Wikibooks Fractals Apollonian fractals In mathematics , an Apollonian gasket or Apollonian net is a fractal ... after Greece Greek mathematician Apollonius of Perga . Construction Image Apollonian gasket.svg thumb An example of an Apollonian gasket An Apollonian gasket can be constructed as follows. Start with three ... these are called Apollonian circles see Descartes theorem . Adding the two Apollonian circles to the original three, we now have five circles. Take one of the two Apollonian circles &ndash say C sub ... 1 sub and C sub 2 sub has its own two Apollonian circles. We already know one of these &ndash it is C ... a total of 3 sup n 1 sup 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket. The Apollonian gasket has a Hausdorff dimension of about 1.3057 http abel.math.harvard.edu .... This circle is in the interior of circle with negative curvature. Variations Image Apollonian spheres.jpg thumb left Apollonian sphere packing An Apollonian gasket can also be constructed ... s. The three dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing . Symmetries ... a radius that is two thirds of this, then the Apollonian gasket has two lines of reflective symmetry ... the Apollonian gasket also has rotational symmetry of degree 2 the symmetry group of this gasket is D sub 2 sub . If all three of the original generating circles have the same radius then the Apollonian ... of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another, so the Apollonian gasket also has rotational symmetry of degree 3 the symmetry group of this gasket ... which maps any two Apollonian gaskets to one another. M bius transformations are also isometries of the Hyperbolic geometry hyperbolic plane , so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, which can be thought of as a tessellation of the hyperbolic plane by circles and hyperbolic triangles. The Apollonian gasket ... more details
File Apollonian network.svg thumb 240px An Apollonian network File Goldner Harary graph.svg thumb The Goldner Harary graph , a non Hamiltonian Apollonian network In combinatorics combinatorial mathematics , an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar ... of Perga , who studied a related circle packing construction. Definition An Apollonian network may ... , are both Apollonian networks. math K sub 3 sub is formed by starting with a triangle and not performing .... The Goldner Harary graph is an Apollonian network that forms the smallest Hamiltonian cycle ... 357. ref Another more complicated Apollonian network was used by harvtxt Nishizeki 1980 to provide ... Apollonian networks are examples of maximal element maximal planar graph s, graphs to which ... that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks they are exactly the chordal maximal planar graphs ... . ref In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing ... tree math k tree , and Apollonian networks are examples of 3 trees. Not every 3 tree is planar, but the planar 3 trees are exactly the Apollonian networks. Every Apollonian network is also a uniquely ..., but also true, that every uniquely 4 colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4 colorable planar graphs. Apollonian ... k 4 . ref For the characterization of Apollonian networks via unique 4 colorability, see harvtxt Fowler 1998 . The fact that Apollonian networks also minimize the number of colorings with larger numbers of colors was shown in a dual form for colorings of maps by harvtxt Birkhoff 1930 . ref The Apollonian ... 2009 . ref The Apollonian networks do not form a family of graphs that is closed under the operation ... more details
The Apollonian and Dionysian is a philosophical and literary concept , or dichotomy , based on certain ... the use of the concepts of the Apollonian and Dionysian is famously related to Nietzsche s The Birth ... of Dionysian and Apollonian Kunsttrieben artistic impulses forms dramatic arts, or tragedies ... and reason robs tragedy of its foundation, namely the fragile balance of the Dionysian and Apollonian ... the Apollonian and Dionysian juxtapositions is apparent, Nietzsche claimed in The Birth of Tragedy ... to make order in the Apollonian sense of his unjust and chaotic Dionysian Fate, though he dies unfulfilled ... in Apollonian ideals, which in turn separate him from his essential connection with self. The Dionysian ... connected with the Apollonian. The Dionysian magnifies man, but only so far as he ... experience is of utmost importance in viewing the Dionysian as it is related to the Apollonian ... American humanities scholar Camille Paglia writes about the Apollonian and Dionysian in her controversial ... significantly. The Apollonian and Dionysian concepts comprise a dichotomy that serves as the basis of Paglia s theory of art and culture. For Paglia, the Apollonian is light and structured while the Dionysian ..., and unconstrained sex procreation. In contrast, the Apollonian is associated with males, clarity ..., 1990, p. 40 ref She argues that there is a biological basis to the Apollonian Dionysian dichotomy ..., and turning instead to the Apollonian trait of ordered creation. The Dionysian is a force of chaos ... of &ndash or combat with &ndash Chthonianism by socially constructed Apollonian virtues accounts ... Apollo program Apollo . As this anecdote shows, the Apollonian tendency would also seem to include ... According to Peter Sloterdijk , the Dionysian and the Apollonian form a dialectic they are contrasting ... being is determined by the Dionysian Apollonian dialectic. Extending the use of the Apollonian and Dionysian .... Stephen King s usage American novelist Stephen King uses Apollonian and Dionysian analysis in Danse ... more details
Image Apollonian circles.svg thumb right 350px Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points, C and D , and every blue circle separates the two points. This article discusses a family of circles sharing a radical axis , and the corresponding family of orthogonal circles. For other circles associated with Apollonius of Perga , please see the disambiguation page, circles of Apollonius . Apollonian circles are two families of circle s such that every circle in the first family intersects every circle in the second family orthogonal ly, and vice versa. These circles form the basis for bipolar coordinates . They were discovered by Apollonius of Perga , a renowned Ancient Greece Greek geometer . Definition The Apollonian circles are defined in two different ways by a line segment denoted CD . Each circle in the first family the blue circles in the figure is associated with a positive real number r , and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r , math left X mid frac d X,C d X,D r right . math For values of r close to zero, the corresponding circle is close to C , while for values of r close to , the corresponding circle is close to D for the intermediate value r 1, the circle degenerates to a line, the perpendicular bisector of CD ... k pi right math Pencils of circles Both of the families of Apollonian circles are called pencils ... parabolic pencil. It is relatively easy to show using inversion that, in the Apollonian circles ... of the blue Apollonian circles with respect to a circle centered on point C results in a pencil of concentric ... coordinates bipolar coordinate system defined by the Apollonian circles into a polar coordinates ..., so the original Apollonian circles also meet at right angles. Alternatively, ref harvtxt Akopyan ... pencil is hyperbolic, and vice versa in this case the two pencils form a set of Apollonian circles ... more details
Image Apollonian spheres.jpg thumb right Apollonian sphere packing Apollonian sphere packing is the three dimensional equivalent of the Apollonian gasket . The principle of construction is very similar with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them. The fractal dimension is 2.473946. ref Citation first M. last Borkovec first2 W. last2 De Paris first3 R. last3 Peikert author link publication date date year 1994 title The Fractal Dimension of the Apollonian Sphere Packing periodical Fractals series publication place place publisher volume 2 issue 4 pages 521 526 url http graphics.ethz.ch peikert papers apollonian.pdf issn doi 10.1142 S0218348X94000739 oclc accessdate 2008 09 15 ref Software for generating and visualization of the apollonian sphere packing ApolFrac. ref http thomasbonner.heliohost.org apolfrac.htm ApolFrac ref References references Category Hyperbolic geometry Category Fractals geometry stub it Impacchettamento di sfere apolloniano ... more details
The Guard may refer to The Guard TV series The Guard TV series , a Canadian drama television series portraying the life of the Canadian Coast Guard along the British Columbia Coast The Guard 1990 film The Guard 1990 film , a 1990 Soviet film The Guard 2001 film The Guard 2001 film , a 2001 Indian Malayalam film The Guard 2011 film The Guard 2011 film , a 2011 Irish film The Guard novel The Guard novel , 2008 Arabic language novel by Ezzat el Kamhawi See also Guard disambiguation Guardian disambiguation disambiguation it The Guard ... more details
On Guard may refer to Le Bossu 1997 film transl. The Hunchback alternative English title, On Guard On Guard novel On Guard , a novel by Upton Sinclair On Guard serial , a 1927 film serial See also En garde disambiguation , various meanings derived from a French phrase used to warn one s fencing opponent to be on guard disambig ... more details
wiktionary guardGuard may refer to TOC right Professional occupations Bodyguard , who protects an individual from personal assault Crossing guard , who stops traffic so pedestrians can cross the street Life guard , who rescues people from drowning Prison guard , who supervises prisoners in a prison or jail Security guard , who protects property, assets, or people conductor transportation Train guard Train conductor , in the UK, Australia, New Zealand, and India Governmental and military see also ..., infantry or cavalry Border guard , a state security agency. Civic guard Civil Guard civil guard , a police force. Coast guard , responsible for coastal defence and offshore rescue. Honor Guard , primarily ceremonial. National Guard disambiguation , primarily domestic defense units which have military or law enforcement missions Sports Guard American football , a player between the center and the tackles on the offensive line of a team Guard grappling , a position in martial arts in which the person ... legs entangling the opponent for control. basketball position Point guard , or playmaker Shooting guard , or off guard In historical swordmanship color guard flag spinning , or guard was a defensive position. Guard colorguard People who toss flags, spin a rifle and a sabre at performances with or without ... to the performance Others Guard computing , in programming language, an expression that directs program execution Guard weapon , part of the handle of a sword designed to stop the user s hand from slipping onto the blade Guard dog , employed to watch for unwanted or unexpected animals or people Guard interval , intervals in transmission, used in telecommunications In buildings, a guard can mean ... surface Abdominal guarding , in medicine, the tensing of the abdominal wall muscles to guard inflamed organs The aircraft emergency frequency , commonly referred to as guard Mate guarding guarding of a potential or former mate from other individuals See also lookfrom intitle The Guard disambiguation ... more details
Conceptdab Presidential Guard may refer to Evzones , the Greek Presidential Guard, formerly the Royal Guard. Presidential Guard Belarus Presidential Guard South Vietnam Presidential Guard Zimbabwe Presidential Guard Brigade Uganda Estado Mayor Presidencial Presidential Guard Brigade Mexico See also Republican Guard disambiguation mil unit dis ... more details
The Changing Of The Guard may refer to the Guard Mounting , a military ceremony A ceremony at Buckingham Palace , Queen s Guard Changing the Queen.27s Life Guard Changing the Queen s Life Guard . a 1967 song by the Marquis of Kensington project an episode of the MTV series Celebrity Deathmatch The Changing of the Guard Starflyer 59 album The Changing of the Guard , the 2010 album by the band Starflyer 59 The Changing of the Guard The Twilight Zone , an episode of the television series The Twilight Zone dab ... more details
The Changing of the Guard or The Change of the Guard can refer to Guard Mounting , or Changing the Guard, a formal ceremony in which sentries providing ceremonial guard duties at important institutions are relieved by a new batch of sentries Star Wars Jedi Quest 8 The Changing of the Guard The Changing of the Guard Starflyer 59 album , an album by indie rock band, Starflyer 59 The Changing of the Guard The Twilight Zone , an episode of The Twilight Zone 1959 TV series The Twilight Zone Changing of the Guards , a song by Bob Dylan Changing of the Guard , a song by Exodus band Exodus from their album Impact is Imminent Change of the Guard , a song by Steely Dan from their album Can t Buy a Thrill disambig ... more details
Unreferenced date December 2009 The quarter guard is a military depot in an Indian Army unit. The quarter guard is the main point of security arrangements for the army camp garrison . The regiment al colours, the Armory military armoury and the treasury would be kept in this building. In addition to this the quarter guard also has a lock up to hold soldier s charged with minor crime s absent without leave , drunkenness being one of the usual crimes . The standard guard strength is 6 guards 1 guard 2 ic 2nd in command 1 guard commander usually a Naik or Havildar . Additionally in some units a bugler would also be present for calling the various duty calls etc. At the time of emergencies threats the strength of the guard would be increased. The Armoury is a part of the quarter guard. A Quarter Guard is a place of pride for the Army Unit and the guards detailed on duty there are always in ceremonials. A visiting dignitary invariably visits the Quarter Guard. A quarter Guard is expected to reflect the high standards of the unit. it has some relation with the number of troops in presidential guard. DEFAULTSORT Quarter Guard Category Indian Army India mil stub ... more details
Image Milizie venete a Verona bis.jpg thumb 180px Verona s Noble Guard Verona Noble Guard in blue and yellow and tricorne s and Schiavoni troops in red jackets and black fez hat fez s during a re enactment of the Pasque Veronesi Noble Guard was an Italian term used to describe many different early modern and modern military guards unit s intended for the personal protection of a sovereign, including Noble Guard Vatican Noble Guard Verona Noble Guard Hungary mil unit dis de Nobelgarde ... more details
wikt Splash guard may refer to An attachment to a snorkel Mudflap , on a vehicle Face shield , on a helmet Splatter guard , on cookware Shower splash guard Any of various devices such as a levee dyke levee used in construction to prevent reduce splash damage See also lookfrom intitle Guard disambiguation Splash disambiguation dab ... more details
Conceptdab Republican Guard is the organization of a republic which serves to protect the President and the government. Usually synonymous with Presidential Guard . Albanian Republican Guard Algerian Republican Guard French Republican Guard called the Garde r publicaine GR Iraqi Republican Guard , the core of the Iraqi military during Saddam Hussein s rule Iraqi Special Republican Guard , a military force formed from the Iraqi Republican Guard and charged with Saddam Hussein s protection Italian National Republican Guard Italian Republican National Guard called the Guardia Nazionale Repubblicana GNR Egyptian Republican Guard , under the control of the Ministry of Defense and numbers about 60,000 personnel and they are mainly for ceremonials and parades. The Republican Guard armored division is not a part of the Corps, it is rather for the defence of the Presidential institution and the Capital. Syrian Republican Guard , an armored division that protects the capital and top Syrian government officials Armed Forces of the Republic of Kazakhstan Kazakhstan Republican Guard , a separate branch of the military than the army Republican Guard Brigade Lebanon Lebanese Republican Guard , a military force attached to the Directorate General of the Presidency of Lebanon Peruvian Republican Guard , a Peruvian security force responsibility for border control, custody of the prisons, and guarding significant government buildings. Portuguese Republican National Guard called the Guarda Nacional Republicana GNR Republican Guard Democratic Republic of the Congo under President Joseph Kabila Guinean Republican Guard South Ossetian Republican Guard Yemeni Republican Guard See also See also Presidential Guard disambig de Republikanische Garde fr Garde r publicaine ko ru ... more details
tocright The term National Guard originally referred to a French citizen militia see National Guard France . The term is now used in many countries. Depending on the country in question, national guard may refer to an organized militia , a military force, a paramilitary force, a gendarmerie , or a police force Forces Americas National Guard of the United States a federally recognized militia organized by each of the 50 U.S. state s and territories that serves both as a military reserve force of the U.S and a Militia United States state militia . It consists of Army National Guard and Air National Guard . National Guard Nicaragua 1925 1979 National Guard El Salvador 1912 1992 Venezuelan National Guard Guardia Nacional de Venezuela officially the Armed Forces of Cooperation Spanish Fuerzas Armadas de Cooperaci n National Guard Dominican Republic Europe National Guards Unit of Bulgaria National Guard of Georgia Cypriot National Guard National Guard France , active on and off from the French Revolution to the Paris Commune Italian National Republican Guard defunct Internal Troops Russia Kaitseliit Estonian National Guard Latvian National Guard Portuguese National Republican Guard National Militia Spain , 19th century White Guard Finland White Guard , Finland. 1918 1944 Africa Tunisian National Guard Asia Sri Lanka National Guard Saudi Arabian National Guard National Guard Bahrain Iraqi National Guard Military of Kuwait Kuwaiti National Guard National Guard of Georgia The National Guard of Pakistan is an army reserve force counted as part of the paramilitary forces of Pakistan Political organizations National Guard, better known as the Blueshirts , a 1930s political movement in Ireland, antecedent to the modern Irish political party, Fine Gael See also List of protective service agencies Republican National Guard disambiguation Republican Guard disambiguation Home Guard disambiguation Civil Guard disambiguation Gendarmerie mil unit dis ar ca Mil cia Nacional ... more details
USS Guard may refer to USS Guard 1857 , purchased by the U.S. Navy in 1861 as USS National Guard renamed in 1866 final ship decommissioning decommissioning in 1878 sold in 1883 USS Guard 1913 , a tug built for the United States Coast Guard at the Mare Island Naval Shipyard served in the U.S. Navy during World War I returned to the Coast Guard in 1919 decommissioned and sold in 1943 USS Guard YP 2384 , a wooden motor launch built in 1902 purchased as Floyd Hurst by the U.S. Navy in 1918 served as dispatch boat between Washington Navy Yard and the Indian Head Naval Proving Ground renamed, then sold in 1921 Shipindex DEFAULTSORT Guard, Uss Category United States Navy ship names ... more details
Home Guard or Home Army may refer to Military Home Guard United Kingdom British Home Guard Combat Groups of the Working Class German Kampfgruppen der Arbeiterklasse, KdA Confederate Home Guard , during the American Civil War Croatian Home Guard disambiguation Croatian Home Guard Danish Home Guard Omakaitse , Estonian Home Guard in WW II. Kaitseliit , Estonian Defence League. Active. Landstorm , Male conscripts over 32 years old of the European legal class in the Royal Dutch East Indies Army KNIL of the Dutch East Indies . Local Defence troops Finland Home Guard Unionist , during the American Civil War Royal Hong Kong Regiment Indian Home Guard . Zemessardze , Latvian National Guard. Active. Lithuanian Territorial Defense Force 1944 Militia National Guard of the United States Norwegian Home Guard Polish Home Army or Armia Krajowa. Slovene Home Guard . Sri Lankan Home Guard State Defense Forces of the various United States of America Swedish Home Guard Territorial Army United Kingdom Territorial Army , British reserve forces Territorial Defense Forces Yugoslavia Volkssturm , German Home Guard Second World War Volunteer Defence Corps Australia Volunteer Defence Corps , Australia Volunteer Fighting Corps Japan Second World War Wachdienst , an auxiliary organisation erected by the Third Reich in Germany during the last months of World War II. Other The Old Home Guard , a song in the movie Bedknobs and Broomsticks Home Guard Shannara , in Terry Brooks Shannara novel series Civil Guard disambiguation See also Home front Dad s Army mil unit dis sk Domobrana rozli ovacia str nka ... more details
Young Guard may refer to The Young Guard Napoleon , a French elite military unit during the time of Napoleon Bonaparte Young Guard , associated with Soviet Union and Russia Molodaya Gvardiya magazine Molodaya Gvardiya magazine , a monthly literary magazine published since 1922 Molodaya Gvardiya publisher , a publishing house in the Soviet Union and Russia, established in 1922 The Young Guard Soviet resistance , a Soviet resistance organisation during the World War II, composed mainly of teenagers The Young Guard novel The Young Guard novel , a 1945 rev. 1951 novel by Alexander Fadeyev about the Soviet resistance organisation The Young Guard film The Young Guard film , a 1949 Soviet film directed by Sergei Gerasimov, based on Alexander Fadeyev s novel The Young Guard of United Russia , a pro Kremlin direction action youth organization, youth wing of United Russia Disambig cs Mlad garda de Junge Garde fr Jeune Garde homonymie he ru sk Mlad garda ... more details
The term White Guard may refer to White Guard Finland , part of the White Army of the 1918 Finnish Civil War White Guard Slovenia , part of the Axis powers Axis collaborators in Slovenia during the Second World War Military arm of the Russian White movement The White Guard , a novel by Mikhail Bulgakov about the Russian White movement disambig hu Feh r g rda egy rtelm s t lap it Guardia Bianca ru sl Bela garda sr ... more details
dabconcept Civil Guard refers to various policing organisations Current Civil Guard Spain The Spanish gendarmerie Civil Guard Israel An Israeli volunteer police reserve Civil Guard Costa Rica Historic Civil Guards now abolished Civil Guard Peru , formed as main police Preventive police preventive police force of Peru in 1924, later became General Police which in 1988 merged into new National Police Civil Guard Colombia , created in 1902 Civil Guard El Salvador , created in 1867, which then gave way to the Guardia Nacional El Salvador Guardia Nacional in 1912. Civil Guard Honduras , a militarized police commanded directly by President of Honduras president Ramon Villeda Morales rather than the chief of the armed forces created in 1957 Civil Guard Panama abolished Civil Guard Philippines , a local gendarmerie organized under the auspices of the Spanish colonial authorities with a contingent of local soldiers that died off with the Spanish American war in 1898, now being reestablished in the city of Ozamiz City Ozamiz http www.sunstar.com.ph static cag 2006 07 05 news .guardia.civil.set.for.revival.html . In the Intramuros district of Manila, security forces are dressed in guardia civil uniforms http www.philippines travel guide.com intramuros.html . Civil Guard South Vietnam Gwardya Sibil Philippine resistance network , a civilian underground network operating during the years 1936 to 1939 to gather intelligence on the activities of the Japanese invaders http www.psdtc.com Pekiti.Tirsia pekiti.tirsia.history.htm . White Guard Finland Suojeluskunta , a Finnish militia for which Civil Guard is one of the many English translations. See also National Guard disambiguation Civic Guard disambiguation Home Guard disambiguation disambig he es Guardia Civil desambiguaci n ... more details
Revolutionary Guard is a military or paramilitary elite unit that plays the role of key protection force of the new regime in a country which has experienced a revolution . Revolutionary Guard Corps of Libya Army of the Guardians of the Islamic Revolution of Iran are also known as the Iranian Revolutionary Guards IRG or predominantly in US media the Iranian Revolutionary Guard Corps IRGC See also Republican Guard disambiguation disamb ... more details
Unreferenced date October 2010 See also Neck guard A throat guard is a piece of protective equipment worn in various sports, including baseball, ice hockey, lacrosse. History In baseball, the guard was invented in 1976 by Los Angeles Dodgers trainer Bill Buhler after his catcher Steve Yeager was injured in the throat by pieces of a broken bat. The guard is required in numerous youth and amateur baseball leagues, including Little League. Category Protective gear ... more details
Refimprove date October 2010 Image Torre Verger Banyalbufar 4.JPG thumb Torre del Verger , guard tower located at Banyalbufar , Majorca , Spain . Guard tower serves as a general term for any military tower providing vigilance, and henceforth guarding a certain area. Famous guard towers Alcatraz Island Alcatraz guard towers ref http www.flickr.com photos 9147703 N03 2180231638 Picture of a guard tower in Alcatraz ref Auschwitz concentration camp Auschwitz II guard towers ref http www.scrapbookpages.com AuschwitzScrapbook Tour Birkenau Camp.html Scrapbookpages.com s article about Auschwitz camp and it s defensive structures ref Tower of London ref http stason.org TULARC education books ghost stories 19 The Tower of London Famous Hauntings And Spooky Spots.html Spooky spots and tales of the Tower of London ref Yuma Territorial Prison 1876 guard tower. Gallery gallery File HollidayUnitHuntsvilleTX.jpg A guard tower at the C.A. Holliday Unit in Huntsville, Texas Huntsville , Texas gallery References Commons category Category Guard towers Guard towers Reflist Fortifications DEFAULTSORT Guard Tower Category Towers Category Prisons History stub ... more details
orphan date April 2010 A Snow Guard is a device used to retain snow , usually on a roof surface. a.k.a. Snow Bracket, Snow Stop, Snow Brake, Ice Guard, Snow Clip, Snow Fence, Snow Rail Category Snow Tool stub ... more details
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