throne', has been advanced by some scholars; This, like the above etymology, is not found in any source materials. It is supported by Saul Lieberman and Peter Schäfer, who give further reasons why this might be a viable etymology. The Latin word metator ('messenger, guide, leader, measurer') had been suggested by Eleazar of Worms (c. 1165–c. 1230), Nachmanides, and brought to light again by Hugo Odeberg. When transliterated into the Hebrew language, it is given as מטיטור or מיטטור. Gershom Scholem argues that there is no data to justify the conversion of metator to Metatron. Philip Alexander also suggests this as a possible origin of Metatron, stating that the word metator also occurs in Greek as – a word for an officer in the Roman army who acted as a forerunner. Using this etymology, Alexander suggests the name may have come about as a description of 'the angel of the Lord who led the Israelites through the wilderness: acting like a Roman army metator guiding the Israelites on their way'.

Other ideas include μέτρον (, 'a measure'). Charles Mopsik believes that the name Metatron may be related to the sentence from Genesis 5:24, "Enoch walked with God, then he was no more, because God took him". The LXX version of the Hebrew word לָקַ֥ח ("took") is μετέθηκεν.

In the entry entitled "Paradigmata" in his study "'The Written' as the Vocation of Conceiving Jewishly", John W. McGinley gives an accounting of how this name functions in the Bavli's version of "four entered pardes".

Origins
In the Hellenistic period, mention of a second divine figure, either beside Yahweh or beneath him, occurs in several Jewish texts, mostly apocryphal. These Jewish traditions implying a divine dualism were most frequently associated with Enoch. In the rabbinic period they center on 'Metatron', often in the context of debates over the heretical doctrine of 'two powers in heaven'. Ultimately, these ideas appear to go back to differing interpretations of the heavenly enthronement passages at Exodus 24:10, Daniel 7:9. and perhaps even Ezekiel 1:26. These different interpretations later came to distinguish what was orthodox from what was heretical in Judaism.

Among the pseudepigrapha 1 Enoch: Book of Parables presents two figures: the son of man and Enoch. At first, these two characters seem to be separate entities. Enoch views the son of man enthroned in Heaven. Later, however, they prove to be the same. Many scholars believe that the final chapters in the Book of Parables are a later addition. Others think they are not and that the son of man is Enoch's heavenly double, similar to the Prayer of Joseph, where Jacob is depicted as an angel. The Book of Daniel displays two similar characters: the Ancient of Days and the one like a man. Parts of the text in Daniel are Aramaic and may have been changed in translation. The Septuagint reads that the son of man came as the Ancient of Days. All other translations say the son of man gained access to the Ancient of Days and was brought before that one.

The identification of Enoch with Metatron in 3 Enoch, where the name first appears, is not explicitly made in the Talmud, although it does refer to a Prince of the World who was young but now is old. However, some of the earliest kabbalists assumed the connection. There also seem to be two Metatrons, one spelled with six letters (מטטרון), and one spelled with seven (מיטטרון). The former may be the transformed Enoch, Prince of the Countenance within the divine palace; the latter, the Primordial Metatron, an sephira "persona" of the "Cause of Causes", specifically the tenth and last persona, identified with the earthly Shekhina. Furthermore, the Merkabah text identifies the Ancient of Days from the Book of Daniel as Metatron.

Scholem's scholastic analysis
Many scholars see a discontinuity between how Enoch is portrayed in the early Enoch literature and how Metatron is portrayed. Scholars commonly see the character of Metatron as being based on an amalgam of Jewish literature; in addition to Enoch, Michael, Melchizedek, and Yahoel among others are seen as influences.

Scholem argues two streams of thought influenced Metatron's character: One linking Metatron with Enoch, and one fusing different obscure entities and mythic motifs. Scholem argues that this second tradition was originally separate but later fused with the Enoch tradition. He points to texts where this second Metatron is a primordial angel and referred to as Metatron Rabbah. Scholem theorizes that the two Hebrew spellings of Metatron's name represent these two separate traditions. In his view, the second Metatron is linked to Yahoel. Scholem also links Yahoel with Michael. In the Apocalypse of Abraham, Yahoel is assigned duties normally reserved for Michael. Yahoel's name is commonly seen as a substitute for the Ineffable Name.

In 2 Enoch, Enoch is assigned titles commonly used by Metatron such as "the Youth, the Prince of the Presence and the Prince of the World." Enoch is not called the Lesser Yahweh. In 3 Enoch, Metatron is called the Lesser Yahweh. This raises a problem since the name Metatron does not seem to be directly related to the name of God, Yahweh. Scholem proposes that this is because the Lesser Yahweh is a reference to Yahoel. In Maaseh Merkabah, the text reasons that Metatron is called the Lesser YHWH because in Hebrew gematria, Metatron is numerically equivalent to another name of God. Scholem does not find this convincing. Scholem points to the fact that both Yahoel and Metatron were known as the Lesser YHWH. In 3 Enoch 48D1, Metatron is called both Yahoel Yah and Yahoel. In addition to being one of the seventy names of Metatron from 3 Enoch 48D, Yahoel and Metatron are also linked in Aramaic incantation bowl inscriptions.

Talmud
The Babylonian Talmud mentions Metatron by name in three places: Hagigah 15a, Sanhedrin 38b, and Avodah Zarah 3b.

Hagigah 15a describes Elisha ben Abuyah in Paradise seeing Metatron sitting down (an action that is not done in the presence of God). Elishah ben Abuyah therefore looks to Metatron as a deity and says heretically: "Perhaps there are, God forbid, two powers in Heaven!" The rabbis explain that Metatron had permission to sit because of his function as the Heavenly Scribe, writing down the deeds of Israel. The Talmud states that it was proved to Elisha that Metatron could not be a second deity by the fact that Metatron received 60 "strokes with fiery rods" to demonstrate that Metatron was not a god, but an angel, and could be punished.

In Sanhedrin 38b, one of the tells Rabbi Idith that Metatron should be worshiped because he has a name like his master. Rabbi Idith uses the same passage Exodus 23:21 to show that Metatron was an angel and not a deity and thus should not be worshiped. Furthermore, as an angel, Metatron has no power to pardon transgressions nor was he to be received even as a messenger of forgiveness.

In Avodah Zarah 3b, the Talmud hypothesizes as to how God spends his day. It is suggested that in the fourth quarter of the day God sits and instructs the school children, while in the preceding three quarters Metatron may take God's place or God may do this among other tasks.

Yevamot 16b records an utterance, "I have been young; also I have been old" found in Psalm 37:25. The Talmud here attributes this utterance to the "chief angel" and "prince of the world", whom the rabbinic tradition identifies as Metatron.

Kirkisani
The tenth century Karaite scholar Jacob Qirqisani believed that rabbinic Judaism was the heresy of Jeroboam of the Kingdom of Israel. He quoted a version of Sanhedrin 38b, which he claimed contained a reference to the "lesser YHWH". Scholem suggests that the name was deliberately omitted from later copies of the Talmud. Extra-talmudic mystical texts such as Sefer Hekhalot do speak of a "lesser YHWH", apparently deriving the concept from Exodus 23:21, which mentions an angel of whom God says "my name [understood as YHWH, the usual divine Proper Name] is in him".

Merkabah, Zohar and other mystical writings
Metatron also appears in the Pseudepigrapha including, and most prominently in the Hebrew Merkabah Book of Enoch, also called 3 Enoch or Sefer Hekhalot (Book of [the Heavenly] Palaces). The book describes the link between Enoch, son of Jared (great-grandfather of Noah) and his transformation into the angel Metatron.

Metatron says, "He [the Holy One] called me, 'The lesser YHVH' in the presence of his whole household in the height, as it is written, 'my name is in him.'" (12:5, Alexander's translation.) The narrator of this book, supposedly Rabbi Ishmael, tells how Metatron guided him through Heaven and explained its wonders. 3 Enoch presents Metatron in two ways: as a primordial angel (9:2–13:2) and as the transformation of Enoch after he was assumed into Heaven.

"And Enoch walked with God: and he was not; for God took him."

"This Enoch, whose flesh was turned to flame, his veins to fire, his eye-lashes to flashes of lightning, his eye-balls to flaming torches, and whom God placed on a throne next to the throne of glory, received after this heavenly transformation the name Metatron."

Metatron "the Youth", a title previously used in 3 Enoch, where it appears to mean "servant". It identifies him as the angel that led the people of Israel through the wilderness after their exodus from Egypt (again referring to Exodus 23:21), and describes him as a heavenly priest.

In the later Ecstatic Kabbalah, Metatron is a messianic figure.

The Zohar describes Metatron as the "King of the angels". and associates the concept of Metatron with that of the divine name. Zohar commentaries such as the by Moses ben Jacob Cordovero explain the Zohar as meaning that Metatron as the head of Yetzira This corresponds closely with Maimonides' description of the Talmudic "Prince of the World", traditionally associated with Metatron, as the core "Active Intellect."

The Zohar describes several biblical figures as metaphors for Metatron. Examples are Enoch, Joseph, Eliezer, Joshua, and others. The Zohar finds the word youth used to describe Joseph and Joshua a hint that the figures are a metaphor to Metatron, and also the concept of servant by Eliezer as a reference to Metatron. The Staff of Moses is also described by the Zohar as a reference to Metatron. The Zohar also states that the two tets in (the Biblical Hebrew term for the phylacteries) are a reference to Metatron. The Zohar draws distinction between Metatron and Michael. While Michael is described repeatedly in the Zohar as the figure represented by the High Priest, Metatron is represented by the structure of the tabernacle itself.

Apocalyptic texts
In the Apocalypse of Zerubbabel, Metatron is not identified as Enoch. Instead he is identified as the archangel Michael. The text also records that Metatron in gematria is the equivalent of. While he also appears in other apocalyptic writings, he is most prominent in the Apocalypse of Zerubbabel. In these writings, he plays the role of heavenly interlocutor delivering knowledge about the coming messianic age.

Islam
The earliest account of Metatron within Islamic scriptures might derive directly from the Quran itself. Uzair, according to Surah 9:30–31 venerated as a Son of God by Jews, commonly interpreted as an Arabic transliteration of the Hebrew name of the prophet Ezra, who was also identified with Enoch and Metatron in Merkabah Mysticism. Islamic heresiologists repeatedly accused Jews for venerating an angel as a lesser god (or an Incarnation of God), especially for celebrating Rosh Hashanah. The name itself is attested early in Islam by Al-Kindi and Al-Masudi. Al-Suyuti identifies him as the angel of the veil and only he knows about that which lies beyond. He is also frequently mentioned in the magical works by Ahmad al-Buni, who describes Metatron as wearing a crown and a lance, probably constituting the Staff of Moses. In other magical practises, he is invoked to ward off evil jinn, devils, sorcerers and other magical threats.

Ibn Hazm mentions that Jews, although regarding Metatron as an angel, would celebrate Metatron as a lesser god ten days each year, perhaps a reference to Rosh Hashanah in connection with Merkabah mysticism that Metatron took part on the creation of the world.

In popular culture

 * In Thomas Pynchon's novel Gravity's Rainbow (1973), a man visiting a dominatrix in an insane asylum hears an inmate proclaiming himself to be Metatron. Critics regard this scene as parodying Kabbalistic accounts of the ascent of the soul.
 * Metatron appears as an angel and the spokesperson of God in the novel Good Omens (1990) and its 2019 TV adaptation, played in the series by Derek Jacobi.
 * Metatron appears in Phillip Pullman's novel series, His Dark Materials (1995–2000), as the main antagonist. He was originally a human named Enoch before becoming a high ranking angel and Regent of the Kingdom of Heaven. In the 2019 BBC/HBO TV adaptation, he is portrayed by Alex Hassell.
 * Metatron appears in the 1999 movie Dogma as an angel and the voice of God, played by Alan Rickman.
 * Guitarist Carlos Santana said in 2000 that he has been in contact with Metatron since 1994, and that the angel gives him messages.
 * Metatron appears in the TV series Supernatural (2005) as the scribe of God, played by Curtis Armstrong.
 * Both the lyrics and title of the second track on The Mars Volta's 2008 album, The Bedlam in Goliath, refer to Metatron.

title: Metatron


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Talk:Tim Wirth

Untitled
Removed from the article:

In 1990, while a U.S. Senator, Wirth said,

 * "We've got to ride the global warming issue. Even if the theory is wrong, we will be doing the right thing in terms of economic and environmental policy." (source)

Until someone provides a primary source for this claim, and preferably more context, this should not be in the article. I have also contacted the UN Foundation (of which Wirth is now president) for feedback on this quote.--Eloquence 20:37 Jan 24, 2003 (UTC)

 * It's in Michael Fumento's book. Is that good enough? --Uncle Ed 15:21, 6 Aug 2004 (UTC)

 * No, it's not - it is not a primary source. It does not state where Wirth supposedly said this and smacks of propaganda. However, since Wirth himself never bothered to reply, I won't bother to remove the statement.--Eloquence*

 * The earliest mention of that quote that I found was from the National Journal quoted by Debra Saunders on RealClearPolitics: "Sen. Timothy E. Wirth, D-Colo., said it in 1988, as the National Journal reported. "What we've got to do in energy conservation is (to) try to ride the global warming issue. Even if the theory of global warming is wrong, to have approached global warming as if it is real means energy conservation, so we will be doing the right thing anyway in terms of economic policy and environmental policy."  From:  http://www.realclearpolitics.com/articles/2007/09/if_global_warming_isnt_real.html --DrDelos (talk) 00:13, 26 June 2011 (UTC)

 * I found another reference from sciencenews.org back to National Journal; As reported in REPORTS - Less Burning, No Tears

By ROCHELLE L. STANFIELD, National Journal © National Journal Group Inc. Saturday, Aug. 13, 1988 ScienceNews, Friday, July 18th, 2008 1:59 pm by Fred Houlihan http://www.sciencenews.org/view/generic/id/34264/title/Comment --DrDelos (talk) 02:17, 26 June 2011 (UTC)

The economy is a wholly owned subsidiary of the environment
This statement has been quoted, paraphrased, co-opted numerous times over the years in speeches, articles and books by people of note in the environmental movement. The earliest utterance I can find of it is from a speech given by Tim Wirth to the National Press Club on July 25, 1994. Here it is in context from that speech: "Stated in the jargon of the business world, the economy is a wholly owned subsidiary of the environment. All economic activity is dependent on the environment and its underlying resource base." --Undersecretary of State Tim Wirth, concluding a speech to the National Press Club, July 25, 1994

My source for the transcript of this speech: http://dosfan.lib.uic.edu/ERC/briefing/dispatch/1994/html/Dispatchv5no30.html This is, or appears to be, from Dispatch Magazine, an organ of the United Stated State Department, wherein his speech to the National Press Club is printed as a "dispatch".

I would like some feedback before I add this quote to his Wikipedia biography. Two specific questions: 1. Does this pass muster as a "primary source"? 2. Can anyone trace this the origin of this quote to a source preceding Tim Wirth's above-noted speech? My main motivation is to find the source of this quote, whether it be Tim Wirth or someone earlier. radioman 64.149.38.135 (talk) 16:15, 14 August 2008 (UTC)

"Home State"?
Small point, but the bio says "returned to his home state" in '70, ran for congress in '74. But he came to Colorado in '70 and the bio details say he was born in Santa Fe NM. This doesn't seem consistent with the "home state" part. I remember when Tim ran for Congress in '74 and there was a fair bit of grumbling about "carpetbaggers" and questioning whether someone who had only lived in Colorado for four years was an appropriate candidate. Can anyone say why this counts Colorado as Wirth's home state? — Charlie (Colorado) (talk) 17:24, 15 February 2010 (UTC)

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title: Talk:Tim Wirth


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Laurence Tomson

Laurence Tomson (1539 – 29 March 1608) was an English politician, author, and translator. He acted as the personal secretary of Sir Francis Walsingham, the secretary of state to Elizabeth I of England.

Tomson revised both the text and the annotations of the New Testament of the Geneva Bible. His revised edition appeared in 1576. Tomson was a Calvinist, and his annotations reflect that system of theology.

Life
He was born in Northamptonshire, and was admitted a demy of Magdalen College, Oxford, in 1553. He graduated B.A. in 1559, was elected a fellow of his college, and commenced M.A. in 1564. He accompanied Sir Thomas Hoby on an embassy to France in 1566, and resigned his Magdalen fellowship in 1569.

Between 1575 and 1587 Tomson represented Weymouth and Melcombe Regis in the House of Commons, and he was member for Downton in 1588–9. In 1582 he was in attendance at court at Windsor Castle. According to his epitaph he travelled in Sweden, Russia, Denmark, Germany, Italy, and France; was conversant with twelve languages; and at one period gave public lectures on the Hebrew language at Geneva. He was employed in political affairs by Sir Francis Walsingham, after whose death he retired into private life.

Tomson died on 29 March 1608, and was buried in the chancel of the church at Chertsey, Surrey, where a black marble was erected to his memory with a curious Latin inscription.

Works
His works are:

 * ‘An Answere to certeine Assertions and Obiections of M. Fecknam,’ London [1570].
 * ‘Statement of Advantages to be obtained by the establishment of a Mart Town in England,’ 1572, manuscript in the Public Record Office.
 * ‘The New Testament … translated out of Greeke by T. Beza. Whereunto are adjoyned brief summaries of doctrine … by the said T. Beza: and also short expositions … taken out of the large annotations of the foresaid authour and J. Camerarius. By P. Loseler, Villerius. Englished by L. Tomson,’ London, 1576, dedicated to Sir Francis Walsingham; again 1580, 1587, 1596. Several other editions of Tomson's revision of the Genevan version of the New Testament were published in the whole Bible. It was based on the work of Pierre Loiseleur.
 * ‘A Treatise of the Excellencie of a Christian Man, and how he may be knowen. Written in French. ... Whereunto is adioyned a briefe description of the life and death of the said authour (set forth by P. de Farnace). ... Translated into English,’ London, 1576, 1577, 1585, dedicated to Mrs. Ursula Walsingham.
 * ‘Sermons of J. Calvin on the Epistles of S. Paule to Timothie and Titus … Translated,’ London, 1579.
 * ‘Propositions taught and mayntained by Mr. R[ichard] Hooker. The same briefly confuted by L. T. in a private letter’ (Harleian MS. 291, f. 183).
 * ‘Treatise on the matters in controversy between the Merchants of the Hanze Towns and the Merchants Adventurers,’ 1590, a Latin manuscript in the Public Record Office.
 * ‘Mary, the Mother of Christ: her tears,’ London, 1596.
 * ‘Brief Remarks on the State of the Low Countries’ (Cottonian MS., Galba D vii. f. 163).

title: Laurence Tomson


text:
Orthogonal group

In mathematics, the orthogonal group in dimension $n$, denoted $O(n)$, is the group of distance-preserving transformations of a Euclidean space of dimension $n$ that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of $n × n$ orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

The orthogonal group in dimension $n$ has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted $SO(n)$. It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see $SO(2)$, $SO(3)$ and $SO(4)$. The other component consists of all orthogonal matrices of determinant $−1$. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.

By extension, for any field $F$, an $n × n$ matrix with entries in $F$ such that its inverse equals its transpose is called an orthogonal matrix over $F$. The $n × n$ orthogonal matrices form a subgroup, denoted $O(n, F)$, of the general linear group $GL(n, F)$; that is $$\operatorname{O}(n, F) = \left\{Q \in \operatorname{GL}(n, F) \mid Q^\mathsf{T} Q = Q Q^\mathsf{T} = I \right\} .$$

More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.

Name
The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space $E$ of dimension $n$, the elements of the orthogonal group $O(n)$ are, up to a uniform scaling (homothecy), the linear maps from $E$ to $E$ that map orthogonal vectors to orthogonal vectors.

In Euclidean geometry
The orthogonal $O(n)$ is the subgroup of the general linear group $GL(n, R)$, consisting of all endomorphisms that preserve the Euclidean norm; that is, endomorphisms $g$ such that $$\|g(x)\| = \|x\|.$$

Let $E(n)$ be the group of the Euclidean isometries of a Euclidean space $S$ of dimension $n$. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point $x ∈ S$ is the subgroup of the elements $g ∈ E(n)$ such that $g(x) = x$. This stabilizer is (or, more exactly, is isomorphic to) $O(n)$, since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural group homomorphism $p$ from $E(n)$ to $O(n)$, which is defined by
 * $$p(g)(y-x) = g(y)-g(x),$$

where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by $g$ (for details, see ).

The kernel of $p$ is the vector space of the translations. So, the translations form a normal subgroup of $E(n)$, the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to $O(n)$.

Moreover, the Euclidean group is a semidirect product of $O(n)$ and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of $O(n)$.

Special orthogonal group
By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that
 * $$ Q Q^\mathsf{T} = I. $$

It follows from this equation that the square of the determinant of $Q$ equals $1$, and thus the determinant of $Q$ is either $1$ or $−1$. The orthogonal matrices with determinant $1$ form a subgroup called the special orthogonal group, denoted $SO(n)$, consisting of all direct isometries of $O(n)$, which are those that preserve the orientation of the space.

$SO(n)$ is a normal subgroup of $O(n)$, as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group $\{−1, +1\}$. This implies that the orthogonal group is an internal semidirect product of $SO(n)$ and any subgroup formed with the identity and a reflection.

The group with two elements $\{±I\}$ (where $I$ is the identity matrix) is a normal subgroup and even a characteristic subgroup of $O(n)$, and, if $n$ is even, also of $SO(n)$. If $n$ is odd, $O(n)$ is the internal direct product of $SO(n)$ and $\{±I\}$.

The group $SO(2)$ is abelian (whereas $SO(n)$ is not abelian when $n > 2$). Its finite subgroups are the cyclic group $C_{k}$ of $k$-fold rotations, for every positive integer $k$. All these groups are normal subgroups of $O(2)$ and $SO(2)$.

Canonical form
For any element of $O(n)$ there is an orthogonal basis, where its matrix has the form
 * $$\begin{bmatrix}

\begin{matrix} R_1 &       & \\ & \ddots & \\ &       & R_k \end{matrix} & 0 \\ 0 & \begin{matrix} \pm 1 &       & \\ & \ddots & \\ &       & \pm 1 \end{matrix}\\ \end{bmatrix},$$ where the matrices $R_{1}, ..., R_{k}$ are 2-by-2 rotation matrices, that is matrices of the form
 * $$\begin{bmatrix}a&b\\-b&a\end{bmatrix},$$

with $a2 + b2 = 1$.

This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to $1$.

The element belongs to $SO(n)$ if and only if there are an even number of $−1$ on the diagonal. A pair of eigenvalues $−1$ can be identified with a rotation by $π$ and a pair of eigenvalues $+1$ can be identified with a rotation by $0$.

The special case of $n = 3$ is known as Euler's rotation theorem, which asserts that every (non-identity) element of $SO(3)$ is a rotation about a unique axis–angle pair.

Reflections
Reflections are the elements of $O(n)$ whose canonical form is
 * $$\begin{bmatrix}-1&0\\0&I\end{bmatrix},$$

where $I$ is the $(n − 1) × (n − 1)$ identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

In dimension two, every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angle $θ$ is the product of two reflections whose axes form an angle of $θ / 2$.

A product of up to $n$ elementary reflections always suffices to generate any element of $O(n)$. This results immediately from the above canonical form and the case of dimension two.

The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.

The reflection through the origin (the map $v ↦ −v$) is an example of an element of $O(n)$ that is not a product of fewer than $n$ reflections.

Symmetry group of spheres
The orthogonal group $O(n)$ is the symmetry group of the $(n − 1)$-sphere (for $n = 3$, this is just the sphere) and all objects with spherical symmetry, if the origin is chosen at the center.

The symmetry group of a circle is $O(2)$. The orientation-preserving subgroup $SO(2)$ is isomorphic (as a real Lie group) to the circle group, also known as $U(1)$, the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number $exp(φ i) = cos(φ) + i sin(φ)$ of absolute value $1$ to the special orthogonal matrix
 * $$\begin{bmatrix}

\cos(\varphi) & -\sin(\varphi) \\ \sin(\varphi) & \cos(\varphi) \end{bmatrix}.$$

In higher dimension, $O(n)$ has a more complicated structure (in particular, it is no longer commutative). The topological structures of the $n$-sphere and $O(n)$ are strongly correlated, and this correlation is widely used for studying both topological spaces.

Group structure
The groups $O(n)$ and $SO(n)$ are real compact Lie groups of dimension $n(n − 1) / 2$. The group $O(n)$ has two connected components, with $SO(n)$ being the identity component, that is, the connected component containing the identity matrix.

As algebraic groups
The orthogonal group $O(n)$ can be identified with the group of the matrices $A$ such that $ATA = I$. Since both members of this equation are symmetric matrices, this provides $n(n + 1) / 2$ equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.

This proves that $O(n)$ is an algebraic set. Moreover, it can be proved that its dimension is
 * $$\frac{n(n - 1)}{2} = n^2 - \frac{n(n + 1)}{2},$$

which implies that $O(n)$ is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, $O(n)$ has two irreducible components, that are distinguished by the sign of the determinant (that is $det(A) = 1$ or $det(A) = −1$). Both are nonsingular algebraic varieties of the same dimension $n(n − 1) / 2$. The component with $det(A) = 1$ is $SO(n)$.

Maximal tori and Weyl groups
A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to $T^{k}$ for some $k$, where $T = SO(2)$ is the standard one-dimensional torus.

In $O(2n)$ and $SO(2n)$, for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form
 * $$\begin{bmatrix}

R_1 &       & 0    \\ & \ddots &     \\ 0  &        & R_n \end{bmatrix},$$ where each $R_{j}$ belongs to $SO(2)$. In $O(2n + 1)$ and $SO(2n + 1)$, the maximal tori have the same form, bordered by a row and a column of zeros, and $1$ on the diagonal.

The Weyl group of $SO(2n + 1)$ is the semidirect product $$\{\pm 1\}^n \rtimes S_n$$ of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each $\{±1\}$ factor of $\{±1\}^{n}$ acts on the corresponding circle factor of $T × {1}$ by inversion, and the symmetric group $S_{n}$ acts on both $\{±1\}^{n}$ and $T × {1}$ by permuting factors. The elements of the Weyl group are represented by matrices in $O(2n) × \{±1\}$. The $S_{n}$ factor is represented by block permutation matrices with 2-by-2 blocks, and a final $1$ on the diagonal. The $\{±1\}^{n}$ component is represented by block-diagonal matrices with 2-by-2 blocks either
 * $$\begin{bmatrix}

1 & 0 \\   0 & 1  \end{bmatrix} \quad \text{or} \quad \begin{bmatrix} 0 & 1 \\   1 & 0  \end{bmatrix}, $$ with the last component $±1$ chosen to make the determinant $1$.

The Weyl group of $SO(2n)$ is the subgroup $$H_{n-1} \rtimes S_n < \{\pm 1\}^n \rtimes S_n$$ of that of $SO(2n + 1)$, where $H_{n−1} < \{±1\}^{n}$ is the kernel of the product homomorphism $\{±1\}^{n} → \{±1\}$ given by $$\left(\varepsilon_1, \ldots, \varepsilon_n\right) \mapsto \varepsilon_1 \cdots \varepsilon_n$$; that is, $H_{n−1} < \{±1\}^{n}$ is the subgroup with an even number of minus signs. The Weyl group of $SO(2n)$ is represented in $SO(2n)$ by the preimages under the standard injection $SO(2n) → SO(2n + 1)$ of the representatives for the Weyl group of $SO(2n + 1)$. Those matrices with an odd number of $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ blocks have no remaining final $−1$ coordinate to make their determinants positive, and hence cannot be represented in $SO(2n)$.

Low-dimensional topology
The low-dimensional (real) orthogonal groups are familiar spaces:
 * $O(1) = S^{0}$, a two-point discrete space
 * $SO(1) = {1}$ is $SO(2)$
 * $S^{1}$ is $SO(3)$
 * $RP^{3}$ is doubly covered by $SO(4)$.
 * $SU(2) × SU(2) = S^{3} × S^{3}$ is doubly covered by $n > 2$.

Fundamental group
In terms of algebraic topology, for $SO(n, R)$ the fundamental group of $Spin(n)$ is cyclic of order 2, and the spin group $n = 2$ is its universal cover. For $Spin(2)$ the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group $π_{k}(O)$ is the unique connected 2-fold cover).

Homotopy groups
Generally, the homotopy groups $S^{n}$ of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:
 * $$\operatorname{O}(0) \subset \operatorname{O}(1)\subset \operatorname{O}(2) \subset \cdots \subset O = \bigcup_{k=0}^\infty \operatorname{O}(k)$$

Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, $O(n + 1)$ is a homogeneous space for $O(n + 1)$, and one has the following fiber bundle:
 * $$\operatorname{O}(n) \to \operatorname{O}(n + 1) \to S^n,$$

which can be understood as "The orthogonal group $S^{n}$ acts transitively on the unit sphere $O(n) → O(n + 1)$, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower." Thus the natural inclusion $(n − 1)$ is $π_{k}(O(n + 1)) = π_{k}(O(n))$-connected, so the homotopy groups stabilize, and $n > k + 1$ for $Ω^{8}O ≅ O$: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain $O$, therefore the homotopy groups of $π_{k + 8}(O) = π_{k}(O)$ are 8-fold periodic, meaning $O$, and one need only to list the lower 8 homotopy groups:
 * $$\begin{align}

\pi_0 (O) &= \mathbf{Z} / 2\mathbf{Z}\\ \pi_1 (O) &= \mathbf{Z} / 2\mathbf{Z}\\ \pi_2 (O) &= 0\\ \pi_3 (O) &= \mathbf{Z}\\ \pi_4 (O) &= 0\\ \pi_5 (O) &= 0\\ \pi_6 (O) &= 0\\ \pi_7 (O) &= \mathbf{Z} \end{align}$$

Relation to KO-theory
Via the clutching construction, homotopy groups of the stable space $π_{k}(O) = π_{k + 1}(BO)$ are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: $KO = BO × Z = Ω^{−1}O × Z$. Setting $π_{0}$ (to make $π_{0}(O) = π_{0}(O(1)) = Z / 2Z$ fit into the periodicity), one obtains:
 * $$\begin{align}

\pi_0 (KO) &= \mathbf{Z}\\ \pi_1 (KO) &= \mathbf{Z} / 2\mathbf{Z}\\ \pi_2 (KO) &= \mathbf{Z} / 2\mathbf{Z}\\ \pi_3 (KO) &= 0\\ \pi_4 (KO) &= \mathbf{Z}\\ \pi_5 (KO) &= 0\\ \pi_6 (KO) &= 0\\ \pi_7 (KO) &= 0 \end{align}$$

Low-dimensional groups
The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
 * $O(2)$, from orientation-preserving/reversing (this class survives to $π_{1}(O) = π_{1}(SO(3)) = Z / 2Z$ and hence stably)
 * $SO(3) = RP^{3} = S^{3} / (Z / 2Z)$, which is spin comes from $π_{2}(O) = π_{2}(SO(3)) = 0$.
 * $π_{2}(SO(4))$, which surjects onto $π_{2}(G)$; this latter thus vanishes.

Lie groups
From general facts about Lie groups, $π_{3}(G)$ always vanishes, and $π_{0}(KO)$ is free (free abelian).

Vector bundles
$S^{0}$ is a vector bundle over $π_{0}(KO) = Z$, which consists of two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so $O$ is the dimension.

Loop spaces
Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of $O$ in terms of simpler-to-analyze homotopies of lower order. Using π0, $O/U$ and $KO = BO × Z$ have two components, $KSp = BSp × Z$ and $π_{0}(KO) = Z$ have countably many components, and the rest are connected.

Interpretation of homotopy groups
In a nutshell:
 * $π_{1}(KO) = Z / 2Z$ is about dimension
 * $π_{2}(KO) = Z / 2Z$ is about orientation
 * $π_{4}(KO) = Z$ is about spin
 * $R$ is about topological quantum field theory.

Let $R$ be any of the four division algebras $C$, $H$, $O$, $L_{R}$, and let $RP^{1}$ be the tautological line bundle over the projective line $[L_{R}]$, and $RP^{1} = S^{1}$ its class in K-theory. Noting that $CP^{1} = S^{2}$, $HP^{1} = S^{4}$, $OP^{1} = S^{8}$, $π_{1}(KO)$, these yield vector bundles over the corresponding spheres, and
 * $[L_{R}]$ is generated by $π_{2}(KO)$
 * $[L_{C}]$ is generated by $π_{4}(KO)$
 * $[L_{H}]$ is generated by $π_{8}(KO)$
 * $[L_{O}]$ is generated by $π_{0}(KO) ≅ π_{8}(KO) = Z$

From the point of view of symplectic geometry, $π_{1}(U/O)$ can be interpreted as the Maslov index, thinking of it as the fundamental group $U/O ≅ Ω^{7}(KO)$ of the stable Lagrangian Grassmannian as $π_{1}(U/O) = π_{1+7}(KO)$, so $p + q = n$.

Whitehead tower
The orthogonal group anchors a Whitehead tower:
 * $$\cdots \rightarrow \operatorname{Fivebrane}(n) \rightarrow \operatorname{String}(n) \rightarrow \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n)$$

which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are the spin group and the string group, and are preceded by the fivebrane group. The homotopy groups that are killed are in turn $\pi$0(O) to obtain SO from O, π1(O) to obtain Spin from SO, π3(O) to obtain String from Spin, and then π7(O) and so on to obtain the higher order branes.

Of indefinite quadratic form over the reals
Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension $n$, such a form can be written as the difference of a sum of $p$ squares and a sum of $q$ squares, with $1$. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with $p$ entries equal to $−1$, and $q$ entries equal to $(p, q)$. The pair $O(p, q)$ called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.

The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted $O(p, q) = O(q, p)$. Moreover, as a quadratic form and its opposite have the same orthogonal group, one has $O(n) = O(n, 0) = O(0, n)$.

The standard orthogonal group is $O(p, q)$. So, in the remainder of this section, it is supposed that neither $p$ nor $q$ is zero.

The subgroup of the matrices of determinant 1 in $SO(p, q)$ is denoted $O(p, q)$. The group $SO+(p, q)$ has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted $O(3, 1)$.

The group $3$ is the Lorentz group that is fundamental in relativity theory. Here the $1$ corresponds to space coordinates, and $C$ corresponds to the time coordinate.

Of complex quadratic forms
Over the field $x12 + ... + xn2$ of complex numbers, every non-degenerate quadratic form in $n$ variables is equivalent to $O(n, C)$. Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension $n$, and one associated orthogonal group, usually denoted $O(n, C)$. It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix.

As in the real case, $1$ has two connected components. The component of the identity consists of all matrices of determinant $O(n, C)$ in $SO(n, C)$; it is denoted $O(n, C)$.

The groups $SO(n, C)$ and $n(n − 1) / 2$ are complex Lie groups of dimension $C$ over $R$ (the dimension over $n ≥ 2$ is twice that). For $SO(n, C)$, these groups are noncompact. As in the real case, $n > 2$ is not simply connected: For $SO(n, C)$, the fundamental group of $SO(2, C)$ is cyclic of order 2, whereas the fundamental group of $Z$ is $Q(w) ≠ 0$.

Characteristic different from two
Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are congruent, that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group.

The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified  into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.

More precisely, Witt's decomposition theorem asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form $Q$ can be decomposed as a direct sum of pairwise orthogonal subspaces
 * $$V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W,$$

where each $Li$ is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of $Q$ to $Li$ has the form $$\textstyle\begin{bmatrix}0&1\\1&0\end{bmatrix}$$), and the restriction of $Q$ to $W$ is anisotropic (that is, $O(2n + 1, q)$ for every nonzero $w$ in $W$).

The Chevalley–Warning theorem asserts that, over a finite field, the dimension of $W$ is at most two.

If the dimension of $V$ is odd, the dimension of $W$ is thus equal to one, and its matrix is congruent either to $$\textstyle\begin{bmatrix}1\end{bmatrix}$$ or to $$\textstyle\begin{bmatrix}\varphi\end{bmatrix},$$ where $phi$ is a non-square scalar. It results that there is only one orthogonal group that is denoted $−1$, where $q$ is the number of elements of the finite field (a power of an odd prime).

If the dimension of $W$ is two and $–I$ is not a square in the ground field (that is, if its number of elements $q$ is congruent to 3 modulo 4), the matrix of the restriction of $Q$ to $W$ is congruent to either $I$ or $−1$, where $I$ is the 2×2 identity matrix. If the dimension of $W$ is two and $O^{+}(2n, q)$ is a square in the ground field (that is, if $q$ is congruent to 1, modulo 4) the matrix of the restriction of $Q$ to $W$ is congruent to $$\textstyle\begin{bmatrix}1&0\\0&\varphi\end{bmatrix},$$ $φ$ is any non-square scalar.

This implies that if the dimension of $V$ is even, there are only two orthogonal groups, depending whether the dimension of $W$ zero or two. They are denoted respectively $O^{−}(2n, q)$ and $O^{ε}(2, q)$.

The orthogonal group $2(q − ε)$ is a dihedral group of order $ε = ±$, where $O^{ε}(2, q)$. $$

When the characteristic is not two, the order of the orthogonal groups are
 * $$\left|\operatorname{O}(2n + 1, q)\right| = 2q^{n^2}\prod_{i=1}^{n}\left(q^{2i} - 1\right),$$
 * $$\left|\operatorname{O}^+(2n, q)\right| = 2q^{n(n-1)}\left(q^n-1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right),$$
 * $$\left|\operatorname{O}^-(2n, q)\right| = 2q^{n(n-1)}\left(q^n+ 1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right).$$

In characteristic two, the formulas are the same, except that the factor $AQAT = Q$ of $a2 – ωb2 = 1$ must be removed.

Dickson invariant
For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group $ac – ωbd = 0$ (integers modulo 2), taking the value $c2 – ωd2 = –ω$ in case the element is the product of an even number of reflections, and the value of 1 otherwise.

Algebraically, the Dickson invariant can be defined as $Fq$, where $c = εωb$ is the identity. Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is $d = εa$ to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant and usually has index 2 in $ε2 = 1$. When the characteristic of $a2 – ωb2 = 1$ is not 2, the Dickson Invariant is $ε = ±1$ whenever the determinant is $Fq$. Thus when the characteristic is not 2, $O^{+}(2, q)$ is commonly defined to be the elements of $Fq2$ with determinant $x = a + αb$. Each element in $y = a – αb$ has determinant $a = a1a2 + ωb1b2$. Thus in characteristic 2, the determinant is always $b = a1b2 + b1a2$.

The Dickson invariant can also be defined for Clifford groups and pin groups in a similar way (in all dimensions).

Orthogonal groups of characteristic 2
Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the hypoabelian groups, but this term is no longer used.)
 * Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the Witt index is 2. A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector $(a, b) ↦ a + αb$ takes a vector $Fq2$ to $O^{+}(2n, q)$ where $Fq$ is the bilinear form and $O^{–}(2n, q)$ is the quadratic form associated to the orthogonal geometry.  Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes $xq+1 = 1$ to $(q + 1)$.
 * The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since $q + 1$.
 * In odd dimensions $gq−1$ in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension $Fq2$. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension $\{1, −1\}$, acted upon by the orthogonal group.
 * In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm
The spinor norm is a homomorphism from an orthogonal group over a field $Fq2$ to the quotient group $Fq2$ (the multiplicative group of the field $2$ up to multiplication by square elements), that takes reflection in a vector of norm $|O(2n + 1, q)|$ to the image of $Z / 2Z$ in $0$.

For the usual orthogonal group over the reals, it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

Galois cohomology and orthogonal groups
In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomenon is concerned. The first point is that quadratic forms over a field can be identified as a Galois $D(f) = rank(I − f) modulo 2$, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the determinant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.
 * $$ 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 $$

Here $I$ is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from $−1$, which is simply the group $O(n, F )$ of $F$-valued points, to  $0$ is essentially the spinor norm, because $1$ is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from $SO(n, F )$ of the orthogonal group, to the $O(n, F )$ of the kernel of the spin covering. The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions.

Lie algebra
The Lie algebra corresponding to Lie groups $1$ and $O(n, F )$ consists of the skew-symmetric $±1$ matrices, with the Lie bracket $1$ given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by $$\mathfrak{o}(n, F)$$ or $$\mathfrak{so}(n, F)$$, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different $u$ are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension $v$, where $v + B(v, u)/Q(u) · u$, while in even dimension $B$, where $Q$.

Since the group $v$ is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ordinary representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups. (The projective representations of $v − 2·B(v, u)/Q(u) · u$ are just linear representations of the universal cover, the spin group Spin(n).) The latter are the so-called spin representation, which are important in physics.

More generally, given a vector space $I = −I$ (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form $2n + 1$, the special orthogonal Lie algebra consists of tracefree endomorphisms $$\varphi$$ which are skew-symmetric for this form ($$(\varphi A, B) + (A, \varphi B) = 0$$). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors $2n$. The correspondence is given by:
 * $$v\wedge w \mapsto (v,\cdot)w - (w,\cdot)v$$

This description applies equally for the indefinite special orthogonal Lie algebras $$\mathfrak{so}(p, q)$$ for symmetric bilinear forms with signature $2n$.

Over real numbers, this characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.

Related groups
The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions $F$ and $F^{×} / (F^{×})^{2}$ are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, $F$ is the Lagrangian Grassmannian.

Lie subgroups
In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:
 * $$\mathrm{O}(n) \supset \mathrm{O}(n - 1)$$ – preserve an axis
 * $$\mathrm{O}(2n) \supset \mathrm{U}(n) \supset \mathrm{SU}(n)$$ – $n$ are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; $n$ also preserves a complex orientation.
 * $$\mathrm{O}(2n) \supset \mathrm{USp}(n)$$
 * $$\mathrm{O}(7) \supset \mathrm{G}_2$$

Lie supergroups
The orthogonal group $F^{×} / (F^{×})^{2}$ is also an important subgroup of various Lie groups:
 * $$\begin{align}

\mathrm{U}(n)   &\supset \mathrm{O}(n) \\ \mathrm{USp}(2n) &\supset \mathrm{O}(n) \\ \mathrm{G}_2 &\supset \mathrm{O}(3) \\ \mathrm{F}_4 &\supset \mathrm{O}(9) \\ \mathrm{E}_6 &\supset \mathrm{O}(10) \\ \mathrm{E}_7 &\supset \mathrm{O}(12) \\ \mathrm{E}_8 &\supset \mathrm{O}(16) \end{align}$$

Conformal group
Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (side-side-side) congruence of triangles and AAA (angle-angle-angle) similarity of triangles. The group of conformal linear maps of $H^{1}$ is denoted $μ_{2}$ for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If $H^{0}(O_{V})$ is odd, these two subgroups do not intersect, and they are a direct product: $O_{V}(F)$, where $F$ is the real multiplicative group, while if $H^{1}(μ_{2})$ is even, these subgroups intersect in $H^{1}(μ_{2})$, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: $H^{1}$.

Similarly one can define $H^{2}$; this is always: $O(n, F )$.

Discrete subgroups
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as point groups and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:
 * Permutation matrices (the Coxeter group $SO(n, F )$)
 * Signed permutation matrices (the Coxeter group $n × n$); also equals the intersection of the orthogonal group with the integer matrices.

Covering and quotient groups
The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively: These are all 2-to-1 covers.
 * Two covering Pin groups, $[, ]$ and $n$,
 * The quotient projective orthogonal group, $B_{k}$.

For the special orthogonal group, the corresponding groups are: Spin is a 2-to-1 cover, while in even dimension, $n = 2k + 1$ is a 2-to-1 cover, and in odd dimension $D_{r}$ is a 1-to-1 cover; i.e., isomorphic to $n = 2r$. These groups, $SO(n)$, $SO(n)$, and $V$ are Lie group forms of the compact special orthogonal Lie algebra, $$\mathfrak{so}(n, \mathbf{R})$$ – $(⋅, ⋅)$ is the simply connected form, while $Λ2V$ is the centerless form, and $(p, q)$ is in general neither.
 * Spin group, $O(n) ⊂ U(n) ⊂ USp(2n)$,
 * Projective special orthogonal group, $USp(n) ⊂ U(n) ⊂ O(2n)$.

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

Principal homogeneous space: Stiefel manifold
The principal homogeneous space for the orthogonal group $U(n)/O(n)$ is the Stiefel manifold $U(n)$ of orthonormal bases (orthonormal $SU(n)$-frames).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds $O(n)$ for $R^{n}$ of incomplete orthonormal bases (orthonormal $CO(n)$-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any $n$-frame can be taken to any other $CO(2k + 1) = O(2k + 1) × R^{∗}$-frame by an orthogonal map, but this map is not uniquely determined.

Specific transforms

 * Coordinate rotations and reflections
 * Reflection through the origin

Specific groups

 * rotation group, $R^{∗} = R∖{0}$

Related groups

 * indefinite orthogonal group
 * unitary group
 * symplectic group

Lists of groups

 * list of finite simple groups
 * list of simple Lie groups

Representation theory

 * Representations of classical Lie groups
 * Brauer algebra

title: Orthogonal group


text:
Talk:Hilary Rosen

Rosen vs. Napster
I think the role Rosen played in declaring war on the internet for the benefit of capitalist middle-men working to shut down Napster needs to have a far more prominent role in the article introduction. Given the significance of the internet on the world level now and the pioneering role that Napster played on the internet (I think at the time it had over 20 million users), if Rosen dropped dead tomorrow, that would be her most notable life "achievement". The LGBTFGHIJKLMNO promotion stuff, while yes it is part of what she does now and so should be mentioned too, is far less significant than Rosen vs. Napster affair. Claíomh Solais (talk) 01:12, 25 December 2017 (UTC)

NYT: Hilary Rosen bought over a half-million fake Twitter followers from 2015 to 2017. Nearly half of her followers are fake.
https://www.nytimes.com/interactive/2018/01/27/technology/social-media-bots.html

Over two years, the Democratic public relations consultant and CNN contributor Hilary Rosen bought more than a half-million fake followers from Devumi. Ms. Rosen previously spent more than a decade as head of the Recording Industry Association of America. In an interview, she described the purchases as “an experiment I did several years ago to see how it worked.” She made more than a dozen purchases of followers from 2015 to 2017, according to company records.

Hilary Rosen, a political commentator and CNN contributor, paid for over a half-million Twitter followers. Many of those accounts have since vanished, but nearly half of her followers — including the fake Jessica Rychly — also follow Mr. Symon.

Xardox (talk) 18:19, 27 January 2018 (UTC)

Time's Up Legal Defense Fund
Hi everyone. My name's John and I am here to disclose that I work for SKDKnickerbocker and would like to submit an edit to this page as a part of my work for the firm. I will not edit the article directly and am seeking others' input and assistance in making this change.

My goals in preparing a new draft is to bring the page up-to-date.

See above for my COI declaration, pledge to make no direct edits, and overall perspective on the current state of this article. I'd like to suggest the following change to the summary section of the page. Below this you'll find the text as it would appear after making the change, followed by the markup itself:

Hilary Rosen (born 1958) is an American communications and political consultant and pundit, former head of the Recording Industry Association of America (RIAA), founding chair-person of the Time's Up Legal Defense Fund, and current partner at SKDKnickerbocker. She was a columnist for The Washington Post, became the first Washington editor-at-large and political director of The Huffington Post, and has provided political commentary for CNN, CNBC, and MSNBC.

If you agree this is an improvement, I hope you'll consider making the change and marking this request as complete. Of course I'm also willing to answer any questions as needed. Best, User:SKDKDigital
 * Do you have some sort of source? A press release should be acceptable to make this change, in the circumstances. Sceptre (talk) 18:27, 4 October 2019 (UTC)

This CNN piece names her as a co-founder:

Reply 4-OCT-2019
Regards, Spintendo  20:07, 4 October 2019 (UTC)
 * The NWLC source states the following:"The TIME’S UP Legal Defense Fund is housed and administered by the National Women’s Law Center Fund, LLC. The TIME’S UP Legal Defense Fund was started by women in Hollywood who were moved by the outpouring of support from the Alianza Nacional de Campesinas, a national farmworker women’s organization. These actresses and activists wanted to show solidarity with survivors of sexual harassment, assault, abuse and related retaliation in all industries — especially low-income women and people of color. They worked together in a historic first to design a structure that would be both inclusive and effective. The actresses and advocates collaborated with super lawyers Tina Tchen and Robbie Kaplan, Fatima Goss Graves of the National Women’s Law Center, and communications specialist Hilary Rosen."That source does not give the status of Ms. Rosen as "founding chairperson". The COI editor's proposal also uses the phrase "co-founder" which is different from the term "founding chairperson". I think at a minimum there needs to be an economy of terms here, with it being decided which term is more appropriate to use based on the sources provided by the COI editor.
 * To that end, an ideal source would be one provided from NWLC, which would then solve all of the issues here by stating her role in the founding of this groundbreaking legal fund. That would allow for the claim to be made in the article. Please advise.
 * When ready to proceed with the requested information, kindly change the  template's answer parameter to read from yes to no. Thank you!

Co-founder status (continued discussion below)
First, apologies for the confusion on titling. I found the following article from Deadline Hollywood that may help clarify, it identifies Rosen as a "Time's Up found co-chair" John at SKDKDigital (talk) 21:19, 4 October 2019 (UTC)

 * I would prefer a source coming from the NWLC which states her position as founding co-chair, as I stated in my review above. To date, none has been provided. Regards, Spintendo  15:05, 8 October 2019 (UTC)

 * Could "founding communications specialist" from the NWLC source above be used instead? "The actresses and advocates collaborated with super lawyers Tina Tchen and Robbie Kaplan, Fatima Goss Graves of the National Women’s Law Center, and communications specialist Hilary Rosen". The implication is that the four of them were involved in founding the Legal Defense Fund arm of Time's Up. John at SKDKDigital (talk) 15:33, 11 October 2019 (UTC)

 * There is also a more recent citation from the New York Times:

 * Ms. Tchen was a founder of the organization’s signature initiative: a legal-defense fund for women in all industries who experienced sexual harassment at work... The other co-founders of the fund are the lawyer Robbie Kaplan; Fatima Goss Graves, the president and chief executive of the National Women’s Law Center; and the communications strategist Hilary Rosen.

 * The New York Times is the preeminent news service today, however, they are not the final authority on whether she was co-founder (that same NYTimes source states that "The group was founded by powerful women in Hollywood", which doesn't include Washington DC). This is information which ought to come from the organization itself (and which should theoretically be easy to provide if the title of co-founder is one that is commonly applied by the organization itself when describing these individuals). I think the best way to describe Ms. Rosen is to use the verbatim description offered by NWLC (without any added implications, and added as a quote). Please advise. Regards, Spintendo  22:03, 18 October 2019 (UTC)

Thank you Spintendo. Understood, can we use the language: "Rosen is the founding communications specialist of the Time's Up (movement) Legal Defense Fund." citing the NWLC source you provided? — Preceding unsigned comment added by John at SKDKDigital (talk • contribs) 17:41, 23 October 2019 (UTC)

title: Talk:Hilary Rosen


text:
Pete Fountain

Pierre Dewey LaFontaine Jr. (July 3, 1930 – August 6, 2016), known professionally as Pete Fountain, was an American jazz clarinetist.

Early life and education
LaFontaine was born to Pierre, Sr. and Madeline, in a small Creole cottage-style frame house on White Street (between Dumaine Street and St. Ann Street) in New Orleans. Pete was the great-grandson of a French immigrant, François Fontaine, who was born in Toulon, circa 1796, and came to the U.S. in the early 19th century, and died on the Mississippi Gulf Coast circa 1885. Pete's father, a truck driver and part-time musician, changed the family name to Fountain.

He started playing clarinet as a child at the McDonogh 28 school located on Esplanade Avenue. As a child, young Pete was very sickly, frequently battling respiratory infections due to weakened lungs. He was given expensive medication but it proved to be not very effective. During a pharmacy visit, Pete's father began a discussion with a neighborhood doctor who was also there shopping and talked with him about his son's condition. The doctor agreed to see the boy the following day. After a short exam, the doctor confirmed the weak lung condition and advised the father to try an unorthodox treatment: purchase the child a musical instrument, anything he has to blow into. The same day, they went to a local music store and, given his choice of instruments, Pete chose the clarinet (after first wanting the drums, which his father declined per the doctor's orders). At first, Pete was unable to produce a sound from the instrument, but he continued to practice and eventually not only made sounds and eventually music, but greatly improved the health of his lungs.

He took private lessons but also learned to play jazz by playing along with phonograph records of first Benny Goodman and then Irving Fazola. By the time he reached his teens, he was playing regular gigs in the nightclubs on Bourbon Street. According to Fountain:
 * When I was a high school senior, my history teacher asked me why I didn’t study more... I answered that I was too busy playing clarinet every night, and when I told him I was making scale — about $125 a week — he said that was more than he made and I should play full time. I guess I was a professional from that point on.

One of Fountain's early engagements were with the bands of Monk Hazel. Fountain founded the Basin Street Six in 1950 with his longtime friend, trumpeter George Girard. In 1954, after the Basin Street Six folded, Fountain briefly went to Chicago to play with the Dukes of Dixieland, then returned to New Orleans and teamed up with Al Hirt to lead a band, playing an extended residence at Dan Levy’s Pier 600.

Lawrence Welk orchestra
A talent scout for Lawrence Welk, who saw Fountain performing at the Pier 600, brought him to the attention of Larry Welk, son of television bandleader Lawrence Welk. Persistent persuasion from the son led the senior Welk to invite Fountain to join the Lawrence Welk orchestra in Los Angeles, where he relocated and lived for two years. Fountain became well known for his many solos on Welk's ABC television show, The Lawrence Welk Show. He was rumored to have quit when Welk refused to let him "jazz up" a Christmas carol on the 1958 Christmas show. Other accounts, including one in Fountain's autobiography A Closer Walk With Pete Fountain, indicate he in fact played a jazzy rendition of "Silver Bells" on the show which upset Welk, leading to Fountain's departure in early 1959. In an interview, Fountain said he left The Lawrence Welk Show because "champagne and bourbon don't mix." Fountain was hired by Decca Records A&R head Charles "Bud" Dant and went on to produce 42 hit albums with Dant. After Welk's death, Fountain would occasionally join with the Welk musical family for reunion shows.

Return to New Orleans
Fountain returned to New Orleans, played with the Dukes of Dixieland, then began leading bands under his own name. He owned his own club in the French Quarter in the 1960s and 1970s. He later acquired "Pete Fountain's Jazz Club" at the Riverside Hilton in downtown New Orleans.

The New Orleans Jazz Club presented "Pete Fountain Day" on October 19, 1959, with celebrations honoring the pride of their city, concluding with a packed concert that evening. His Quintett was made up of his studio recording musicians, Stan Kenton's bassist Don Bagley, vibeist Godfrey Hirsch, pianist Merle Koch, and the double bass drummer Jack Sperling. Fountain brought these same players together in 1963 when they played the Hollywood Bowl. Pete would make the trek to Hollywood many times, appearing on The Tonight Show Starring Johnny Carson 56 times.

Fountain opened his club, the French Quarter Inn, located in the heart of the famed French Quarter district, at 800 Bourbon Street, in the spring of 1960. His group members were Oliver "Stick" Felix on bass, John Probst on piano, Paul Guma on guitar, Godfrey Hirsch on vibes, and Jack Sperling on drums. In no time at all, major entertainers found their way there. Cliff Arquette, popularly known as Charley Weaver, and Jonathan Winters were there on opening night and performed their comedy routines. Over the next few years Frank Sinatra, Phil Harris, Carol Lawrence and Robert Goulet, Keely Smith, Robert Mitchum, and Brenda Lee, among many others, came to the club. Many would perform with the band, and Brenda Lee's sit-in resulted in a duet record album recorded by her and Pete. Benny Goodman came to the club twice, but without bringing his clarinet.

His greatest friendly rivalry was with trumpeter Al Hirt, whose club was down the street from Fountain's. They stole musicians from each other, and sometimes came into each other's clubs and played together. They were good friends who came up together and later recorded several albums together.

In 2003, Fountain closed his club at the Hilton with a performance before a packed house filled with musical friends and fans. He began performing two nights a week at Casino Magic in Bay St. Louis, Mississippi, where he also had a home (later destroyed by Hurricane Katrina).

After heart surgery in 2006, he performed at the New Orleans Jazz and Heritage Festival and helped reopen the Bay St. Louis casino. It has since been renamed the Hollywood Casino. He performed his last show at Hollywood Casino on December 8, 2010, before returning to help reopen the resort in 2014, by which point he was mostly retired.

Half-Fast Walking Club
Fountain was a founder and the most prominent member of the Half-Fast Walking Club, one of the best known freelance marching units that parade in New Orleans on Mardi Gras Day. The original name was "The Half-Assed Walking Club," and it was an excuse to take a "lubricated" musical stroll down the parade route. Pete changed the name under pressure exerted by the parade organizers. On Mardi Gras Day 2007, Pete again joined his Half-Fast Walking Club, having missed the event in 2006 due to illness.

Musical style
Fountain's clarinet work was noted for his sweet fluid tone. He recorded over 100 LPs and CDs under his own name, some in the Dixieland style, many others essentially instrumental pop records with only peripheral relevance to any type of jazz. Nonetheless, in 1997 critic Doug Ramsey of Jazz Times magazine included Fountain on a list of underrated musicians, writing: "His ear for harmony and mastery of time are among the best-kept secrets in jazz because all these years he has chosen to stick with the repertoire and sidemen that make him comfortable."

The distinctive Fountain sound — more woody than most — came from the crystal mouthpieces he played with since 1949 rather than the more common rubber mouthpieces. His first crystal mouthpiece was actually Irving Fazola's, given to Pete by Fazola's mother after Faz's death, because she had heard him play and noted how he played like her son. That mouthpiece was shattered on the bandstand one night when Pete had played his solo and was standing by as trumpeter George Girard played his [own solo], and Girard brought his trumpet down suddenly on top of the mouthpiece. Pete kept the shattered mouthpiece, and played other crystal mouthpieces from then on.

Fountain led the Pete Fountain Quintett, a New Orleans French Quarter jazz band of Fountain and his Creole-style music. The "Quintett" had many musicians over the years, but primarily recorded with Jack Sperling on drums, bassists Don Bagley or Morty Corb, vibraphonist Godfrey Hirch, and pianists Merle Koch or Stan Wrightsman.

Personal life
Fountain married Beverly Lang on October 27, 1951; they remained married for sixty-five years until his death. They have two sons and a daughter: Kevin, Jeffrey, and Dahra. Dahra's husband, Benny Harrell, was Fountain's manager in his later years.

Fountain died of heart failure in his home town on August 6, 2016, at the age of eighty-six. He had suffered from heart problems and was in hospice care when he died.

Awards and honors

 * 2006, Loyola University New Orleans awarded Fountain an honorary degree.
 * On March 18, 2007, Pete Fountain was inducted into the Louisiana Music Hall of Fame.
 * On April 5, 2008, Fountain was inducted into the Delta Music Museum Hall of Fame in Ferriday, Louisiana.

title: Pete Fountain


text:
Center of mass

In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.

In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid. The center of mass may be located outside the physical body, as is sometimes the case for hollow or open-shaped objects, such as a horseshoe. In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.

The center of mass is a useful reference point for calculations in mechanics that involve masses distributed in space, such as the linear and angular momentum of planetary bodies and rigid body dynamics. In orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass (see Barycenter (astronomy) for details). The center of mass frame is an inertial frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system.

History
The concept of center of gravity or weight was studied extensively by the ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. In his work On Floating Bodies, Archimedes demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.

Other ancient mathematicians who contributed to the theory of the center of mass include Hero of Alexandria and Pappus of Alexandria. In the Renaissance and Early Modern periods, work by Guido Ubaldi, Francesco Maurolico, Federico Commandino, Evangelista Torricelli, Simon Stevin, Luca Valerio, Jean-Charles de la Faille, Paul Guldin, John Wallis, Christiaan Huygens, Louis Carré, Pierre Varignon, and Alexis Clairaut expanded the concept further.

Newton's second law is reformulated with respect to the center of mass in Euler's first law.

Definition
The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space.

A system of particles
In the case of a system of particles $P_{i}, i = 1, ..., n$, each with mass $m_{i}$ that are located in space with coordinates $r_{i}, i = 1, ..., n$, the coordinates R of the center of mass satisfy the condition

Solving this equation for R yields the formula

A continuous volume
If the mass distribution is continuous with the density ρ(r) within a solid Q, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass R over the volume V is zero, that is $$\iiint_{Q} \rho(\mathbf{r}) \left(\mathbf{r} - \mathbf{R}\right) dV = 0.$$

Solve this equation for the coordinates R to obtain $$\mathbf R = \frac 1 M \iiint_{Q}\rho(\mathbf{r}) \mathbf{r} \, dV,$$ Where M is the total mass in the volume.

If a continuous mass distribution has uniform density, which means that ρ is constant, then the center of mass is the same as the centroid of the volume.

Barycentric coordinates
The coordinates R of the center of mass of a two-particle system, P1 and P2, with masses m1 and m2 is given by $$ \mathbf{R} = {{m_1 \mathbf{r}_1 + m_2\mathbf{r}_2} \over m_1 + m_2}.$$

Let the percentage of the total mass divided between these two particles vary from 100% P1 and 0% P2 through 50% P1 and 50% P2 to 0% P1 and 100% P2, then the center of mass R moves along the line from P1 to P2. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, and are termed barycentric coordinates. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively.

Systems with periodic boundary conditions
For particles in a system with periodic boundary conditions two particles can be neighbours even though they are on opposite sides of the system. This occurs often in molecular dynamics simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, x and y and/or z, as if it were on a circle instead of a line. The calculation takes every particle's x coordinate and maps it to an angle, $$\theta_i = \frac{x_i}{x_\max} 2 \pi $$ where xmax is the system size in the x direction and $$x_i \in [0, x_\max)$$. From this angle, two new points $$(\xi_i, \zeta_i)$$ can be generated, which can be weighted by the mass of the particle $$x_i$$ for the center of mass or given a value of 1 for the geometric center: $$\begin{align}   \xi_i &= \cos(\theta_i) \\  \zeta_i &= \sin(\theta_i) \end{align}$$

In the $$(\xi, \zeta)$$ plane, these coordinates lie on a circle of radius 1. From the collection of $$\xi_i$$ and $$\zeta_i$$ values from all the particles, the averages $$\overline{\xi}$$ and $$\overline{\zeta}$$ are calculated.

$$\begin{align} \overline{\xi} &= \frac 1 M \sum_{i=1}^n m_i \xi_i, \\ \overline{\zeta} &= \frac 1 M \sum_{i=1}^n m_i \zeta_i, \end{align}$$ where $M$ is the sum of the masses of all of the particles.

These values are mapped back into a new angle, $$\overline{\theta}$$, from which the x coordinate of the center of mass can be obtained:

$$\begin{align} \overline{\theta} &= \operatorname{atan2}\left(-\overline{\zeta}, -\overline{\xi}\right) + \pi \\ x_\text{com} &= x_\max \frac{\overline{\theta}}{2 \pi} \end{align}$$

The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using cluster analysis to "unfold" a cluster straddling the periodic boundaries. If both average values are zero, $$\left(\overline{\xi}, \overline{\zeta}\right) = (0, 0)$$, then $$\overline{\theta}$$ is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their x coordinates are mathematically identical in a periodic system.

Center of gravity
A body's center of gravity is the point around which the resultant torque due to gravity forces vanishes. Where a gravity field can be considered to be uniform, the mass-center and the center-of-gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer-to and further-from the planet (stronger and weaker gravity respectively) can lead to a torque that will tend to align the satellite such that its long axis is vertical. In such a case, it is important to make the distinction between the center-of-gravity and the mass-center. Any horizontal offset between the two will result in an applied torque.

The mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center-of-gravity will always be located somewhat closer to the main attractive body as compared to the mass-center, and thus will change its position in the body of interest as its orientation is changed.

In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass-center as the center-of-gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably.

In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the resultant of the gravity forces on a continuous body. Consider a body Q of volume V with density ρ(r) at each point r in the volume. In a parallel gravity field the force f at each point r is given by, $$ \mathbf{f}(\mathbf{r}) = -dm\, g\mathbf{\hat{k}} = -\rho(\mathbf{r}) \, dV\,g\mathbf{\hat{k}},$$ where dm is the mass at the point r, g is the acceleration of gravity, and $\mathbf{\hat{k}}$ is a unit vector defining the vertical direction.

Choose a reference point R in the volume and compute the resultant force and torque at this point,

and

If the reference point R is chosen so that it is the center of mass, then $$ \iiint_{Q} \rho(\mathbf{r}) \left(\mathbf{r} - \mathbf{R}\right) dV = 0, $$ which means the resultant torque T = 0. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass.

Linear and angular momentum
The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles Pi, i = 1, ..., n of masses mi be located at the coordinates ri with velocities vi. Select a reference point R and compute the relative position and velocity vectors, $$ \mathbf{r}_i = (\mathbf{r}_i - \mathbf{R}) + \mathbf{R}, \quad \mathbf{v}_i = \frac{d}{dt}(\mathbf{r}_i - \mathbf{R}) + \mathbf{v}.$$

The total linear momentum and angular momentum of the system are

and

If R is chosen as the center of mass these equations simplify to $$ \mathbf{p} = m\mathbf{v},\quad \mathbf{L} = \sum_{i=1}^n m_i (\mathbf{r}_i - \mathbf{R}) \times \frac{d}{dt}(\mathbf{r}_i - \mathbf{R}) + \sum_{i=1}^n m_i \mathbf{R} \times \mathbf{v}$$ where m is the total mass of all the particles, p is the linear momentum, and L is the angular momentum.

The law of conservation of momentum predicts that for any system not subjected to external forces the momentum of the system will remain constant, which means the center of mass will move with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that cancel in accordance with Newton's Third Law.

Determination
The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center of gravity in the parallel gravity field near the earth's surface.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.

In two dimensions
An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass.

The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers. This method can even work for objects with holes, which can be accounted for as negative masses.

A direct development of the planimeter known as an integraph, or integerometer, can be used to establish the position of the centroid or center of mass of an irregular two-dimensional shape. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to compare with the required displacement and center of buoyancy of a ship, and ensure it would not capsize.

In three dimensions
An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, F1, F2, and F3 that resist the weight of the object, $$\mathbf{W} = -W\mathbf{\hat{k}}$$ ($$\mathbf{\hat{k}}$$ is the unit vector in the vertical direction). Let r1, r2, and r3 be the position coordinates of the support points, then the coordinates R of the center of mass satisfy the condition that the resultant torque is zero, $$\mathbf{T} = (\mathbf{r}_1 - \mathbf{R}) \times \mathbf{F}_1 + (\mathbf{r}_2 - \mathbf{R}) \times \mathbf{F}_2 + (\mathbf{r}_3 - \mathbf{R}) \times \mathbf{F}_3 = 0,$$ or $$\mathbf{R} \times \left(-W\mathbf{\hat{k}}\right) = \mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times \mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3. $$

This equation yields the coordinates of the center of mass R* in the horizontal plane as, $$ \mathbf{R}^* = -\frac{1}{W} \mathbf{\hat{k}} \times (\mathbf{r}_1 \times \mathbf{F}_1 + \mathbf{r}_2 \times\mathbf{F}_2 + \mathbf{r}_3 \times \mathbf{F}_3).$$

The center of mass lies on the vertical line L, given by $$ \mathbf{L}(t) = \mathbf{R}^* + t\mathbf{\hat{k}}.$$

The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object. The center of mass will be the intersection of the two lines L1 and L2 obtained from the two experiments.

Automotive applications
Engineers try to design a sports car so that its center of mass is lowered to make the car handle better, which is to say, maintain traction while executing relatively sharp turns.

The characteristic low profile of the U.S. military Humvee was designed in part to allow it to tilt farther than taller vehicles without rolling over, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the horizontal.

Aeronautics
The center of mass is an important point on an aircraft, which significantly affects the stability of the aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within specified limits. If the center of mass is ahead of the forward limit, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing. If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly unstable enough so as to be impossible to fly. The moment arm of the elevator will also be reduced, which makes it more difficult to recover from a stalled condition.

For helicopters in hover, the center of mass is always directly below the rotorhead. In forward flight, the center of mass will move forward to balance the negative pitch torque produced by applying cyclic control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.

Astronomy

The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the barycenter. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies orbit each other. When a moon orbits a planet, or a planet orbits a star, both bodies are actually orbiting a point that lies away from the center of the primary (larger) body. For example, the Moon does not orbit the exact center of the Earth, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km (1,062 miles) below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the Sun. If the masses are more similar, e.g., Pluto and Charon, the barycenter will fall outside both bodies.

Rigging and safety
Knowing the location of the center of gravity when rigging is crucial, possibly resulting in severe injury or death if assumed incorrectly. A center of gravity that is at or above the lift point will most likely result in a tip-over incident. In general, the further the center of gravity below the pick point, the safer the lift. There are other things to consider, such as shifting loads, strength of the load and mass, distance between pick points, and number of pick points. Specifically, when selecting lift points, it is very important to place the center of gravity at the center and well below the lift points.

Body motion
In kinesiology and biomechanics, the center of mass is an important parameter that assists people in understanding their human locomotion. Typically, a human's center of mass is detected with one of two methods: the reaction board method is a static analysis that involves the person lying down on that instrument, and use of their static equilibrium equation to find their center of mass; the segmentation method relies on a mathematical solution based on the physical principle that the summation of the torques of individual body sections, relative to a specified axis, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.

Optimization
The Center-of-gravity method is a method for convex optimization, which uses the center-of-gravity of the feasible region.

title: Center of mass


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Talk:Infantry

Air force and naval infantry
Regarding that section of the article, it currently mentions particular air force infantry forces. Considering that they and others are already listed in the Air force ground forces and special forces article, keeping them in this article is not necessary. I suggest removing them. Dreddmoto (talk) 03:44, 11 March 2024 (UTC)

title: Talk:Infantry