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<unk> subdivision is an example of a subdivision rule with one edge type ( that gets subdivided into two edges ) and one tile type ( a triangle that gets subdivided into 6 smaller triangles ) . Any triangulated surface is a barycentric subdivision complex .
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The Penrose tiling can be generated by a subdivision rule on a set of four tile types ( the curved lines in the table below only help to show how the tiles fit together ) :
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Certain rational maps give rise to finite subdivision rules . This includes most <unk> maps .
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Every prime , non @-@ split alternating knot or link complement has a subdivision rule , with some tiles that do not subdivide , corresponding to the boundary of the link complement . The subdivision rules show what the night sky would look like to someone living in a knot complement ; because the universe wraps around itself ( i.e. is not simply connected ) , an observer would see the visible universe repeat itself in an infinite pattern . The subdivision rule describes that pattern .
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The subdivision rule looks different for different geometries . This is a subdivision rule for the trefoil knot , which is not a hyperbolic knot :
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And this is the subdivision rule for the Borromean rings , which is hyperbolic :
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In each case , the subdivision rule would act on some tiling of a sphere ( i.e. the night sky ) , but it is easier to just draw a small part of the night sky , corresponding to a single tile being repeatedly subdivided . This is what happens for the trefoil knot :
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And for the Borromean rings :
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= = Subdivision Rules in Higher Dimensions = =
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Subdivision rules can easily be generalized to other dimensions . For instance , barycentric subdivision is used in all dimensions . Also , binary subdivision can be generalized to other dimensions ( where <unk> get divided by every <unk> ) , as in the proof of the Heine @-@ Borel theorem .
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= = Rigorous definition = =
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A finite subdivision rule <formula> consists of the following .
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1 . A finite 2 @-@ dimensional CW complex <formula> , called the subdivision complex , with a fixed cell structure such that <formula> is the union of its closed 2 @-@ cells . We assume that for each closed 2 @-@ cell <formula> of <formula> there is a CW structure <formula> on a closed 2 @-@ disk such that <formula> has at least two vertices , the vertices and edges of <formula> are contained in <formula> , and the characteristic map <formula> which maps onto <formula> restricts to a homeomorphism onto each open cell .
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2 . A finite two dimensional CW complex <formula> , which is a subdivision of <formula> .
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3.A continuous cellular map <formula> called the subdivision map , whose restriction to every open cell is a homeomorphism .
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Each CW complex <formula> in the definition above ( with its given characteristic map <formula> ) is called a tile type .
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An <formula> <unk> for a subdivision rule <formula> is a 2 @-@ dimensional CW complex <formula> which is the union of its closed 2 @-@ cells , together with a continuous cellular map <formula> whose restriction to each open cell is a homeomorphism . We can subdivide <formula> into a complex <formula> by requiring that the induced map <formula> restricts to a homeomorphism onto each open cell . <formula> is again an <formula> <unk> with map <formula> . By repeating this process , we obtain a sequence of subdivided <formula> <unk> <formula> with maps <formula> .
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Binary subdivision is one example :
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The subdivision complex can be created by gluing together the opposite edges of the square , making the subdivision complex <formula> into a torus . The subdivision map <formula> is the doubling map on the torus , wrapping the meridian around itself twice and the longitude around itself twice . This is a four @-@ fold covering map . The plane , tiled by squares , is a subdivision complex for this subdivision rule , with the structure map <formula> given by the standard covering map . Under subdivision , each square in the plane gets subdivided into squares of one @-@ fourth the size .
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= = Quasi @-@ isometry properties = =
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Subdivision rules can be used to study the quasi @-@ isometry properties of certain spaces . Given a subdivision rule <formula> and subdivision complex <formula> , we can construct a graph called the history graph that records the action of the subdivision rule . The graph consists of the dual graphs of every stage <formula> , together with edges connecting each tile in <formula> with its subdivisions in <formula> .
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The quasi @-@ isometry properties of the history graph can be studied using subdivision rules . For instance , the history graph is quasi @-@ isometric to hyperbolic space exactly when the subdivision rule is conformal , as described in the combinatorial Riemann mapping theorem .
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= = Applications = =
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Islamic Girih tiles in Islamic architecture are self @-@ similar tilings that can be modeled with finite subdivision rules . In 2007 , Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self @-@ similar fractal <unk> tilings such as Penrose tilings ( presentation 1974 , predecessor works starting in about 1964 ) predating them by five centuries .
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Subdivision surfaces in computer graphics use subdivision rules to refine a surface to any given level of precision . These subdivision surfaces ( such as the Catmull @-@ Clark subdivision surface ) take a polygon mesh ( the kind used in 3D animated movies ) and refines it to a mesh with more polygons by adding and shifting points according to different recursive formulas . Although many points get shifted in this process , each new mesh is <unk> a subdivision of the old mesh ( meaning that for every edge and vertex of the old mesh , you can identify a corresponding edge and vertex in the new one , plus several more edges and vertices ) .
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Subdivision rules were applied by Cannon , Floyd and Parry ( 2000 ) to the study of large @-@ scale growth patterns of biological organisms . Cannon , Floyd and Parry produced a mathematical growth model which demonstrated that some systems determined by simple finite subdivision rules can results in objects ( in their example , a tree trunk ) whose large @-@ scale form oscillates wildly over time even though the local subdivision laws remain the same . Cannon , Floyd and Parry also applied their model to the analysis of the growth patterns of rat tissue . They suggested that the " negatively curved " ( or non @-@ <unk> ) nature of microscopic growth patterns of biological organisms is one of the key reasons why large @-@ scale organisms do not look like crystals or polyhedral shapes but in fact in many cases resemble self @-@ similar fractals . In particular they suggested that such " negatively curved " local structure is manifested in highly folded and highly connected nature of the brain and the lung tissue .
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= = Cannon 's conjecture = =
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Cannon , Floyd , and Parry first studied finite subdivision rules in an attempt to prove the following conjecture :
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Cannon 's conjecture : Every Gromov hyperbolic group with a 2 @-@ sphere at infinity acts geometrically on hyperbolic 3 @-@ space .
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Here , a geometric action is a <unk> , properly discontinuous action by isometries . This conjecture was partially solved by Grigori Perelman in his proof of the <unk> conjecture , which states ( in part ) than any Gromov hyperbolic group that is a 3 @-@ manifold group must act geometrically on hyperbolic 3 @-@ space . However , it still remains to show that a Gromov hyperbolic group with a 2 @-@ sphere at infinity is a 3 @-@ manifold group .
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Cannon and Swenson showed that a hyperbolic group with a 2 @-@ sphere at infinity has an associated subdivision rule . If this subdivision rule is conformal in a certain sense , the group will be a 3 @-@ manifold group with the geometry of hyperbolic 3 @-@ space .
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= = Combinatorial Riemann Mapping Theorem = =
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Subdivision rules give a sequence of tilings of a surface , and tilings give an idea of distance , length , and area ( by letting each tile have length and area 1 ) . In the limit , the distances that come from these tilings may converge in some sense to an analytic structure on the surface . The Combinatorial Riemann Mapping Theorem gives necessary and sufficient conditions for this to occur .
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Its statement needs some background . A tiling <formula> of a ring <formula> ( i.e. , a closed annulus ) gives two invariants , <formula> and <formula> , called approximate moduli . These are similar to the classical modulus of a ring . They are defined by the use of weight functions . A weight function <formula> assigns a non @-@ negative number called a weight to each tile of <formula> . Every path in <formula> can be given a length , defined to be the sum of the weights of all tiles in the path . Define the height <formula> of <formula> under <formula> to be the infimum of the length of all possible paths connecting the inner boundary of <formula> to the outer boundary . The circumference <formula> of <formula> under <formula> is the infimum of the length of all possible paths circling the ring ( i.e. not <unk> in R ) . The area <formula> of <formula> under <formula> is defined to be the sum of the squares of all weights in <formula> . Then define
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<formula>
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<formula> .
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Note that they are invariant under scaling of the metric .
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A sequence <formula> of tilings is conformal ( <formula> ) if mesh approaches 0 and :
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For each ring <formula> , the approximate moduli <formula> and <formula> , for all <formula> sufficiently large , lie in a single interval of the form <formula> ; and
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Given a point <formula> in the surface , a neighborhood <formula> of <formula> , and an integer <formula> , there is a ring <formula> in <formula> separating x from the complement of <formula> , such that for all large <formula> the approximate moduli of <formula> are all greater than <formula> .
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= = = Statement of theorem = = =
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If a sequence <formula> of tilings of a surface is conformal ( <formula> ) in the above sense , then there is a conformal structure on the surface and a constant <formula> depending only on <formula> in which the classical moduli and approximate moduli ( from <formula> for <formula> sufficiently large ) of any given annulus are <formula> <unk> , meaning that they lie in a single interval <formula> .
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= = = Consequences = = =
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The Combinatorial Riemann Mapping Theorem implies that a group <formula> acts geometrically on <formula> if and only if it is Gromov hyperbolic , it has a sphere at infinity , and the natural subdivision rule on the sphere gives rise to a sequence of tilings that is conformal in the sense above . Thus , Cannon 's conjecture would be true if all such subdivision rules were conformal .
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= Ash Crimson =
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Ash Crimson ( <unk> ・ <unk> , <unk> <unk> ) is a video game character in The King of Fighters fighting game series developed by SNK Playmore . His first appearance was in The King of Fighters 2003 as leader of its Hero Team . Ash , a teenager , participates in the series ' fighting tournaments . He employs a personal fighting style that involves pyrokinesis with green flames . Despite being the series ' protagonist since its third story arc , Ash 's behavior is generally antagonistic and he tries to steal powers from several recurring characters , leaving them powerless . His identity and intentions are further explored in the most @-@ recent title , The King of Fighters XIII , and he has been featured in print adaptations and a drama CD based on the games .
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SNK Playmore staff created Ash as an evil main character , in contrast to previous main characters . His moves have been reworked in every game in which he has appeared , gaining new techniques as a result of his actions or modified to balance the character roster . Critical response to Ash has been largely negative due to his androgynous appearance and his technique . In interviews and press releases , video game publisher UTV Ignition Entertainment noted that he was unpopular with Western gamers and his design was better suited to Japanese fans .
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= = Appearances = =
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Introduced in The King of Fighters 2003 , Ash leads the New Hero Team of Duo Lon and Shen Woo which participates in a new King of Fighters tournament . After the tournament , he ambushes host Chizuru Kagura and steals her powers , telling Iori Yagami he will be his next opponent . In The King of Fighters XI , the character teams up with Oswald and Shen Woo . At the game 's end Ash leaves his teammates to fight each other , after disclosing that Oswald 's price for joining him was finding a drug he can only obtain after defeating their pharmacist 's enemy : Shen Woo . He later finds a berserker , Iori , defeats him and steals his powers . When Ash 's comrade Elisabeth <unk> accuses him of abandoning his original mission and becoming obsessed with power , he leaves after saying that Kyo Kusanagi will be his next victim . Ash also appears as an available fighter in the KOF : Maximum Impact 2 update KOF Maximum Impact Regulation A and The King of Fighters XII , games without a storyline .
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In The King of Fighters XIII Ash is the only character not part of a team , and it is learned that he works for Those From the Past ( seen in 2003 and XI ) . However , he has a conflict with organization leader Saiki at the end of the game . Ash is the final boss in The King of Fighters XIII as " Ash driven insane by the Spiral of Blood " ( <unk> , Chi no Rasen ni <unk> <unk> ) ( also known as Evil Ash ) with power stolen from Saiki , who is trying to possess him . He ultimately sacrifices himself to destroy Saiki .
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Ash has made appearances in various other King of Fighters media as well . In the fourth chapter of the anime The King of Fighters : Another Day , he sets fire to a city to lure Kyo into a trap . He is also prominent in the manhua adaptations of The King of Fighters 2003 by Wing Yang and King Tung . In the 2003 tournament , Ash 's team is defeated in the final but later helps defeat the demon Mukai . Although he is not part of the recurring tournament in The King of Fighters XII manhua , he briefly confronts Kyo in an attempt to steal his powers . Ash also appears in the CD drama KOF : Mid Summer Struggle , which has a mock King of Fighters tournament . Outside The King of Fighters , the character appears in a dating sim part of the Days of Memories series .
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= = Creation and development = =
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Ash was designed as an " attractive evil character " , in contrast to previous King of Fighters heroes . The supervising designer created the character as desired , with few changes since conception . When The King of Fighters 2003 was released and Ash introduced , the staff did not want to disclose information about him , instead telling fans to look forward to his " exploits . " His canonical height is <unk> ( 5 ' 10 " ) and he weighs <unk> ( 130 lbs ) . According to a staff member , Falcoon , the goal was to make Ash an ambiguous protagonist who made players " feel bad " for cheering for him . Although Falcoon did not design the character , he added details while illustrating him . Due to Ash 's late appearance in The King of Fighters 2003 , the staff joked that teammate Shen Woo seemed more like the series ' main character than Ash did . For The King of Fighters XIII , producer Kei Yamamoto wanted gamers who had played previous titles with Ash to consider the character 's motivation and whether they could relate to him . One of three reasons for the title of KOF : Maximum Impact 2 Regulation " A " was Ash 's introduction to the spin @-@ off series . Falcoon called the character " really wild " because of his role in the series and his personality . Nona ( another artist in charge ) said that of the original King of Fighters characters , he liked Ash and looked forward to his development .
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In The King of Fighters XII Ash was the character the staff worked on the most , reworking his movements and speech to be more consistent with the rest of the cast . The King of Fighters XIII producer Kei Yamamoto jokingly called him a character players could use to " show off " . His EX version in the game makes more multiple hits than his regular form , and the " disgusting " style of his " Genie " ( <unk> ) move is said to fit his character . Since his introduction , Ash has acquired new moves in accordance with his action in the series . After The King of Fighters 2003 Ash gained Chizuru Kagura 's " <unk> " ( <unk> ) , a move allowing him to steal his opponents ' special moves . After defeating Iori Yagami , the character acquired the " Fructidor " ( <unk> ) Neo Max move ( his strongest attack in The King of Fighters XIII ) . During the game 's development , the staff considered returning Fructidor 's animation style to an earlier version they had tested .
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= = Reception = =
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Ash Crimson 's character had a mixed reception in video @-@ game publications . When he was introduced , GameSpy 's Christian Nutt called him a " clone " of Guile from the Street Fighter series because of their similar movesets but Ash 's team was praised as new characters with new elements in the series . Lucas M. Thomas of IGN wrote that a major fan complaint was the character 's strength in the games , which made him one of the strongest opponents despite an apparent lack of effort . According to Thomas , Ash is " nothing if not strange " and the writer lamented his many appearances in the KOF 2003 manhua compared with more @-@ popular characters . When playable characters for the arcade version of The King of Fighters XIII were introduced , Anime News Network 's Todd Ciolek speculated about Ash 's lack of teammates and wondered if he was alone because " no one likes Ash now . " Marissa Meli of UGO Networks ranked Ash sixteenth on a list of " The Most Androgynous Video Game Characters " , with Meli jokingly attributing his androgynous appearance to a desire by Japanese designers to confuse Western gamers about their sexuality . Although he called Ash 's voice one of " Gaming 's Top 10 <unk> Voices " , Dan <unk> of Now Gamer referred to him as a female character and writers for Game Informer joked about their surprise at the discovery that Ash is a male character . After UTV Ignition Entertainment polled fans to choose an artbox for console versions of The King of Fighters XII , the company announced that Ash 's unpopularity reduced the number of potential covers to two ( featuring Kyo and Iori ) .
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In an interview with Ignition Entertainment director of business development Shane Bettehausen , Alex <unk> of Diehard GameFan said that North American SNK fans detested Ash and complained about his inclusion in The King of Fighters XII without a storyline while popular series characters were overlooked . Bettehausen defended the character , calling him " nuanced and improved " and his moves " incredibly effective " . In June 2009 , Stephen Totilo of Kotaku interviewed Bettehausen , who asked him while Totilo was playing to guess Ash 's gender . When Totilo said the character was female , <unk> called KOF " progressive " in introducing a cross @-@ dressing character and said that fans were apparently " warming up to him " . On Destructoid , Jonathan Holmes praised Ash 's character as " defying traditional gender roles while kicking some ass " and called him " the perfect mascot for the <unk> series as a whole . " Merchandising related to Ash 's appearance has also been released .
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= Cunard Building =
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Not to be confused with the 1921 Cunard Building ( New York City )
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The Cunard Building is a Grade II * listed building in Liverpool , England . It is located at the Pier Head and along with the neighbouring Liver Building and Port of Liverpool Building is one of Liverpool 's Three Graces , which line the city 's waterfront . It is also part of Liverpool 's UNESCO designated World Heritage Maritime Mercantile City .
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It was designed by William Edward Willink and Philip Coldwell Thicknesse and was constructed between 1914 and 1917 . The building 's style is a mix of Italian Renaissance and Greek Revival , and its development has been particularly influenced by Italian palace design . The building is noted for the ornate sculptures that adorn its sides .
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The building was , from its construction until the 1960s , the headquarters of the Cunard Line , and the building still retains the name of its original tenants . It was also home to Cunard 's passenger facilities for trans @-@ Atlantic journeys that departed from Liverpool . Today , the building is owned by the Merseyside Pension Fund and is home to numerous public and private sector organisations . It is located directly opposite from Albion House , the former headquarters of White Star Line .
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= = History = =
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In 1914 the Cunard Steamship Company commissioned the construction of new headquarters for the company . Cunard 's expansion had meant that they had outgrown their previous offices , which were also in Liverpool , and the site chosen for construction was at the former George 's Dock , in between the Liver Building and Port of Liverpool Building . The building was designed by the architects William Edward Willink and Philip Coldwell Thicknesse and was inspired by the grand palaces of Renaissance Italy . It was constructed by Holland , Hannen & Cubitts between 1914 and 1917 , with Arthur J. Davis , of Mewes and Davis , acting as consultant on the project .
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In 1934 the Cunard Steamship Company merged with the White Star Line to form Cunard White Star Line , which became the largest passenger steamship company in the world , helping to make Liverpool one of the most important centres of the British trans @-@ Atlantic ocean liner industry . The Cunard building subsequently acted as the central headquarters for the newly merged firm , with both administrative and ship @-@ designing facilities located within the building . Many ships and liners were developed and designed at the Cunard Building , including the RMS Queen Mary and RMS Queen Elizabeth . Given that Liverpool was a major trans @-@ Atlantic port and due to the building 's proximity to the River Mersey , the lower floors of the Cunard Building were allocated to provide space for cruise liner passengers , both prior to and after sailing . Within the building there were passenger facilities including separate waiting rooms for first , second and third class passengers , a booking hall , luggage storage space , and a currency exchange . The building also provided facilities for both land and sea based Cunard employees . During the Second World War , the sub @-@ basement level of the Cunard Building was used as an air raid shelter for workers in the building and also for those from adjacent premises . The basement levels also served as the central Air Raid Precautions headquarters for the City of Liverpool during the war . Additional reinforced steel joists were fitted to further strengthen the basement in case of a direct hit on the building .
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The building remained the headquarters of Cunard until the 1960s , when they decided to relocate their UK operations to Southampton on England 's south coast and their global headquarters to New York . Cunard subsequently sold the building to Prudential plc in 1969 . In 1965 the Cunard Building was awarded Grade II * listed building status by the English Heritage . Initially , it was listed together with the Liver Building and Port of Liverpool Building under Pier Head , but in 1985 each building gained its own listing . In November 2001 the building was sold to the Merseyside Pension Fund , an organisation that provides pension services to public sector workers in Liverpool . Today , the building provides a range of office accommodation for a variety of public and private sector organisations , including Government Office North West . In November 2008 it was announced that the building managers had appointed the local architects firm Buttress Fuller Alsop Williams to draw up a conservation plan to preserve the building . The plan involved collaboration with English Heritage and the Local Authority Conservation Officer and would be used to control any modification and repairs made to the building .
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In October 2013 Liverpool City Council approved the acquisition of the Cunard Building for use as offices and as a cruise liner terminal . The council projected that the building would accommodate 1000 staff relocated from Millennium House and leases in the Capital Building , saving an estimated £ 1 @.@ 3 million . The anticipated use as cruise terminal however had to be abandoned due to the high costs associated with security and border control .
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= = Architectural design = =
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