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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
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 )
certain rational maps give rise to finite subdivision rules this includes most <unk> maps
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 ( ie is not simply connected ) an observer would see the visible universe repeat itself in an infinite pattern the subdivision rule describes that pattern
the subdivision rule looks different for different geometries this is a subdivision rule for the trefoil knot which is not a hyperbolic knot
and this is the subdivision rule for the borromean rings which is hyperbolic
in each case the subdivision rule would act on some tiling of a sphere ( ie 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
and for the borromean rings
= = subdivision rules in higher dimensions = =
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
= = rigorous definition = =
a finite subdivision rule <formula> consists of the following
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
2 a finite two dimensional cw complex <formula> which is a subdivision of <formula>
3a continuous cellular map <formula> called the subdivision map whose restriction to every open cell is a homeomorphism
each cw complex <formula> in the definition above ( with its given characteristic map <formula> ) is called a tile type
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>
binary subdivision is one example
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
= = quasi @@ isometry properties = =
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>
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
= = applications = =
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
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 )
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
= = cannon 's conjecture = =
cannon floyd and parry first studied finite subdivision rules in an attempt to prove the following conjecture
cannon 's conjecture every gromov hyperbolic group with a 2 @@ sphere at infinity acts geometrically on hyperbolic 3 @@ space
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
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
= = combinatorial riemann mapping theorem = =
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
its statement needs some background a tiling <formula> of a ring <formula> ( ie 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 ( ie 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
<formula>
<formula>
note that they are invariant under scaling of the metric
a sequence <formula> of tilings is conformal ( <formula> ) if mesh approaches 0 and
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
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>
= = = statement of theorem = = =
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>
= = = consequences = = =
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
= ash crimson =
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
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
= = appearances = =
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
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
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
= = creation and development = =
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
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
= = reception = =
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 )
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
= cunard building =
not to be confused with the 1921 cunard building ( new york city )
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
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
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
= = history = =
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
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
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
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
= = architectural design = =