text:
Talk:Tina Modotti

Untitled
I've moved the below comments to here, they don't belong in an article. --[[User:Bodnotbod| bodnotbod »  .....TALK Q uietly  )  ]] 17:52, Aug 7, 2004 (UTC)

Images
Desperately needed for this entry are usable [see copyright issues elsewhere] images in at least these categories.

 * Photographs by Tina Modotti

 * Photographs of Tina Modotti

 * Photograph of Diego Rivera’s murals with Modotti as a model.

 * Many of her photographs still under copyright, mostly held by her estate. According to the Masters of Photography website, provided that we attribute them as a source and say who holds the copyright. Shiri — Talk 19:43, Mar 26, 2005 (UTC)

I added a (presumably) PD image of her in The Tiger's Coat. Not terrific, but all the photographs I could find of her are still under copyright. Shiri — Talk 19:46, Mar 26, 2005 (UTC)

 * I just added this link to Modotti's photographs at the Italian Fondazione Italiana per la Fotografia. I think all of these photos are from the 1920s, and I've heard that EU copyright law is less stringent than in the U.S. Would anyone with EU copyright knoweldge and/or Italian skills be interested in checking this out? –dablaze  02:35, Mar 27, 2005 (UTC)

 * I just added Woman from Tehuantepec by Modotti.Smlarthist (talk) 21:05, 8 March 2017 (UTC)

Links
These will be left for someone else.

more cleanup needed
I cleaned up the first six paragraphs of this article. After that, the author begins to drop names not otherwised referenced and makes use of unexplained abbreviations. Much of this makes little sense. Someone with better knowledge of the subject will have to complete the cleanup.

Rosquet 23:19, 5 Jan 2005 (UTC)
 * Rosquet - you should register as a user and we can talk more about this. Carptrash 05:35, 6 Jan 2005 (UTC)

Did a Fair Amount
I did a fair amount of stuff, although I added no new content. Mainly reorganizing, wikifying, and trying to clarify.

Superm401 03:35, 11 Jan 2005 (UTC)

a funny coincidence, if there is such a thing as coincidence.
When I first wrote this piece I emersed myself in Modotti, a person that I have long been infatuated with, but burned out after writing a very sketchy piece and could not get up the feelings needed to return to it. Then a couple of weeks ago I was interviewed by Daniel Pink for his Wired story on Wikipedia. At that time I mentioned to him about my incomplete Modotti article. Almost immediately folks began showing up and doing the work that I seemed incapable of performing. This makes me feel good about myself, good about Tina and mostly, good about wikipedia. [postdated] Carptrash 22:26, 14 Jan 2005 (UTC)

Maybe the copyedit tag helped...

At any rate, the part about the painting too close to Mello and the affair and the peering from behind...it's very confusing; someone who knows the facts should restructure it to indicate at the least what "with whom she was trying to hide the affair" means: was Mello her beard while she was schtupping Rivera? Was she hiding her sex with Rivera from Mello? Was she doing Mello and hiding it from Rivera, who she was also doing? And it means nothing to us that Vidali is peering at her, as he's not been mentioned in the context of "the affair".

The phrase "...continuing to work under the cover of the Red Aid organization..." implies that we've been told she had been before, but there is no indication of that; perhaps an earlier statement of her joining the organization has gone missing. I've adjusted it to "...for whom she worked under cover..." just to make it flow as-is.

Decamped seems such a prissy word, but it fits perfectly here in the wide-brimmed-hatted, artsy-commie spy world she inhabited. Blair P. Houghton 03:57, 16 Jan 2005 (UTC)

Timeline & Stalinist purge questions
I commented out the following text (in mid-edit, so there may be some fragments in there):

 * Sometime in 1923 Modotti met muralist Diego Rivera, modeled for several of his paintings. began a yearlong affair with him after separating from Weston. During this time she turned more and more towards radical politics. Also in 1926, Diego Rivera's wife Lupe Marín asserted that her separation from her husband was caused by his affair with Modotti, who was modeling for him.  The Modotti-Rivera affair lasted for about a year.

 * She did not care for the fact that in one painting Rivera depicted her in close proximity to Mella, with whom Modotti was trying to hide her affair. Peering at her from behind is the ominous face of Vidali. About this Tina wrote to Weston, "Recently Diego has taken to painting details with an exaggerated precision. He leaves nothing to the imagination."  This marked the end of her admiration of him as an artist, and perhaps as a communist as well.  Later, after Rivera was expelled from the Communist Party for his Trotskyist leanings, she would denounce him as a traitor.

I tried a lot with this, but there's no real timeline to work with -- I can't really figure out what happened when. Can anyone clarify the whole Diego Rivera affair timeline? In general terms? :-)

Also, the sentence that claims Modotti was involved in the Stalinist purges, was this in Spain or in Moscow? (The claim appeared in the Spanish Civil War paragraph, but I cut it for now.) What kind of involvement was it, or was it just a rumor?

--dablaze  05:12, Jan 16, 2005 (UTC)

 * Well I started this article, so I guess i can . . . . if not finish it at least address some of these issues.  So first thing tomorrow AM [after two cups of coffee].......................... Carptrash 05:27, 16 Jan 2005 (UTC)

Almost two years (and how many cups of coffee?) later.... I've added some more, mostly by expanding the section on 'In the Arsenal', this narrative is really messy and confusing not least because it involves the concatanation of the time lines of umpteen different relationships Frida and Diego, Lupa and Diego, Tina and Diego, Tina and Weston, Tina and Mella, Tina and Vidalli all of which contained great passion and great animosity and conflict (in many cases both personal and political!) so I'm sure that others can improve on my contribution, and I'm not sure that all of my material sits best where it is. I think that IF my additions to the section on the the Arsenal stay there then an image of the mural could be added to the article, there is one in the Diego Rivera article but its is a detail which omits the image of Tina. Green ribbon (talk) 06:04, 3 January 2008 (UTC)

Kevincof has made a number of corrections and improvements to my additions. I have made one further amendment to his. I accept that my initial contribution probably reflected an element of POV (broadly Trotskyist) on my part, some of Kevincof's amendment was justfied in that respect, however I think his amendment tended towards an opposite POV (equating internationalism with adherence to the then line of the CPM and the the Communist International) which is equally inappropriate, my amendment I think restores neutrality by recognising that this was Moddoti's POV without accepting it. Perhaps someone with less of an axe to grind than myself (and maybe Kevincof)should look at this to avoid a deletion war! Green ribbon (talk) 14:50, 6 January 2008 (UTC)

I edited the statement for two reasons. Firstly, there is ample documentary evidence to associate Modotti's motivations with an internationalist as well as 'humanist' perspective. There is also ample historical evidence that these motivations operated broadly during that period (just as today). By labeling her 'Stalinist' (or Rivera 'Trotskyist' for that matter), the statement tends to obscure with political invective. Certainly by virture of his position Stalin was very influential among Marxists in the 20s and 30s, but I have found no documentary evidence that Modotti evaluated herself (politically or otherwise) in regard to Joseph Stalin. Kevincof (talk) 15:21, 6 January 2008 (UTC)

 * I am re-reading all the Stalinist references in Alber's Fire, Shadows, Snow ( bunch) where the Trotsky Stalin split comes to the fore regarding the assassination of the maverick (if not Trotskyite, then certainly not Stalinist) Mella by (perhaps) Vidali, Tina next lover and clearly a Stalinist.  Would she have spent the rest of her life with a Stalinist (including going to Stalin's Soviet Union with him) were she not in sympathy with that POV?   However I just ran across the statement that if Mella's "Assassination is shrouded in confusion, so is the matter of Tina's political affiliation.''  (p. 216)  this at the time of the murder.  Albers follows this with "she flew back into the Party fold." Stalin's party that is.  Carptrash (talk) 16:45, 6 January 2008 (UTC)

 * And so, that is Alber's view, not Modotti's, isn't it? Again, among Marxists and many other political leftists during that period (1920-1945 at least), the Russian Revolution and ensuing Soviet government were considered representative, including of the 'anti-fascist struggle'. To assert that everyone who supported either was ideologically a "Stalinist' is tenuous. Kevincof (talk) 17:21, 6 January 2008 (UTC)

Well wikipedia is largely a collection of published POV - that fact thay they are published making them okay. We will probably never get Tina's version but Vidale writes that Tina was, '' more reserved. less talkative, more rigid in her judgement about the relationships between Party and comrades and Albers adds Vittorio's observations bespeak Tina's further drift toward physiological capitulation to Stalinist doctrine as ultimate truth.''  You might not agree with Albers, but there it is. Carptrash (talk) 17:36, 6 January 2008 (UTC)

I don't think that there's a very big difference between any of us, from my POV i'd say that, whilst I've no doubt that Tina was motivated by a conviction that support for the line of the Comintern was the form which internationalism took in her day, in fact the line of the Comintern was far from internationalist, placing the national interests of the Russian bureaucracy over the needs of the world revolution (in Trotsky's words Stalin had converted the Comintern 'from the vanguard of the revolution to the border guard of the Soviet Union'), and I guess that what motivated me was that Kevincof's amendment appeared to equate internationalism with the Comintern line and thus advance an opposite POV, but I now feel that was rather an over-reaction, I feel that my addition of the adjective 'Stalinist' was unnecessary and have deleted it, although I hope its agreed my other amendments should stand. Kevincof also says 'but I have found no documentary evidence that Modotti evaluated herself (politically or otherwise) in regard to Joseph Stalin', I'm not sure that 'documentary evidence' is needed in this case: after 1927 all of Comintern's policies were dictated by Stalin and it declarations addressed him in hagiographic terms. By being an adherent of the Comintern and as far as we can see supporting, or at least not opposing, Vidale in his efforts to purge the MCP of its 'maverick' elements, including her past and present lovers, Tina showed where she stood in relation to uncle Joe.

As to whether Mella was 'Trotskyist' the evidence is vague, all oppositionist in the Comintern were branded Trotskyist at time, even those such as the American Lovestonites who actually attacked Stalin from the Right, on the other hand Mella does seem to have discussed the position of the Left Opposition with the Spanish oppositionist Andres Nin, who would then have been regarded as a Trotskyist, and of course Diego and Frida both found their way back into the MCP and adhered to the Stalinist line by 1942. Green ribbon (talk) 00:01, 7 January 2008 (UTC)

This is a very interesting line of discussion on several levels, but still, the topic at hand is Modotti's motivations. As for the Third International, and the assertion that "after 1927 all of Comintern's policies were dictated by Stalin and it declarations addressed him in hagiographic terms", although I would agree that Stalin and his group were the predominant political force within it, there was not complete unanimity. The various purges and expulsions are clear indication of that. So too, the Chinese Communist Party very obviously did not treat Stalin hagiographically and neither did the Yugoslavian group. My basic argument is that one should not downplay the 'agency' role of any activists or organizations on the basis of their programmatic alignment with other Comintern parties or organizations; social reality is more complex than that. I think that based on correspondence that I've read, Modotti was not primarily motivated by a specific "theoretical" position. For example, it doesn't appear that she was that familiar with Marx or Lenin when she emigrated to Moscow. Rather, her statements indicate an empathy for the poor and oppressed, and a sense that the injustice of that oppression should be opposed (and that certain groups were or could be instrumental in that). Kevincof (talk) 00:36, 7 January 2008 (UTC)

I'd agree that the Chinese and Yugoslav CP's were able to pursue indpendent lines, BUT only after they had achieved a position of indpendence at the head of a state machine from which they were able challenge the dominance of the Soviet Bureaucracy within the Communist movement.

So more accurately my statement should read 'between 1927 and the mid-late 1940's.....' Ironically both Tito and Mao had been installed at the head of their parties as Moscow loyalists in the late 1920s after the defeat of oppositionist leaderships within their parties and the extermination/imprisonment of significant Left Oppositionist currents amongst the Yugoslav and Chinese CP activists in Russia, (see Ante Ciliga's 'Russian Enigma' and Wang Fan Hsi's 'Chinese Revolutionary: Memoirs 1919-1949' for graphi accounts of the process in both parties). Once installed they both unswervingly followed the twists and turns of the Comintern line throughout the 1930's, it was only when the interests of their particular national bureaucracies came into conflict with those of the Russian bureaucracy once they'd achieved state power that they discovered their disagreements with Stalin (incidentally, after Stalin had disbanded the Comintern at the behest of his British and American allies in 1943).

As to your comments about Tina, I'd largely agree, but then the political level amongst and capacity for independent political thinking of Comintern activists in the 1930's was appallingly low and few of them could be said to be 'familiar with Marx or Lenin' in the way that CP members had been in the early 1920's.

Supporting Diego's expulsion was an (I'd argue key) point in the process as a result of which by 1937-9 she was willing to knowingly stand by and witness, if not actively participate in, Vidali's role in the liquidation of many of the best fighters in the Spanish revolution (whether or not she later came to regret that, and I'd like to think she did). I also think that this was linked to her abandonment of art as a means of struggle. Green ribbon (talk) 07:30, 7 January 2008 (UTC)

Final point: 'So too, the Chinese Communist Party very obviously did not treat Stalin hagiographically' Well actually, they did, their dispute with the Moscow 'revisionists' was that they'd abandoned Stalinism! Green ribbon (talk) 07:49, 7 January 2008 (UTC)

I would be interested to learn (here or off-line) the (documentary) basis for your assertion that Modotti knew of or was active in Vidali's activities with the Fifth Regiment. As you may know, she chose not to return to Moscow from Spain, and she did not renew her ties with the Mexican CP upon her return to Mexico. I suggest that this reflects some type of critical analysis.

Regarding one of the other examples, many historians have commented on the divergence of the Chinese CP from the CPSU & Comintern following the failed attempts at urban insurrection and in regard to the CCP adopting a strategy grounded in a peasant-based political movement. While the record shows that there was little or no public criticism of Stalin during the 30s and 40s — and that in some parts of the world the penalty for same could be severe — documents made public since that time show that Chinese communists did have significant political and ideological differences with 'Stalinism' in various areas. One document that comes to mind is "A critique of Soviet economics" which is attributed to Mao Tsetung (1977, Monthly Review Press), in which he/they criticize (specifically CPSU) 'personnel of the Party and the state, beset by bureaucratism ... using the machinery of the state to take arbitrary action ... violating Party and state democracy.'Kevincof (talk) 04:18, 8 January 2008 (UTC)

Again, fair questions Kevin, and I'd an concede an element of validity in your points about the CCP I'm highly tempted to respond now but it would require the best part an hour of time which I haven't really got at the moment. I will reply here and if I haven't by the end of the week by all means remind me! Paul Green ribbon (talk) 09:56, 8 January 2008 (UTC)

Modotti as photographer
The article concentrates mostly on the fact that Modotti hung out and/or slept with several prominent artists and leftist political figures of her time, which doesn't seem in itself all that significant to me.

The one accomplishment she seems to have, though, is her work as a photographer. I think this article needs a meaty section on that area of her life, especially since the web searches I've done on her all talk about her primarily as a photographer. I don't know too much about her life, so I'm hoping someone else with more knowledge of Modotti could write this (Carptrash? :-), but it's definitely the missing piece of the puzzle here, AFAIC.

--dablaze  18:08, Jan 16, 2005 (UTC)

Movies
Why was the movies list removed? Vandalism or was something wrong? I vote keep, if it comes down to democracy. Blair P. Houghton 21:52, 21 Jan 2005 (UTC)

Italian-American activist
it appears that someone has added, and then another removed Modotti from this list. I figure that she does not really belong on it because she was never an American. Now a list of Italian - Mexican activists. . .  . . . . . ..........  Carptrash 17:33, 25 Feb 2005 (UTC)

broken external link to George Eastman House
http://www.eastman.org/ has no active link as given in the External links section as   http://www.eastman.org/ar/strip87/htmlsrc3/modotti_sld00001.html

There are many iamges at modotti.com

Roger Moore's 2007 book "Obsidian 22" concerns 10 of her Oaxacan photos.

G. Robert Shiplett 15:42, 21 April 2011 (UTC) — Preceding unsigned comment added by Grshiplett (talk • contribs)

Suggestion of separate gallery of photographs by Tina Modotti
I suggest that there be a separate gallery of photographs by Tina ModottiSmlarthist (talk) 21:06, 8 March 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 4 external links on Tina Modotti. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20081010171350/http://www.getty.edu/art/gettyguide/artMakerDetails?maker=1830 to http://www.getty.edu/art/gettyguide/artMakerDetails?maker=1830
 * Added archive https://web.archive.org/web/20150905134154/http://www.sfmoma.org/explore/multimedia/interactive_features/30 to http://www.sfmoma.org/explore/multimedia/interactive_features/30
 * Added archive https://web.archive.org/web/20160304061516/https://www.kunsthauswien.com/en/exhibitions/archive/24-ausstellungs-archiv-2010/40-tina-modotti to https://www.kunsthauswien.com/en/exhibitions/archive/24-ausstellungs-archiv-2010/40-tina-modotti
 * Added archive https://web.archive.org/web/20060628055725/http://eastman.org/ar/strip87/htmlsrc3/modotti_sld00001.html to http://www.eastman.org/ar/strip87/htmlsrc3/modotti_sld00001.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 00:44, 20 December 2017 (UTC)

Category:Italian artists' models
I cut out the category Category:Italian artists' models because I am not aware of any Italian artists that Modotti modeled for. Feel free to convince me otherwise. Carptrash (talk) 20:34, 24 February 2018 (UTC)

title: Talk:Tina Modotti


text:
Otway

Otway may refer to:

Australia

 * Cape Otway, a geographical feature on the coast of Victoria, Australia
 * Shire of Colac Otway, Victoria, Australia
 * Great Otway National Park, a national park in Victoria, Australia
 * Otway Basin, a geological feature with natural gas reserves spanning the coast of South Australia and Victoria
 * Otway Ranges in Victoria, Australia
 * Parish of Otway, a cadastral land division in the County of Polwarth in Western Victoria

Elsewhere

 * Otway, Ohio, a village in the United States
 * Otway Massif, massif in Antarctica
 * Seno Otway, inland sound in southern Chile

People

 * Otway Burns (1775–1850), American privateer
 * Otway (surname)

Other uses

 * HMAS Otway, two submarines of the Royal Australian Navy
 * Otway (1800 ship), British sailing ship in the slave trade
 * Otway-Rees protocol, computer network authentication protocol
 * SS Otway A passenger liner sunk in 1917

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

title: Otway


text:
Wikipedia:Articles for deletion/Declaration and Resolves of the First Continental Congress (0th nomination)

Declaration and Resolves of the First Continental Congress was proposed for deletion. This page is an archive of the discussion about the proposed deletion. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was transwiki to Wikisource. As of 17:18, 21 Nov 2004 (UTC), this article was still in the queue to be moved. Rossami (talk)

Transwiki to Wikisource. Dunc_Harris|☺ 17:57, 7 Aug 2004 (UTC)

 * Also get George Washington July 9 1776 General Orders. [[User:Meelar|Meelar (talk)]] 18:02, 2004 Aug 7 (UTC)
 * Transwiki to Wikisource for both of them, if the latter isn't alredy there. Geogre 02:59, 8 Aug 2004 (UTC)
 * Agreed. Skyler 21:17, Aug 9, 2004 (UTC)
 * Ditto all. Neutrality 00:23, 12 Aug 2004 (UTC)

This page is now preserved as an archive of the debate and, like other '/delete' pages is no longer 'live'. Subsequent comments on the issue, the deletion or on the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


text: Template:War

title: Template:War


text:
Frank Capra Jr.

Frank Warner Capra (March 20, 1934 – December 19, 2007), known as Frank Capra Jr., was an American film and television producer. He was one of the three children of film director Frank Capra and his second wife, Lucille Warner. His own sons, Frank Capra III and Jonathan Capra, are assistant directors.

At the time of his death, Capra Jr. was president of EUE/Screen Gems studio, which he had helped to found in Wilmington, North Carolina, in the mid-1980s, and a member of the North Carolina Film Council.

Death
Capra Jr. died on December 19, 2007, aged 73, at a hospital in Philadelphia, Pennsylvania, after a long battle with prostate cancer.

Filmography

 * Mark Twain's Greatest Adventure: 'It's a Matter of Time' (2005) (pre-production) (producer)
 * Queen City Blowout (2003) (executive producer)
 * Waterproof (2000) (producer)
 * Death Before Dishonor (1987) (executive producer)
 * Marie (1985) (producer)
 * Firestarter (1984) (producer)
 * The Seduction (1982) (executive producer)
 * Vice Squad (1982) (executive producer)
 * High Hopes: The Capra Years (1981) (TV) (producer)
 * An Eye for an Eye (1981) (producer)
 * The Black Marble (1980) (producer)
 * Born Again (1978) (producer)
 * Billy Jack Goes to Washington (1977) (producer)
 * Trapped Beneath the Sea (1974) (TV) (producer)
 * Battle for the Planet of the Apes (1973) (associate producer)
 * Tom Sawyer (1973/I) (associate producer)
 * Conquest of the Planet of the Apes (1972) (associate producer)
 * Play It Again, Sam (1972) (associate producer)
 * Escape from the Planet of the Apes (1971) (associate producer)
 * Marooned (1969) (associate producer)

title: Frank Capra Jr.


text:
User talk:Sp7412

Your edits to Pensacola Christian College
Please do not remove information from articles. Wikipedia is not censored, and content is not removed even if some believe it to be contentious. Please discuss this issue on the article's talk page to reach consensus rather than continuing to remove the disputed material. You also have the option to configure Wikipedia to hide the images that you may find offensive. --Kralizec! (talk) 02:21, 31 December 2008 (UTC)

Your submission at Articles for creation
Ransom Dunn, which you submitted to Articles for creation, has been created. Thank you for helping Wikipedia!  Chzz  ► 00:15, 25 January 2011 (UTC)
 * Please continue making quality contributions to Wikipedia. Note that because you are a logged-in user, you can create articles yourself, and don't have to post a request.
 * If you would like to help us improve this process, please consider

Speedy deletion nomination of Ransom Dunn

A tag has been placed on Ransom Dunn, requesting that it be speedily deleted from Wikipedia. This has been done for the following reason:

it contains little information/sources and shows no real significance of a living person

Under the criteria for speedy deletion, articles that do not meet basic Wikipedia criteria may be deleted at any time.

If you think that this notice was placed here in error, you may contest the deletion by adding to the top of the page that has been nominated for deletion (just below the existing speedy deletion, or "db", tag; if no such tag exists, then the page is no longer a speedy delete candidate and adding a hang-on tag is unnecessary), coupled with adding a note on the talk page explaining your position, but be aware that once tagged for speedy deletion, if the page meets the criterion, it may be deleted without delay. Please do not remove the speedy deletion tag yourself, but don't hesitate to add information to the page that would render it more in conformance with Wikipedia's policies and guidelines. If the page is deleted, you can contact one of these administrators to request that the administrator userfy the page or email a copy to you. SilverSoul91911 (talk) 19:15, 26 January 2011 (UTC)

DYK for Ransom Dunn
— HJ Mitchell |  Penny for your thoughts?   18:03, 1 February 2011 (UTC)

Notification: changes to "Mark my edits as minor by default" preference
Hello there. This is an automated message to tell you about the gradual phasing out of the preference entitled "Mark all edits minor by default", which you currently have enabled.

On 13 March 2011, this preference was hidden from the user preferences screen as part of efforts to prevent its accidental misuse (consensus discussion). This had the effect of locking users in to their existing preference, which, in your case, was. To complete the process, your preference will automatically be changed to  in the next few days. This does not require any intervention on your part and you will still be able to manually mark your edits as being 'minor'. The only thing that's changed is that you will no longer be able to have them marked as minor by default. For more information on what a minor edit is, see WP:MINOR or feel to get in touch.

Thank you for your understanding and happy editing :) Editing on behalf of User:Jarry1250, LivingBot (talk) 21:20, 13 March 2011 (UTC)

title: User talk:Sp7412


text:
Overfield

Overfield may refer to:
 * Overfield (band)
 * Overfield Township, Pennsylvania
 * Overfield, West Virginia

People with the name Overfield

 * Al Overfield
 * Jackie Overfield
 * Pete Overfield

title: Overfield


text: Country codes: G

title: Country codes: G


text:
The Ashlee Simpson Show

The Ashlee Simpson Show is a television reality show about the life of Ashlee Simpson. The first season, taped from late 2003 to mid-2004, focused on the beginnings of her career as a singer and the recording of her debut album, Autobiography. A second season, taped from late 2004 to early 2005, focused on her career after the album's release.

Consisting of eight half-hour episodes, the first season ran on MTV in the U.S. in the summer of 2004. The show proved quite successful and helped to establish widespread recognition for Simpson, whose album was released on July 20, 2004, a month after the show began airing, and debuted at number one in sales. Jordan Schur, the president of Simpson's record label, Geffen, said that "There's no question that it really helped expose her and her music. She's got a tremendous personality; people gravitate toward that. They want to watch her and listen to her music." The second and final season, consisting of ten episodes, aired in early 2005.

Series overview
The show begins with Simpson signing her record deal with Geffen; the first season ends with the U.S. release of her album, and the second season ends with the launch of her first tour.

Much of the show deals with the process of writing and recording songs. Simpson expresses her preference for rock music and that she wants her album to reflect that; she says she does not want her album to sound like the pop music of Hilary Duff or her sister Jessica. The show ties Simpson's music together with her personal life in that many of the songs are inspired by events in her personal life, both good and bad.

The show also chronicles minor but often humorous tribulations of Simpson's life. Simpson's parents, Joe and Tina, appear frequently, as well as her sister Jessica.

Season 1 (2004)

 * 1) "Ashlee Moves Onward and Upward" – June 16, 2004
 * 2) "Ashlee Verses Her Label" – June 23, 2004
 * 3) "Ashlee Rocks Ryan's World" – June 30, 2004
 * 4) "Valentine's Bummer" – July 7, 2004
 * 5) "Ashlee Strikes a Pose" – July 14, 2004
 * 6) "Ashlee Performs Live" – July 21, 2004
 * 7) "Ashlee Hits the Big Time" – July 28, 2004
 * 8) "Ashlee Goes Platinum" – August 4, 2004

Season 2 (2005)

 * 1) "Ashlee Heads to the Big Apple" – January 26, 2005
 * 2) "Ashlee's Notorious Performance" – February 2, 2005
 * 3) "Ashlee Turns 20" – February 9, 2005
 * 4) "Ashlee Backs Up Her Vocals" — February 16, 2005
 * 5) "New Female Artist of the Year" – February 23, 2005
 * 6) "Jingle Ball Rock" – March 2, 2005
 * 7) "Happy New Year!" – March 9, 2005
 * 8) "The Orange Bowl" – March 16, 2005
 * 9) "Puppy Love" – March 23, 2005
 * 10) "The Show Must Go On..." – March 30, 2005

Broadcast
The first season of The Ashlee Simpson Show debuted in the U.S. on June 16, 2004 and aired every Wednesday until the last episode on August 4. The show aired at 10:30 PM ET, the time slot just after Ashlee's sister Jessica Simpson's already successful reality show Newlyweds: Nick and Jessica. The second season of the show, airing in the same time slot as before and again following Newlyweds, began in the U.S. on January 26, 2005 and ended on March 30.

In Canada, the first season of The Ashlee Simpson Show began showing on MTV Canada on July 1, 2004, airing on Thursdays at 9:00 ET; the second season began airing at the same time on February 10, 2005. Also in Canada, the first season began airing on MuchMusic beginning on November 14, 2004 at 4:30 p.m. ET (having been moved up from the previously planned date of November 21). In the United Kingdom, the first episode of the first season aired on MTV UK on September 27, 2004.

title: The Ashlee Simpson Show


text:
Josia Thugwane

Josia Thugwane (born 15 April 1971) is a South African retired long-distance runner, best known for winning the gold medal in the marathon at the 1996 Summer Olympics. Thugwane, who is of Ndebele heritage, is the first black athlete to earn an Olympic gold for South Africa.

Born in Bethal, Thugwane ran his first marathon in 1991, but his breakthrough to the international athletics scene came in 1995, when he won the Honolulu Marathon.

Just five months before the Games commenced, Thugwane was carjacked and shot; the bullet grazed his chin, leaving an inch-long scar, and he injured his back as a result of jumping from his moving car. The coalmine that employed him paid for his medical care and rehabilitation.

At Atlanta, in the 1996 Olympic marathon, a large leading pack stayed in contact with each other for most of the race, until at the 35 km mark when Thugwane initiated a break away and he along with Lee Bong-Ju from South Korea and Erick Wainaina from Kenya. They stayed together until entering the stadium, when Thugwane got a slight lead. Thugwane finished three seconds ahead of Lee for the closest Olympic marathon finish ever.

Thugwane had a very successful year in 1997 by winning the Fukuoka Marathon and he won the AIMS Best Marathon Runner Award that year. After that point his career performance dipped. He failed to finish in three successive marathons, and finished only twentieth in the 2000 Sydney Olympic marathon despite top ten finishes in the New York Marathon and London Marathon that year. In 2002 he won the Nagano Olympic Memorial Marathon in Japan.

He was awarded the Silver Order of Ikhamanga, South Africa's second highest cultural honour, in 2011.

National titles

 * South African Athletics Championships
 * Marathon: 1993, 1996

title: Josia Thugwane


text:
Four Horsemen (Supreme Court)

The "Four Horsemen" (in allusion to the Four Horsemen of the Apocalypse) was the nickname given by the press to four conservative members of the United States Supreme Court during the 1932–1937 terms, who opposed the New Deal agenda of President Franklin D. Roosevelt. They were Justices Pierce Butler, James Clark McReynolds, George Sutherland, and Willis Van Devanter. They were opposed by the liberal "Three Musketeers"—Justices Louis Brandeis, Benjamin Cardozo, and Harlan Stone. Chief Justice Charles Evans Hughes and Justice Owen J. Roberts controlled the balance. Hughes was more inclined to join the liberals, but Roberts was often swayed to the side of the conservatives.

Opposition to New Deal
Though at first the court had accepted some of the New Deal legislation over the objections of the four conservative justices, in the 1935 term, the Four Horsemen, together with Roberts and Hughes, voided the Agricultural Adjustment Act of 1933 (United States v. Butler, 297 U.S. 1 [1936]), along with the Federal Farm Bankruptcy Act, the Railroad Act, and the Coal Mining Act. In Carter v. Carter Coal Company, 298 U.S. 238 (1936), the Four together with Roberts voided legislation regulating the coal industry; the same line-up voided a New York minimum wage law for women and children in Morehead v. New York, 298 U.S. 587 (1936). The court had also struck down the National Industrial Recovery Act in Schechter Poultry Corp. v. United States 295 U.S. 495 (1935) the previous May. Though the latter decision was unanimous, Cardozo did not join the opinion for the court (written by Hughes and joined by Brandeis, Roberts, and the Four Horsemen), and wrote a separate concurrence, joined by Stone, to state the view of why the delegated power of legislation in the code at issue was "not canalized within banks that keep it from overflowing."

The Four Horsemen would ride in a car to and from the court together to coordinate positions and arguments. They were vehemently opposed to the New Deal policies for unemployment and economic recovery, and they invalidated state laws regulating labor and business relations. The Four's votes kept Congress and the states from regulating the economy. These actions led many observers to the conclusion that the court was likely to be obstructive to all legislative efforts to cope with the depression, and to remain wedded to the precedents of the Lochner era. Some academics argued that the court's aversion to "regulated capitalism" confronted the country with "the question not how governmental functions shall be shared, but whether in substance we shall govern at all."

The result of these dynamics was a steady drift of the court towards a crisis; the 1935 term was labeled by Stone "one of the most disastrous in [the court's] history." New Dealers decried the court's actions as "economic dictatorship" and some communities even hanged the justices in effigy.

Reform Bill of 1937
When Roosevelt rolled out his reform plan in early 1937, all Four Horsemen of Reaction were over the age of 70. Roosevelt, who himself would never make it to the age of 70, indicated that the elderly justices lacked energy. He said, "A lower mental or physical vigor leads men to avoid an examination of complicated and changed conditions."

It was the success of the Horsemen in striking down New Deal legislation that led to Roosevelt's court-packing scheme, a controversial proposal in February 1937 to appoint more justices in order to change the composition of the court. This was rendered unnecessary when Roberts, who had supported the Four Horsemen on several decisions during the 1935–1936 term, sided with the Three Musketeers in a landmark minimum wage case in March 1937 (known as "the switch in time that saved nine"). This, together with the retirement of Van Devanter in June 1937 and his replacement by Hugo Black ended the Four Horsemen's domination of the court. Black and Roosevelt considered the Four the "direct descendants of Darwin and Spencer."

Sutherland retired from the Court in 1938 and Butler died in 1939. McReynolds was the last of the Four Horsemen to step down, retiring in 1941.

title: Four Horsemen (Supreme Court)


text:
Sigurður Kári Kristjánsson

Sigurður Kári Kristjánsson (born May 9, 1973) is an Icelandic Member of Parliament for the Independence Party (Sjálfstæðisflokkurinn).

Kristjánsson graduated with a law degree from the University of Iceland in 1998. He was chairman of The National Youth Organisation of the Independence Party (Samband ungra sjálfstæðismanna) from 1999 to 2001. He has served as Member of the Icelandic parliament Althing since the parliament elections in 2003 and as vice chairman of Heimssýn, the cross-political organisation of Icelandic Eurosceptics, since 2004.

title: Sigurður Kári Kristjánsson


text:
Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations.

Geometrically, a simple bivector can be interpreted as characterizing an directed plane segment, much as vectors can be thought of as characterizing directed line segments. The bivector $a ∧ b$ has an attitude (direction) of the plane spanned by $a ∧ b$ and $a$, has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, it has a magnitude equal to the area of the parallelogram with edges $b$ and $a$), and has an orientation being the side of $b$ on which $a$ lies within the plane spanned by $b$ and $a$.

In layman terms, any surface defines the same bivector if it is parallel to the same plane (same attitude), has the same area, and same orientation (see figure).

Bivectors are generated by the exterior product on vectors: given two vectors $b$ and $a$, their exterior product $b$ is a bivector, as is any sum of bivectors. Not all bivectors can be expressed as an exterior product without such summation. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product of two vectors is alternating, so $a ∧ b$ is the zero bivector, and $a ∧ a$ is the negative of the bivector $b ∧ a$, producing the opposite orientation. Concepts directly related to bivector are rank-2 antisymmetric tensor and skew-symmetric matrix.

History
The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors. Just the previous year, in Ireland, William Rowan Hamilton had discovered quaternions. Hamilton coined both vector and bivector, the latter in his Lectures on Quaternions (1853) as he  introduced biquaternions, which have bivectors  for their vector parts. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector of this article arose. Henry Forder used the term bivector to develop exterior algebra in 1941.

In the 1890s Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication. The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, had the effect that the insights of Hamilton and Clifford were overlooked for a long time, since much of 20th century mathematics and physics was formulated in vector terms. Gibbs used vectors to fill the role of bivectors in three dimensions, and used bivector in Hamilton's sense, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a quadratic form. Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics.

Derivation
For this article, the bivector will be considered only in real geometric algebras, which may be applied in most areas of physics. Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form.

Geometric algebra and the geometric product
The bivector arises from the definition of the geometric product over a vector space with an associated quadratic form sometimes called the metric. For vectors $a ∧ b$, $a$ and $b$, the geometric product satisfies the following properties: \mathbf{a}(\mathbf{b} + \mathbf{c}) &= \mathbf{ab} + \mathbf{ac} \\ (\mathbf{b} + \mathbf{c})\mathbf{a} &= \mathbf{ba} + \mathbf{ca} \end{align}$$
 * Associativity : $$ (\mathbf{ab})\mathbf{c} = \mathbf{a}(\mathbf{bc}) $$
 * Left and right distributivity : $$\begin{align}
 * Scalar square : $\mathbf{a}^2 = Q(\mathbf{a})$, where $c$ is the quadratic form, which need not be positive-definite.

Scalar product
From associativity, $Q$, is a scalar times $a(ab) = a^{2}b$. When $b$ is not parallel to and hence not a scalar multiple of $b$, $a$ cannot be a scalar. But
 * $$\tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}) = \tfrac{1}{2} \left((\mathbf{a} + \mathbf{b})^2 - \mathbf{a}^2 - \mathbf{b}^2\right)$$

is a sum of scalars and so a scalar. From the law of cosines on the triangle formed by the vectors its value is $ab$, where $θ$ is the angle between the vectors. It is therefore identical to the scalar product between two vectors, and is written the same way,
 * $$\mathbf{a} \cdot \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}).$$

It is symmetric, scalar-valued, and can be used to determine the angle between two vectors: in particular if $|a| |b| cos θ$ and $a$ are orthogonal the product is zero.

Exterior product
Just as the scalar product can be formulated as the symmetric part of the geometric product of another quantity, the exterior product (sometimes known as the "wedge" or "progressive" product) can be formulated as its antisymmetric part:
 * $$\mathbf{a} \wedge \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} - \mathbf{ba})$$

It is antisymmetric in $b$ and $a$
 * $$\mathbf{b} \wedge \mathbf{a} = \tfrac{1}{2}(\mathbf{ba} - \mathbf{ab}) = -\tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) = -\mathbf{a} \wedge \mathbf{b}$$

and by addition:
 * $$\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \wedge \mathbf{b} = \tfrac{1}{2}(\mathbf{ab} + \mathbf{ba}) + \tfrac{1}{2}(\mathbf{ab} - \mathbf{ba}) = \mathbf{ab}$$

That is, the geometric product is the sum of the symmetric scalar product and alternating exterior product.

To examine the nature of $b$, consider the formula
 * $$(\mathbf{a} \cdot \mathbf{b})^2 - (\mathbf{a} \wedge \mathbf{b})^2 = \mathbf{a}^2\mathbf{b}^2,$$

which using the Pythagorean trigonometric identity gives the value of $a ∧ b$
 * $$(\mathbf{a} \wedge \mathbf{b})^2 = (\mathbf{a} \cdot \mathbf{b})^2 - \mathbf{a}^2\mathbf{b}^2 = \left|\mathbf{a}\right|^2\left|\mathbf{b}\right|^2( \cos^2 \theta - 1) = -\left|\mathbf{a}\right|^2\left|\mathbf{b}\right|^2\sin^2 \theta$$

With a negative square, it cannot be a scalar or vector quantity, so it is a new sort of object, a bivector. It has magnitude $(a ∧ b)^{2}$, where $|a| |b$ is the angle between the vectors, and so is zero for parallel vectors.

To distinguish them from vectors, bivectors are written here with bold capitals, for example:
 * $$\mathbf{A} = \mathbf{a} \wedge \mathbf{b} = -\mathbf{b} \wedge \mathbf{a} \,,$$

although other conventions are used, in particular as vectors and bivectors are both elements of the geometric algebra.

Properties
The algebra generated by the geometric product (that is, all objects formed by taking repeated sums and geometric products of scalars and vectors) is the geometric algebra over the vector space. For an Euclidean vector space, this algebra is written $$\mathcal{G}_n$$ or $θ$, where $n$ is the dimension of the vector space $Cl_{n}(R)$. $R^{n}$ is both a vector space and an algebra, generated by all the products between vectors in $Cl_{n}(R)$, so it contains all vectors and bivectors. More precisely, as a vector space it contains the vectors and bivectors as linear subspaces, though not as subalgebras (since the geometric product of two vectors is not generally another vector).

The space ⋀2Rn
The space of all bivectors has dimension $R^{n}$ and is written $1⁄2n(n − 1)$, and is the second exterior power of the original vector space.

Even subalgebra
The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written $⋀^{2}R^{n}$. This algebra results from considering all repeated sums and geometric products of scalars and bivectors. It has dimension $Cl[0] n(R)$, and contains $2^{n−1}$ as a linear subspace. In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest. In two dimensions, the even subalgebra is isomorphic to the complex numbers, $⋀^{2}R^{n}$, while in three it is isomorphic to the quaternions, $C$. The even subalgebra contains the rotations in any dimension.

Magnitude
As noted in the previous section the magnitude of a simple bivector, that is one that is the exterior product of two vectors $H$ and $a$, is $b$, where $θ$ is the angle between the vectors. It is written $|a| |b| sin θ$, where $|B|$ is the bivector.

For general bivectors, the magnitude can be calculated by taking the norm of the bivector considered as a vector in the space $B$. If the magnitude is zero then all the bivector's components are zero, and the bivector is the zero bivector which as an element of the geometric algebra equals the scalar zero.

Unit bivectors
A unit bivector is one with unit magnitude. Such a bivector can be derived from any non-zero bivector by dividing the bivector by its magnitude, that is
 * $$\frac{\mathbf{B}}{\left\vert\mathbf{B}\right\vert}.$$

Of particular utility are the unit bivectors formed from the products of the standard basis of the vector space. If $⋀^{2}R^{n}$ and $e_{i}$ are distinct basis vectors then the product $e_{j}$ is a bivector. As $e_{i} ∧ e_{j}$ and $e_{i}$ are orthogonal, $e_{j}$, written $e_{i} ∧ e_{j} = e_{i}e_{j}$, and has unit magnitude as the vectors are unit vectors. The set of all bivectors produced from the basis in this way form a basis for $e_{ij}$. For instance, in four dimensions the basis for $⋀^{2}R^{n}$ is ($⋀^{2}R^{4}$, $e_{1}e_{2}$, $e_{1}e_{3}$, $e_{1}e_{4}$, $e_{2}e_{3}$, $e_{2}e_{4}$) or ($e_{3}e_{4}$, $e_{12}$, $e_{13}$, $e_{14}$, $e_{23}$, $e_{24}$).

Simple bivectors
The exterior product of two vectors is a bivector, but not all bivectors are exterior products of two vectors. For example, in four dimensions the bivector
 * $$\mathbf{B} = \mathbf{e}_1 \wedge \mathbf{e}_2 + \mathbf{e}_3 \wedge \mathbf{e}_4 = \mathbf{e}_1\mathbf{e}_2 + \mathbf{e}_3\mathbf{e}_4 = \mathbf{e}_{12} + \mathbf{e}_{34}$$

cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by a directed plane area.

Product of two bivectors
The geometric product of two bivectors, $e_{34}$ and $A$, is
 * $$\mathbf{A}\mathbf{B} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B} + \mathbf{A} \wedge \mathbf{B}.$$

The quantity $B$ is the scalar-valued scalar product, while $A · B$ is the grade 4 exterior product that arises in four or more dimensions. The quantity $A ∧ B$ is the bivector-valued commutator product, given by
 * $$\mathbf{A} \times \mathbf{B} = \tfrac{1}{2}(\mathbf{AB} - \mathbf{BA}),$$

The space of bivectors $A × B$ is a Lie algebra over $⋀^{2}R^{n}$, with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra.

Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to
 * $$\mathbf{A}\mathbf{A} = \mathbf{A} \cdot \mathbf{A} + \mathbf{A} \wedge \mathbf{A}.$$

If the bivector is simple the last term is zero and the product is the scalar-valued $R$, which can be used as a check for simplicity. In particular the exterior product of bivectors only exists in four or more dimensions, so all bivectors in two and three dimensions are simple.

General bivectors and matrices
Bivectors are isomorphic to skew-symmetric matrices in any number of dimensions. For example, the general bivector $A · A$ in three dimensions maps to the matrix
 * $$M_B = \begin{pmatrix} 0 & B_{12} & -B_{31} \\ -B_{12} & 0 & B_{23}\\ B_{31} & -B_{23} & 0  \end{pmatrix}.$$

This multiplied by vectors on both sides gives the same vector as the product of a vector and bivector minus the exterior product; an example is the angular velocity tensor.

Skew symmetric matrices generate orthogonal matrices with determinant $B_{23}e_{23} + B_{31}e_{31} + B_{12}e_{12}$ through the exponential map. In particular, applying the exponential map to a bivector that is associated with a rotation yields a rotation matrix. The rotation matrix $1$ given by the skew-symmetric matrix above is
 * $$M_R = \exp{M_B}.$$

The rotation described by $M_{R}$ is the same as that described by the rotor $R$ given by
 * $$R = \exp{\tfrac{1}{2} B},$$

and the matrix $M_{R}$ can be also calculated directly from rotor $R$. In three dimensions, this is given by
 * $$M_R = \begin{pmatrix} (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_1 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_1 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_1 \\ (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_2 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_2 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_2 \\ (R\mathbf{e}_1R^{-1}) \cdot \mathbf{e}_3 & (R\mathbf{e}_2R^{-1}) \cdot \mathbf{e}_3 & (R\mathbf{e}_3R^{-1}) \cdot \mathbf{e}_3 \end{pmatrix}.$$

Bivectors are related to the eigenvalues of a rotation matrix. Given a rotation matrix $M$ the eigenvalues can be calculated by solving the characteristic equation for that matrix $M_{R}$. By the fundamental theorem of algebra this has three roots (only one of which is real as there is only one eigenvector, i.e., the axis of rotation). The other roots must be a complex conjugate pair. They have unit magnitude so purely imaginary logarithms, equal to the magnitude of the bivector associated with the rotation, which is also the angle of rotation. The eigenvectors associated with the complex eigenvalues are in the plane of the bivector, so the exterior product of two non-parallel eigenvectors results in the bivector (or a multiple thereof).

Two dimensions
When working with coordinates in geometric algebra it is usual to write the basis vectors as ($0 = det(M − λI)$, a convention that will be used here.

A vector in real two-dimensional space $e_{1}, e_{2}, ...)$ can be written $R^{2}$, where $a = a_{1}e_{1} + a_{2}e_{2}$ and $a_{1}$ are real numbers, $a_{2}$ and $e_{1}$ are orthonormal basis vectors. The geometric product of two such vectors is
 * $$\begin{align} \mathbf{a}\mathbf{b} &= (a_1\mathbf{e}_1 + a_2\mathbf{e}_2)(b_1\mathbf{e}_1 + b_2\mathbf{e}_2) \\&= a_1b_1\mathbf{e}_1\mathbf{e}_1 + a_1b_2\mathbf{e}_1\mathbf{e}_2 + a_2b_1\mathbf{e}_2\mathbf{e}_1 + a_2b_2\mathbf{e}_2\mathbf{e}_2 \\&= a_1b_1 + a_2b_2 + (a_1b_2 - a_2b_1)\mathbf{e}_1\mathbf{e}_2. \end{align}$$

This can be split into the symmetric, scalar-valued, scalar product and an antisymmetric, bivector-valued exterior product:
 * $$\begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1b_1 + a_2b_2, \\ \mathbf{a} \wedge \mathbf{b} &= (a_1b_2 - a_2b_1)\mathbf{e}_1\mathbf{e}_2 = (a_1b_2 - a_2b_1)\mathbf{e}_{12}. \end{align}$$

All bivectors in two dimensions are of this form, that is multiples of the bivector $e_{2}$, written $e_{1}e_{2}$ to emphasise it is a bivector rather than a vector. The magnitude of $e_{12}$ is $e_{12}$, with
 * $$\mathbf{e}_{12}^2 = -1,$$

so it is called the unit bivector. The term unit bivector can be used in other dimensions but it is only uniquely defined (up to a sign) in two dimensions and all bivectors are multiples of $1$. As the highest grade element of the algebra $e_{12}$ is also the pseudoscalar which is given the symbol $i$.

Complex numbers
With the properties of negative square and unit magnitude, the unit bivector can be identified with the imaginary unit from complex numbers. The bivectors and scalars together form the even subalgebra of the geometric algebra, which is isomorphic to the complex numbers $e_{12}$. The even subalgebra has basis $C$, the whole algebra has basis $(1, e_{12})$.

The complex numbers are usually identified with the coordinate axes and two-dimensional vectors, which would mean associating them with the vector elements of the geometric algebra. There is no contradiction in this, as to get from a general vector to a complex number an axis needs to be identified as the real axis, $(1, e_{1}, e_{2}, e_{12})$ say. This multiplies by all vectors to generate the elements of even subalgebra.

All the properties of complex numbers can be derived from bivectors, but two are of particular interest. First as with complex numbers products of bivectors and so the even subalgebra are commutative. This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions.

Second a general bivector can be written
 * $$\theta\mathbf{e}_{12} = i\theta,$$

where $θ$ is a real number. Putting this into the Taylor series for the exponential map and using the property $e_{1}$ results in a bivector version of Euler's formula,
 * $$\exp{\theta\mathbf{e}_{12}} = \exp{i\theta} = \cos{\theta} + i\sin{\theta},$$

which when multiplied by any vector rotates it through an angle $e_{12}^{2} = −1$ about the origin:
 * $$(x'\mathbf{e}_1 + y'\mathbf{e}_2) = (x\mathbf{e}_1 + y\mathbf{e}_2) \exp {i\theta}.$$

The product of a vector with a bivector in two dimensions is anticommutative, so the following products all generate the same rotation
 * $$\mathbf{v}' = \mathbf{v} \exp{i\theta} = \exp(-i\theta)\,\mathbf{v} = \exp({-i\theta}/{2})\,\mathbf{v} \exp({i\theta}/{2}).$$

Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol $R$, so in two dimensions a rotor that rotates through angle $θ$ can be written
 * $$R = \exp({-\tfrac{1}{2} i\theta}) = \exp({-\tfrac{1}{2}\theta\mathbf{e}_{12}}),$$

and the rotation it generates is
 * $$\mathbf{v}' = R\mathbf{v}R^{-1}.$$

Three dimensions
In three dimensions the geometric product of two vectors is
 * $$\begin{align} \mathbf{ab} &= (a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3)(b_1\mathbf{e}_1 + b_2\mathbf{e}_2 + b_3\mathbf{e}_3) \\ &= a_1 b_1{\mathbf{e}_1}^2 + a_2 b_2{\mathbf{e}_2}^2 + a_3 b_3{\mathbf{e}_3}^2 + (a_2 b_3 - a_3 b_2)\mathbf{e}_2\mathbf{e}_3 + (a_3 b_1 - a_1 b_3)\mathbf{e}_3\mathbf{e}_1 + (a_1 b_2 - a_2 b_1)\mathbf{e}_1\mathbf{e}_2. \end{align}$$

This can be split into the symmetric, scalar-valued, scalar product and the antisymmetric, bivector-valued, exterior product:
 * $$\begin{align} \mathbf{a} \cdot \mathbf{b} &= a_1b_1 + a_2b_2 + a_3b_3 \\ \mathbf{a} \wedge \mathbf{b} &= (a_2 b_3 - a_3 b_2)\mathbf{e}_{23} + (a_3 b_1 - a_1 b_3)\mathbf{e}_{31} + (a_1 b_2 - a_2 b_1)\mathbf{e}_{12}. \end{align}$$

In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors $θ$, $e_{23}$ and $e_{31}$ form a basis for the space of bivectors $e_{12}$, which is itself a three-dimensional linear space. So if a general bivector is:
 * $$\mathbf{A} = A_{23}\mathbf{e}_{23} + A_{31}\mathbf{e}_{31} + A_{12}\mathbf{e}_{12}, $$

they can be added like vectors
 * $$\mathbf{A} + \mathbf{B} = (A_{23} + B_{23})\mathbf{e}_{23} + (A_{31} + B_{31})\mathbf{e}_{31} + (A_{12} + B_{12})\mathbf{e}_{12}.$$

while when multiplied they produce the following
 * $$\mathbf{A} \mathbf{B} = -A_{23}B_{23} - A_{31}B_{31} - A_{12}B_{12} + (A_{12}B_{31} - A_{31}B_{12})\mathbf{e}_{23} + (A_{23}B_{12} - A_{12}B_{23})\mathbf{e}_{31} + (A_{31}B_{23} - A_{23}B_{31})\mathbf{e}_{12}$$

which can be split into symmetric scalar and antisymmetric bivector parts as follows
 * $$\begin{align} \mathbf{A} \cdot \mathbf{B} &= -A_{12}B_{12} - A_{31}B_{31} - A_{23}B_{23} \\ \mathbf{A} \times \mathbf{B} &= (A_{23}B_{31} - A_{31}B_{23})\mathbf{e}_{12} + (A_{12}B_{23} - A_{23}B_{12})\mathbf{e}_{13} + (A_{31}B_{12} - A_{12}B_{31})\mathbf{e}_{23}. \end{align}$$

The exterior product of two bivectors in three dimensions is zero.

A bivector $⋀^{2}R^{3}$ can be written as the product of its magnitude and a unit bivector, so writing $β$ for $B$ and using the Taylor series for the exponential map it can be shown that
 * $$\exp{\mathbf{B}} = \exp({\beta\frac{\mathbf{B}}{\beta}}) = \cos{\beta} + \frac{\mathbf{B}}{\beta}\sin{\beta}.$$

This is another version of Euler's formula, but with a general bivector in three dimensions. Unlike in two dimensions bivectors are not commutative so properties that depend on commutativity do not apply in three dimensions. For example, in general $|B|$ in three (or more) dimensions. The full geometric algebra in three dimensions, $exp(A + B) ≠ exp(A) exp(B)$, has basis ($Cl_{3}(R)$, $1$, $e_{1}$, $e_{2}$, $e_{3}$, $e_{23}$, $e_{31}$, $e_{12}$). The element $e_{123}$ is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors to which they are related, as discussed below.

Quaternions
Bivectors are not closed under the geometric product, but the even subalgebra is. In three dimensions it consists of all scalar and bivector elements of the geometric algebra, so a general element can be written for example $e_{123}$, where $a$ is the scalar part and $a + A$ is the bivector part. It is written $A$ and has basis $Cl[0] 3$. The product of two general elements of the even subalgebra is
 * $$(a + \mathbf{A})(b + \mathbf{B}) = ab + a\mathbf{B} + b\mathbf{A} + \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}.$$

The even subalgebra, that is the algebra consisting of scalars and bivectors, is isomorphic to the quaternions, $(1, e_{23}, e_{31}, e_{12})$. This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector scalar product $H$. Other quaternion properties can be similarly related to or derived from geometric algebra.

This suggests that the usual split of a quaternion into scalar and vector parts would be better represented as a split into scalar and bivector parts; if this is done the quaternion product is merely the geometric product. It also relates quaternions in three dimensions to complex numbers in two, as each is isomorphic to the even subalgebra for the dimension, a relationship that generalises to higher dimensions.

Rotation vector
The rotation vector, from the axis–angle representation of rotations, is a compact way of representing rotations in three dimensions. In its most compact form, it consists of a vector, the product of a unit vector $A · B$ that is the axis of rotation with the (signed) angle of rotation $ω$, so that the magnitude of the overall rotation vector $θ$ equals the (unsigned) rotation angle.

The quaternion associated with the rotation is
 * $$q = \left(\cos{\tfrac{2} \theta}, \omega \sin{\tfrac{1}{2} \theta}\right)$$

In geometric algebra the rotation is represented by a bivector. This can be seen in its relation to quaternions. Let $θω$ be a unit bivector in the plane of rotation, and let $θ$ be the angle of rotation. Then the rotation bivector is $Ω$. The quaternion closely corresponds to the exponential of half of the bivector $Ωθ$. That is, the components of the quaternion correspond to the scalar and bivector parts of the following expression: $$\exp{\tfrac{1}{2} \boldsymbol{\Omega} \theta} = \cos{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sin{\tfrac{1}{2} \theta} $$

The exponential can be defined in terms of its power series, and easily evaluated using the fact that $Ωθ$ squared is $Ω$.

So rotations can be represented by bivectors. Just as quaternions are elements of the geometric algebra, they are related by the exponential map in that algebra.

Rotors
The bivector $−1$ generates a rotation through the exponential map. The even elements generated rotate a general vector in three dimensions in the same way as quaternions: $$\mathbf{v}' = \exp(-\tfrac{1}{2} \boldsymbol{\Omega} \theta)\,\mathbf{v} \exp(\tfrac{1}{2} \boldsymbol{\Omega} \theta).$$

As in two dimensions, the quantity $Ωθ$ is called a rotor and written $R$. The quantity $exp(−1⁄2Ωθ)$ is then $exp(1⁄2Ωθ)$, and they generate rotations as

This is identical to two dimensions, except here rotors are four-dimensional objects isomorphic to the quaternions. This can be generalised to all dimensions, with rotors, elements of the even subalgebra with unit magnitude, being generated by the exponential map from bivectors. They form a double cover over the rotation group, so the rotors $R$ and $R^{−1}$ represent the same rotation.

Axial vectors

The rotation vector is an example of an axial vector. Axial vectors, or pseudovectors, are vectors with the special feature that their coordinates undergo a sign change relative to the usual vectors (also called "polar vectors") under inversion through the origin, reflection in a plane, or other orientation-reversing linear transformation. Examples include quantities like torque, angular momentum and vector magnetic fields. Quantities that would use axial vectors in vector algebra are properly represented by bivectors in geometric algebra. More precisely, if an underlying orientation is chosen, the axial vectors are naturally identified with the usual vectors; the Hodge dual then gives the isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa; that is
 * $$\mathbf{A} = {\star} \mathbf{a} \,,\quad \mathbf{a} = {\star} \mathbf{A}$$

where $m$ is the Hodge star. Note that if the underlying orientation is reversed by inversion through the origin, both the identification of the axial vectors with the usual vectors and the Hodge dual change sign, but the bivectors don't budge. Alternately, using the unit pseudoscalar in $−R$, $x$ gives
 * $$\mathbf{A} = \mathbf{a}i \,,\quad \mathbf{a} = - \mathbf{A} i.$$

This is easier to use as the product is just the geometric product. But it is antisymmetric because (as in two dimensions) the unit pseudoscalar $n$ squares to $p$, so a negative is needed in one of the products.

This relationship extends to operations like the vector-valued cross product and bivector-valued exterior product, as when written as determinants they are calculated in the same way:
 * $$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix} \,,\quad \mathbf{a} \wedge \mathbf{b} = \begin{vmatrix} \mathbf{e}_{23} & \mathbf{e}_{31} & \mathbf{e}_{12}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3 \end{vmatrix}\,,$$

so are related by the Hodge dual:
 * $${\star} (\mathbf a \wedge \mathbf b ) = \mathbf {a \times b} \,,\quad {\star} (\mathbf {a \times b} ) = \mathbf a \wedge \mathbf b\,.$$

Bivectors have a number of advantages over axial vectors. They better disambiguate axial and polar vectors, that is the quantities represented by them, so it is clearer which operations are allowed and what their results are. For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalise to other dimensions; in particular bivectors can be used to describe quantities like torque and angular momentum in two as well as three dimensions. Also, they closely match geometric intuition in a number of ways, as seen in the next section.

Geometric interpretation

As suggested by their name and that of the algebra, one of the attractions of bivectors is that they have a natural geometric interpretation. This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions. In two dimensions the geometric interpretation is trivial, as the space is two-dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor.

All bivectors can be interpreted as planes, or more precisely as directed plane segments. In three dimensions there are three properties of a bivector that can be interpreted geometrically:
 * The arrangement of the plane in space, precisely the attitude of the plane (or alternately the rotation, geometric orientation or gradient of the plane), is associated with the ratio of the bivector components. In particular the three basis bivectors, $GL(n, R)$, $A → A det(A) / |det(A)|$ and $Cl_{3}(R)$, or scalar multiples of them, are associated with the $i = e_{1}e_{2}e_{3}$-plane, $−1$-plane and $a ∧ b$-plane respectively.
 * The magnitude of the bivector is associated with the area of the plane segment. The area does not have a particular shape so any shape can be used. It can even be represented in other ways, such as by an angular measure. But if the vectors are interpreted as lengths the bivector is usually interpreted as an area with the same units, as follows.
 * Like the direction of a vector a plane associated with a bivector has a direction, a circulation or a sense of rotation in the plane, which takes two values seen as clockwise and counterclockwise when viewed from viewpoint not in the plane. This is associated with a change of sign in the bivector, that is if the direction is reversed the bivector is negated. Alternately if two bivectors have the same attitude and magnitude but opposite directions then one is the negative of the other.
 * If imagined as a parallelogram, with the origin for the vectors at 0, then signed area is the determinant of the vectors' Cartesian coordinates ($e_{23}$).

In three dimensions all bivectors can be generated by the exterior product of two vectors. If the bivector $e_{31}$ then the magnitude of $e_{12}$ is
 * $$|\mathbf{B}| = |\mathbf{a}||\mathbf{b}|\sin\theta,$$

where ${\star}$ is the angle between the vectors. This is the area of the parallelogram with edges $yz$ and $zx$, as shown in the diagram. One interpretation is that the area is swept out by $xy$ as it moves along $a_{x} b_{y} − b_{x }a_{y}$. The exterior product is antisymmetric, so reversing the order of $a × b$ and $a ∧ b$ to make $B = a ∧ b$ move along $B$ results in a bivector with the opposite direction that is the negative of the first. The plane of bivector $a$ contains both $b$ and $b$ so they are both parallel to the plane.

Bivectors and axial vectors are related by Hodge dual. In a real vector space the Hodge dual relates a subspace to its orthogonal complement, so if a bivector is represented by a plane then the axial vector associated with it is simply the plane's surface normal. The plane has two normals, one on each side, giving the two possible orientations for the plane and bivector.

This relates the cross product to the exterior product. It can also be used to represent physical quantities, like torque and angular momentum. In vector algebra they are usually represented by vectors, perpendicular to the plane of the force, linear momentum or displacement that they are calculated from. But if a bivector is used instead the plane is the plane of the bivector, so is a more natural way to represent the quantities and the way they act. It also unlike the vector representation generalises into other dimensions.

The product of two bivectors has a geometric interpretation. For non-zero bivectors $a$ and $a$ the product can be split into symmetric and antisymmetric parts as follows:
 * $$\mathbf{AB} = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \times \mathbf{B}.$$

Like vectors these have magnitudes $b$ and $a$, where $i$ is the angle between the planes. In three dimensions it is the same as the angle between the normal vectors dual to the planes, and it generalises to some extent in higher dimensions.

Bivectors can be added together as areas. Given two non-zero bivectors $b$ and $a ∧ b$ in three dimensions it is always possible to find a vector that is contained in both, $a$ say, so the bivectors can be written as exterior products involving $b$:
 * $$\begin{align}\mathbf{B} &= \mathbf{a} \wedge \mathbf{b}\\ \mathbf{C} &= \mathbf{a} \wedge \mathbf{c}\end{align}$$

This can be interpreted geometrically as seen in the diagram: the two areas sum to give a third, with the three areas forming faces of a prism with $F$, $τ$, $p$ and $L$ as edges. This corresponds to the two ways of calculating the area using the distributivity of the exterior product:
 * $$\begin{align} \mathbf{B} + \mathbf{C} &= \mathbf{a} \wedge \mathbf{b} + \mathbf{a} \wedge \mathbf{c} \\ &= \mathbf{a} \wedge (\mathbf{b} + \mathbf{c}).\end{align}$$

This only works in three dimensions as it is the only dimension where a vector parallel to both bivectors must exist. In higher dimensions bivectors generally are not associated with a single plane, or if they are (simple bivectors) two bivectors may have no vector in common, and so sum to a non-simple bivector.

Four dimensions
In four dimensions, the basis elements for the space $A$ of bivectors are ($B$, $|A · B| = |A| |B| cos θ$, $|A × B| = |A| |B| sin θ$, $B$, $C$, $a$), so a general bivector is of the form
 * $$\mathbf{A} = a_{12}\mathbf{e}_{12} + a_{13}\mathbf{e}_{13} + a_{14}\mathbf{e}_{14} + a_{23}\mathbf{e}_{23} + a_{24}\mathbf{e}_{24} + a_{34}\mathbf{e}_{34}.$$

Orthogonality
In four dimensions, the Hodge dual of a bivector is a bivector, and the space $a$ is dual to itself. Normal vectors are not unique, instead every plane is orthogonal to all the vectors in its Hodge dual space. This can be used to partition the bivectors into two 'halves', in the following way. We have three pairs of orthogonal bivectors: $a$, $b$ and $c$. There are four distinct ways of picking one bivector from each of the first two pairs, and once these first two are picked their sum yields the third bivector from the other pair. For example, $b + c$ and $⋀^{2}R^{4}$.

Simple bivectors in 4D
In four dimensions bivectors are generated by the exterior product of vectors in $e_{12}$, but with one important difference from $e_{13}$ and $e_{14}$. In four dimensions not all bivectors are simple. There are bivectors such as $e_{23}$ that cannot be generated by the exterior product of two vectors. This also means they do not have a real, that is scalar, square. In this case
 * $$(\mathbf{e}_{12} + \mathbf{e}_{34})^2 =\mathbf{e}_{12} \mathbf{e}_{12} + \mathbf{e}_{12} \mathbf{e}_{34} + \mathbf{e}_{34} \mathbf{e}_{12} + \mathbf{e}_{34} \mathbf{e}_{34} = -2 + 2 \mathbf{e}_{1234}.$$

The element $e_{24}$ is the pseudoscalar in $e_{34}$, distinct from the scalar, so the square is non-scalar.

All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as
 * $$\mathbf{e}_{12} + \mathbf{e}_{34} = \mathbf{e}_{1} \wedge \mathbf{e}_{2} + \mathbf{e}_{3} \wedge \mathbf{e}_{4}.$$

Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique. The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero.

Rotations in R4
As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions if $⋀^{2}R^{4}$ is a bivector then the rotor $θ$ is $(e_{12}, e_{34})$ and rotations are generated in the same way:
 * $$v' = RvR^{-1}.$$

The rotations generated are more complex though. They can be categorised as follows:
 * simple rotations are those that fix a plane in 4D, and rotate by an angle "about" this plane.
 * double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified
 * isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique.

These are generated by bivectors in a straightforward way. Simple rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors that the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique.

Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector $(e_{13}, e_{24})$, where $(e_{14}, e_{23})$ and $(e_{12}, e_{13}, e_{14})$ are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows:
 * $$R = \exp(\tfrac{1}{2} (\mathbf{B}_1 + \mathbf{B}_2)) = \exp(\tfrac{1}{2} \mathbf{B}_1)\,\exp(\tfrac{1}{2} \mathbf{B}_2) = \exp(\tfrac{1}{2} \mathbf{B}_2)\,\exp(\tfrac{1}{2} \mathbf{B}_1)$$

It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors.

Spacetime rotations
Spacetime is a mathematical model for our universe used in special relativity. It consists of three space dimensions and one time dimension combined into a single four-dimensional space. It is naturally described using geometric algebra and bivectors, with the Euclidean metric replaced by a Minkowski metric. That algebra is identical to that of Euclidean space, except the signature is changed, so
 * $$\mathbf{e}_i^2 = \begin{cases} 1, & i = 1, 2, 3 \\  -1, & i = 4 \end{cases}$$

(Note the order and indices above are not universal – here $(e_{23}, e_{24}, e_{34})$ is the time-like dimension). The geometric algebra is $R^{4}$, and the subspace of bivectors is $R^{3}$.

The simple bivectors are of two types. The simple bivectors $R^{2}$, $e_{12} + e_{34}$ and $e_{1234}$ have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, $Cl_{4}$. These bivectors generate ordinary rotations in $B$.

The simple bivectors $exp 1⁄2B$, $B = B_{1} + B_{2}$ and $B_{1}$ have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:
 * $$\exp{\tfrac{1}{2}{\boldsymbol{\Omega}\theta}} = \cosh{\tfrac{1}{2} \theta} + \boldsymbol{\Omega}\sinh{\tfrac{1}{2} \theta},$$

where $B_{2}$ is the bivector ($e_{4}$, etc.), identified via the metric with an antisymmetric linear transformation of $Cl_{3,1}(R)$. These are Lorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in $⋀^{2}R^{3,1}$ and $e_{23}$.

In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector $e_{31}$ is of the form
 * $$R = \exp{\tfrac{1}{2} \mathbf{A}}.$$

The set of all rotations in spacetime form the Lorentz group, and from them most of the consequences of special relativity can be deduced. More generally this show how transformations in Euclidean space and spacetime can all be described using the same kind of algebra.

Maxwell's equations
(Note: in this section traditional 3-vectors are indicated by lines over the symbols and spacetime vector and bivectors by bold symbols, with the vectors $e_{12}$ and $R^{3}$ exceptionally in uppercase)

Maxwell's equations are used in physics to describe the relationship between electric and magnetic fields. Normally given as four differential equations they have a particularly compact form when the fields are expressed as a spacetime bivector from $R^{3}$. If the electric and magnetic fields in $e_{14}$ are $e_{24}$ and $e_{34}$ then the electromagnetic bivector is
 * $$\mathbf{F} = \frac{1}{c}\overline{E}\mathbf{e}_4 + \overline{B}\mathbf{e}_{123},$$

where $Ω$ is again the basis vector for the time-like dimension and $θ$ is the speed of light. The product $e_{14}$ yields the bivector that is Hodge dual to $R^{3,1}$ in three dimensions, as discussed above, while $R^{3}$ as a product of orthogonal vectors is also bivector-valued. As a whole it is the electromagnetic tensor expressed more compactly as a bivector, and is used as follows. First it is related to the 4-current $R^{4}$, a vector quantity given by
 * $$\mathbf{J} = \overline{j} + c\rho\mathbf{e}_4,$$

where $A$ is current density and $R$ is charge density. They are related by a differential operator ∂, which is
 * $$\partial = \nabla - \mathbf{e}_4\frac{1}{c}\frac{\partial}{\partial t}.$$

The operator ∇ is a differential operator in geometric algebra, acting on the space dimensions and given by $J$. When applied to vectors $A$ is the divergence and $⋀^{2}R^{3,1}$ is the curl but with a bivector rather than vector result, that is dual in three dimensions to the curl. For general quantity $R^{3}$ they act as grade lowering and raising differential operators. In particular if $\overline{E}$ is a scalar then this operator is just the gradient, and it can be thought of as a geometric algebraic del operator.

Together these can be used to give a particularly compact form for Maxwell's equations with sources:
 * $$\partial\mathbf{F} = \mathbf{J}.$$

This equation, when decomposed according to geometric algebra, using geometric products which have both grade raising and grade lowering effects, is equivalent to Maxwell's four equations. It is also related to the electromagnetic four-potential, a vector $\overline{B}$ given by
 * $$\mathbf{A} = \overline{A} + \frac{1}{c}V\mathbf{e}_4,$$

where $e_{4}$ is the vector magnetic potential and $c$ is the electric potential. It is related to the electromagnetic bivector as follows
 * $$\partial\mathbf{A} = -\mathbf{F},$$

using the same differential operator $\overline{B}e_{123}$.

Higher dimensions
As has been suggested in earlier sections much of geometric algebra generalises well into higher dimensions. The geometric algebra for the real space $\overline{B}$ is $\overline{E}e_{4}$, and the subspace of bivectors is $J$.

The number of simple bivectors needed to form a general bivector rises with the dimension, so for $ρ$ odd it is $\overline{j}$, for $V$ even it is $∇M = ∇·M + ∇∧M$. So for four and five dimensions only two simple bivectors are needed but three are required for six and seven dimensions. For example, in six dimensions with standard basis ($∇·M$, $∇∧M$, $M$, $M$, $A$, $\overline{A}$) the bivector
 * $$\mathbf{e}_{12} + \mathbf{e}_{34} + \mathbf{e}_{56}$$

is the sum of three simple bivectors but no less. As in four dimensions it is always possible to find orthogonal simple bivectors for this sum.

Rotations in higher dimensions
As in three and four dimensions rotors are generated by the exponential map, so
 * $$\exp{\tfrac{1}{2} \mathbf{B}}$$

is the rotor generated by bivector $∂$. Simple rotations, that take place in a plane of rotation around a fixed blade of dimension $R^{n}$ are generated by simple bivectors, while other bivectors generate more complex rotations which can be described in terms of the simple bivectors they are sums of, each related to a plane of rotation. All bivectors can be expressed as the sum of orthogonal and commutative simple bivectors, so rotations can always be decomposed into a set of commutative rotations about the planes associated with these bivectors. The group of the rotors in $n$ dimensions is the spin group, $Cl_{n}(R)$.

One notable feature, related to the number of simple bivectors and so rotation planes, is that in odd dimensions every rotation has a fixed axis – it is misleading to call it an axis of rotation as in higher dimensions rotations are taking place in multiple planes orthogonal to it. This is related to bivectors, as bivectors in odd dimensions decompose into the same number of bivectors as the even dimension below, so have the same number of planes, but one extra dimension. As each plane generates rotations in two dimensions in odd dimensions there must be one dimension, that is an axis, that is not being rotated.

Bivectors are also related to the rotation matrix in $n$ dimensions. As in three dimensions the characteristic equation of the matrix can be solved to find the eigenvalues. In odd dimensions this has one real root, with eigenvector the fixed axis, and in even dimensions it has no real roots, so either all or all but one of the roots are complex conjugate pairs. Each pair is associated with a simple component of the bivector associated with the rotation. In particular, the log of each pair is the magnitude up to a sign, while eigenvectors generated from the roots are parallel to and so can be used to generate the bivector. In general the eigenvalues and bivectors are unique, and the set of eigenvalues gives the full decomposition into simple bivectors; if roots are repeated then the decomposition of the bivector into simple bivectors is not unique.

Projective geometry
Geometric algebra can be applied to projective geometry in a straightforward way. The geometric algebra used is $⋀^{2}R^{n}$, the algebra of the real vector space $(n − 1) / 2$. This is used to describe objects in the real projective space $n / 2$. The non-zero vectors in $e_{1}$ or $e_{2}$ are associated with points in the projective space so vectors that differ only by a scale factor, so their exterior product is zero, map to the same point. Non-zero simple bivectors in $e_{3}$ represent lines in $e_{4}$, with bivectors differing only by a (positive or negative) scale factor representing the same line.

A description of the projective geometry can be constructed in the geometric algebra using basic operations. For example, given two distinct points in $e_{5}$ represented by vectors $e_{6}$ and $B$ the line containing them is given by $(n − 2)$ (or $Spin(n)$). Two lines intersect in a point if $Cl_{n}(R), n ≥ 3$ for their bivectors $R^{n}$ and $RP^{n−1}$. This point is given by the vector
 * $$\mathbf{p} = \mathbf{A} \lor \mathbf{B} = (\mathbf{A} \times \mathbf{B}) J^{-1}.$$

The operation "$Cl_{n}(R)$" is the meet, which can be defined as above in terms of the join, $R^{n}$ for non-zero $⋀^{2}R^{n}$. Using these operations projective geometry can be formulated in terms of geometric algebra. For example, given a third (non-zero) bivector $RP^{n−1}$ the point $RP^{n−1}$ lies on the line given by $a$ if and only if
 * $$\mathbf{p} \land \mathbf{C} = 0.$$

So the condition for the lines given by $b$, $a ∧ b$ and $b ∧ a$ to be collinear is
 * $$(\mathbf{A} \lor \mathbf{B}) \land \mathbf{C} = 0,$$

which in $A ∧ B = 0$ and $A$ simplifies to
 * $$\langle \mathbf{ABC} \rangle = 0,$$

where the angle brackets denote the scalar part of the geometric product. In the same way all projective space operations can be written in terms of geometric algebra, with bivectors representing general lines in projective space, so the whole geometry can be developed using geometric algebra.

Tensors and matrices
As noted above a bivector can be written as a skew-symmetric matrix, which through the exponential map generates a rotation matrix that describes the same rotation as the rotor, also generated by the exponential map but applied to the vector. But it is also used with other bivectors such as the angular velocity tensor and the electromagnetic tensor, respectively a 3×3 and 4×4 skew-symmetric matrix or tensor.

Real bivectors in $B$ are isomorphic to $∨$ skew-symmetric matrices, or alternately to antisymmetric tensors of degree 2 on $J = A ∧ B$. While bivectors are isomorphic to vectors (via the dual) in three dimensions they can be represented by skew-symmetric matrices in any dimension. This is useful for relating bivectors to problems described by matrices, so they can be re-cast in terms of bivectors, given a geometric interpretation, then often solved more easily or related geometrically to other bivector problems.

More generally, every real geometric algebra is isomorphic to a matrix algebra. These contain bivectors as a subspace, though often in a way which is not especially useful. These matrices are mainly of interest as a way of classifying Clifford algebras.

title: Bivector


text:
User:Denni/User talk:Denni/2004 July Archive

''. . Hi there. How are you today?''

SAY THANKEE, SAI
Click here to leave a message

<:> ACHTUNG <:>
'''This here's my talk page.

'''I refactor at will.

'''I put things where I can find them.

'''If what you say is important, I keep it. If it's not, I don't.

'''I read everything here. I reply to the sane. I definitely reply if you've remembered to take your meds. '''My IQ is 176 and I have read all of Voltaire in his original Norwegian. Sometimes I lie.'''

MY PERSONAL SANDBOX
 Invited Guests Only  Please do not poop in the sand.

User:Dwindrim/Testaroonies!!!

User:Dwindrim/Wildflowers of the Canadian Rocky Mountains

THE DARK SIDE OF WIKI
 There is no dark side of Wiki. Wiki is all dark.

ARCHIVES
2004 February Archive - 2004 February - 2004 April 10 Archive - 2004 April Archive - 2004 May Archive - 2004 June Archive -2004 July Archive -

Sysop
Congratualtions! You're a sysop. You might want to read the Administrators' reading list before unleashing your new powers upon the Wikiworld. Cheers! -- Cecropia | Talk 15:32, 1 Jul 2004 (UTC)

Excuse me?
Unfortunately, your first encounter was with someone who ought to have his virtual butt kicked into a pixel pit - nice. Rick'''K 23:56, Jul 4, 2004 (UTC)

Yes, I'm new to Wikipedia. I realized what to do, and I left a message on article's talk page. Thank you very much for guidelines.

Grzes 00:36, 21 Jul 2004 (UTC)

Holocaust
Although the two events can certainly fairly be considered a Holocaust, I think the issue is much more whether they are considered a Holocaust. "Dresden Holocaust" and "Hiroshima Holocaust" both come in at 200 Google hits, which is pretty low. Unless you can think of a better term to search for, I don't think the content is appropriate for a disambiguation page, honestly. Snowspinner 01:49, Jul 25, 2004 (UTC)

 * I'm just not comfortable with the use of Wikipedia as a vehicle for establishing truth. We document current usage. The problem is that the Holocaust disambig page is already open to being overloaded. (In fact, I'm half-tempted to re-cull it on a harsher standard) I agree that Dresden shouldn't be forgotten. I'm actually kind of surprised List of mass deaths doesn't exist, because I think an article like that is needed (And would be a good see also from the disambig page). But I don't think this particular page is the right place for this. Snowspinner 02:19, Jul 25, 2004 (UTC)

Wild flowers
Well, it certainly is money well spent ! I am glad and I thank you :-) Any additional flower in Wikipedia is worth this. I put one on my talk page. Orange and blue are my favorite colors. I took the lily :-) Keep on with the good work SweetLittleFluffyThing 09:53, 1 Aug 2004 (UTC)

¿What do you have against Genital Integrity and Intact Day?
Ŭalabio 21:57, 2004 Aug 1 (UTC)

How blackholes give information back.
Thank you "Dwindrim" for your reaction to my article-information. I accept your point of view. Anyhow I feel fine that several of you now know about my findings. Thank you for your wise comment. Sincerely Dan Visser, Netherlands.

circumcision
dennis,hello! great photos! thanks for sharing them. just wanted to let you know that mr.ualabio is pretty upset over circumcision for a good cause. he was not able to orgasm due to his very extreme cut until he restored his forskin by tugging methods. i don't know if men believe in compassion toward one another, but i hope you do. i am female, born in kansas and i too was circumcised. so the genital integrity movement makes a lot of sense to me. i wish my parents had allowed me to remain natural, whole. as a therapist, i am well aware of the trauma that occurs when someone is hurt in that way. that is what involved me in the issue to begin with. when i learned about my own circumcision, i was very surprised. it was a family secret. all i have to show for it is scar tissue. but from my perspective as a healer, i see not a lot of difference between hurting a girl and hurting a boy. either is unnecessary and can have longterm detrimental effects to the quality of life of that individual and ... well, all i can say is that i don't believe the fact that the three cultures at war in the middle east are all three circumcisers of their young, whereas 83-85% of the men in the world are intact... ??? coincidence? i don't know. i DO know that i was an angry young woman until i discovered and healed my own sexual wounds. i think circumcision of the young may be playing with fire. not all will react in the same way, but when someone is inclined in the direction of anger/violence, they just might manifest that in their: family, city, state, country, world. i think the question deserves some very concentrated study before circumcising even one more baby in the US. best wishes! sincerely, patricia


text:
User talk:Europekid2oo4

I appreciate the addition of an image to Ashlee Simpson, but are you sure it isn't a copyvio? Everyking 18:18, 7 Aug 2004 (UTC)
 * Oh, I'm sorry. I wasn't sure if it is copyrighted or not. How do you know if the photo is or not? Becuase I couldn't find anything official on the page I retrieved it from originally.
 * I have to assume it is, just because it's from a commercial photo shoot for the album and all. Would be great if we could find a noncopyrighted picture, though. Everyking 18:47, 7 Aug 2004 (UTC)
 * Ok, but I'm not sure how to remove the picture from the whole Wikipedia database...
 * You'd have to list it on the copyvio page, along with the source. Conceivably, the image might be fair use? I don't know enough to say, but album covers are explicitly mentioned as fair use, at least, so I suppose an image of the cover would be fine, and could go in the Autobiography (album) article as well. (Although I think that picture on the album cover does not do her justice at all!) Everyking 20:13, 7 Aug 2004 (UTC)

Unverified image
Thanks for uploading the image I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GFDL, or  if you claim fair use, etc.) If you don't know what any of this means, just let me know where you got the image and I'll tag it for you. Thanks, Kbh3rd 05:50, 29 Dec 2004 (UTC)
 * Image:Central Europe.bmp

title: User talk:Europekid2oo4


text:
John McLean (disambiguation)

John McLean (1785–1861) was an American jurist and politician.

John McLean may also refer to:

Politics

 * John McLean (Illinois politician) (1791–1830), Illinois politician and U.S. Senator
 * John McLean Jr. (1793–1858), New York politician
 * John McLean (New Zealand politician) (1818–1902), member of the New Zealand Legislative Council
 * John Donald McLean (1820–1866), politician and colonial Treasurer of Queensland
 * John McLean (Canadian politician) (1846–1936), Canadian politician from Prince Edward Island
 * John R. McLean (Canadian politician) (1906–1964), merchant and political figure on Prince Edward Island

Sports

 * John McLean (athlete) (1878–1955), American Olympic athlete and head football coach at Missouri, 1903–1905
 * John McLean (rower) (1859–1925), Australian sculler
 * Jock McLean (John Calderwood McLean, 1908–1988), Scottish footballer
 * John McLean (footballer, born 1872) (1872–?), Scottish footballer
 * John McLean (footballer, born 1877) (1877–1958), Scottish footballer
 * Johnny McLean (1881–?), Scottish footballer

Other

 * John McLean (merchant) (1761–1823), Massachusetts merchant, major donor to the Massachusetts General Hospital and Harvard University
 * John McLean (furniture maker) (1770–1825), English Upholder and cabinet maker
 * John McLean (explorer) (c. 1799–1890), Scoto-Canadian trapper, explorer, and author
 * John McLean (bishop) (1828–1866), first Bishop of Saskatchewan
 * John R. McLean (publisher) (1848–1916), owner of The Washington Post, The Cincinnati Enquirer, and the Great Falls and Old Dominion Railroad
 * John Walford McLean (1925–2009), British dentist
 * Jackie McLean (John Lenwood McLean, 1931–2006), American jazz saxophonist
 * John McLean (1851–1928), Scoto-Canadian Methodist minister and author who later changed his name to John Maclean

title: John McLean (disambiguation)


text:
Catherine (1999 TV series)

Catherine was a Quebec sitcom that aired on Radio-Canada from 1999 to 2003. It tells the story of Catherine (Sylvie Moreau), a sexy, epicurean, man-crazy Montrealer in her thirties working at the advertising agency Mirage-Image, as well as the story of her best friend and orderly flatmate Sophie (Marie-Hélène Thibault), her landlord Rachel (Dominique Michel), her ex-boyfriend Pierre (Charles Lafortune) and other friends and co-workers.

title: Catherine (1999 TV series)


text:
Current wars

Current wars can refer to:

 * Wars presently being waged, see List of ongoing armed conflicts
 * War of the currents, a late 19th century commercial battle over the choice of AC or DC for electricity supply
 * The Current War, a 2017 historical drama film about the war of the currents

title: Current wars


text: Category:Canadian painters

title: Category:Canadian painters