finite subdivision rule mathematics finite subdivision rule way dividing dimensional shape smaller smaller pieces subdivision rules sense instead repeating exactly design slight variations stage allowing richer structure maintaining elegant style subdivision rules architecture biology computer science study studied type subdivision rule definition subdivision rule takes plane turns new smaller finite ways way tile called tile type tile type represented label usually letter tile type smaller tile types edge gets subdivided according edge types finite subdivision rules labelled tile types called subdivision complexes subdivision rule given subdivision complex subdivision rule sequence instance subdivision tile type edge type tile type subdivision means subdivision complexes regular doesn start complex twice type tiles examples finite subdivision rules subdivision example subdivision rule edge type gets subdivided edges tile type triangle gets subdivided smaller surface subdivision complex generated subdivision rule set tile types curved lines table help tiles fit certain rational maps rise finite subdivision rules includes maps prime non split alternating knot link complement subdivision rule tiles corresponding boundary link complement subdivision rules night sky look like living knot complement universe simply connected observer visible universe repeat infinite pattern subdivision rule describes pattern subdivision rule looks different different subdivision rule knot knot subdivision rule rings case subdivision rule act sphere night sky easier draw small night sky corresponding single tile repeatedly subdivided happens knot rings subdivision rules higher dimensions subdivision rules easily dimensions instance subdivision dimensions subdivision dimensions divided proof rigorous definition finite subdivision rule formula consists following finite dimensional complex formula called subdivision complex fixed cell structure formula union closed cells assume closed cell formula formula structure formula closed disk formula edges formula contained formula characteristic map formula maps formula open cell finite dimensional complex formula subdivision formula continuous cellular map formula called subdivision map restriction open cell complex formula definition given characteristic map formula called tile type formula subdivision rule formula dimensional complex formula union closed cells continuous cellular map formula restriction open cell formula complex formula requiring induced map formula open cell formula formula map formula repeating process obtain sequence subdivided formula formula maps formula subdivision example subdivision complex created opposite edges square making subdivision complex formula subdivision map formula doubling map wrapping twice twice fold covering map plane squares subdivision complex subdivision rule structure map formula given standard covering map subdivision square plane gets subdivided squares fourth size quasi properties subdivision rules study quasi properties certain spaces given subdivision rule formula subdivision complex formula construct called history records action subdivision rule consists dual stage formula edges connecting tile formula subdivisions formula quasi properties history studied subdivision rules instance history quasi space exactly subdivision rule described mapping applications islamic tiles islamic architecture self similar modeled finite subdivision rules harvard_university professor princeton_university published paper journal science suggesting possessed properties consistent self similar presentation predecessor works starting centuries subdivision surfaces computer graphics use subdivision rules surface given level precision subdivision surfaces subdivision surface kind animated movies adding shifting points according different points shifted process new subdivision old meaning edge old identify corresponding edge new plus edges subdivision rules applied cannon floyd study large scale growth patterns biological organisms cannon floyd produced mathematical growth model demonstrated systems determined simple finite subdivision rules results objects example tree trunk large scale form wildly time local subdivision laws remain cannon floyd applied model analysis growth patterns rat tissue suggested negatively curved non nature microscopic growth patterns biological organisms key reasons large scale organisms look like crystals shapes fact cases resemble self similar particular suggested negatively curved local structure manifested highly folded highly connected nature brain lung tissue cannon cannon floyd studied finite subdivision rules attempt prove following cannon group sphere infinity acts space geometric action properly action partially solved proof states group group act space remains group sphere infinity group cannon showed group sphere infinity associated subdivision rule subdivision rule certain sense group group geometry space mapping subdivision rules sequence surface idea distance length area letting tile length area limit distances come converge sense structure surface mapping gives necessary sufficient conditions occur statement needs background formula ring formula closed gives formula formula called approximate similar classical ring defined use weight functions weight function formula assigns non negative number called weight tile formula path formula given length defined sum weights tiles path define height formula formula formula length possible paths connecting inner boundary formula outer boundary formula formula formula length possible paths ring area formula formula formula defined sum squares weights formula define formula formula note scaling metric sequence formula formula approaches ring formula approximate formula formula formula sufficiently large lie single interval form formula given point formula surface neighborhood formula formula formula ring formula formula separating complement formula large formula approximate formula greater formula statement sequence formula surface formula sense structure surface constant formula depending formula classical approximate formula formula sufficiently large given formula meaning lie single interval formula consequences mapping implies group formula acts formula sphere infinity natural subdivision rule sphere gives rise sequence sense cannon true subdivision rules
Finite subdivision rule