
aperiodic koottam prototylukal

saadhyamaaya ellah tesilation paattenukalum noonn-periodic aayathinal, tilingukal srishtikkaan prototylukalude pakarppukal kootticherkkaan kazhiyumenkil oru koottam prototylukal aperiodic aanu. paraamarsikkunna "aperiodicity" ennathu oru prathyeka koottam protottailukalude oru savisheshathayaanu; thathfalamaayundaakunna vividha cherivukal avarkk mathramullathum avarkk aavarthikkappedaathathum.
euclidean thalatthilo mattethenkilum jyaamitheeya krameekaranatthilo oru koottam tailukal, settile tailukalude overlapping allatha pakarppukal muzhuvan sthalavum ulkkollaan orumichu gadippikkaan kazhiyumenkil "oru tiling amgeekarikkunnu". oru nischitha tyle koottam aavarthichulla cherivukal sweekarikkaam-athaayath, oru vivarthanathiloode maattiyathin sesham maattamillathe thudarunna cherivukal (udaaharanathinu, chathuraakrithiyilulla tailukalude oru laattis aavarthichullathaanu). aanukaalikamallaatha cherivukalum aanukaalika cherivukalum sweekarikkunna oru koottam tailukal roopakalppana cheiyunnathu budhimuttulla kaaryamalla. (udaaharanathinu, 2x2 chathuravum 2x1 deerghachathuravum upayogichu kramarahithamayi krameekarichirikkunna cherivukal saadhaaranayaayi kaalakramamillaathavayaanu.)
ennirunnaalum, oru koottam aperiodic tailukalkk noonn-periodic tilingukal maathrame utppaadippikkaan kazhiyoo. orotta aperiodic tailukalil ninnu ananthamaaya niravadhi vyathyastha tilingukal labikkum.
vividha penros tailukalaanu aperiodic tailukalude ettavum ariyappedunna udaaharanagal. ariyappedunna aperiodic settu prototylukal aperiodic settu tailukalude pattikayil kaanapedunnu. domino prashnatthinte adisthaana anishchithathvam soochippikkunnathu oru koottam tailukalkk playinine tyle cheyyaan kazhiyumo ennu theerumaanikkunnathinu chittayaya nadapadikramangalonnumille soochippikkunnu.
charithram.
nerrekhaasampradaayangalaal chuttapetta samathalachithrangalaan bahubhujangal. saadhaarana bahubhujangalkk ellah vashangalum thulya neelavum ellah konukalum thulya alavumundu. 325-ll thanne, alexandriayile papasinu oru euclidean thalatthile aavarthichulla tessellationukalil 3 tharam saadhaarana bahubhujangalkk (chathuram, samabahu trikonavum shadbhujavum) maathrame orumichu yojikkan kazhiyoo ennu ariyamaayirunnu. aaa thalathinullil, oro trikonavum, kramam pariganikkathe, tesilettu cheyyum. ithinu vipareethamaayi, saadhaarana pentagonukal tesilettu cheiyunnilla. ennirunnaalum, vyathyastha vashangalum konukalumulla kramarahithamaya pentagonukalkku tesilettu cheyyaan kazhiyum. samatalam tyle cheiyunna 15 kramarahithamaya convex pentagonukalundu.
bahubhujangalude trimana parasparabandhangalaan polyheadra. paranna mukhangalil ninnum neraaya arikukalil ninnum nirmmicha ivaykku konukalil moorchayulla moolakalulla valavukalundu. cube maathramaan tesilation amgeekarikkunna eaka saadhaarana polyhedron aanenkilum, vettimattappetta octahedron polulla niravadhi noonn-regular trimana roopangalkku tesilettu cheyyaan kazhiyum.
hilbertinte pathinettaam prashnatthinte randaam bhagam oru polyhedron tiling euclidean 3-space aavasyappettu, athinte tiling isohedral (oru anisohedral tyle) allaathavidham. 1928ll kaal reinhard eee prashnam pariharichuvenkilum oru koottam aperiodic tailukal swaabhaavika vipuleekaranamaayi kanakkaakkappedunnu.
1961-ll logician haavo wang domino prashnam nirnnaayakamaano ennu nirnnayikkaan shramichappol aperiodic tailukalude prathyeka chodyam aadyamaayi uyarnnu-athaayath, thannirikkunna parimithamaaya oru koottam prototylukal tiling amgeekarikkunnundo ennu theerumaanikkunnathinu oru algoritam nilavilundo ennu. thalam tyle cheyyaan kazhiyaatha tilesettukalum athu idaykkide tyle cheiyunna tilesettukalum kanakkakkaan wang algorithangal kandethi; thalam tyle cheyyaan kazhiyunna oro parimithamaaya koottam prototylukalum oru periodic tiling amgeekarikkukayaanenkil atharamoru theerumaana algoritam nilavilundennu ithiloode addeham kaanichu.
athinaal, 1966-ll roobert berger oru aperiodic koottam prototylukal kandethiyappol ithu tiling prashnam yadhaarthathil nirnnaayakamallenn theliyichu. (athinaal vaanginte nadapadikramangal ellah tyle settukalilum pravarthikkunnilla, ennirunnaalum athu praayogika aavasyangalkku upayogashoonyamaakkunnil.) atharathilulla aadyathe settinaayi berger thante anishchithathvathinte thelivaayi 20,426 wang tailukal aavashyamaayi vannu. berger pinneedu thante settu 104 aayi kurachu, hans lukely pinneedu 40 wang tailukal maathram aavasyamulla oru aperiodic settu kandethi. valathuvashathulla thithreekaranatthil nalkiyirikkunna 13 tailukalude koottam 1996 ll carrell kulik, II prasidheekaricha oru aperiodic settaanu.
ennirunnaalum, aarr noonn-wang tailukalulla oru cheria aperiodic settu rafale emm kandethi. 1971ll robinson. 1973lum 1974lum rogger penros moonnu settukal koodi kandethi, aavashyamaaya tailukalude ennam randaayi kurachu, 1977ll roobert amman niravadhi puthiya settukal kandethi. orotta prototyl maathram ulla oru aperiodic settu nilavilundo enna chodyathe ainsteen prashnam ennu vilikkunnu.
nirmmanangal.
bergerinte thakarppan nirmmaanathinu naalpat varshathinu sheshavum aperiodic tillingukalude nirmmanangal valare kuravaanu. chila nirmmanangal aperiodic tailukalude ananthamaaya kudumbangalaanu. kandethiya aaa nirmmanangal pradhaanamaayum chila reethikalil onnaayaan nirmmichirikkunnathu-praadhamikamaayi eathenkilum tharatthilulla kaalakramamillaatha sreniparamaaya ghadanaye nirbandhithamaakkunnathiloodu. ithokkeyaanengilum, domino prashnatthinte anishchithathvam nirmmaanathinu ananthamaaya niravadhi vyathyastha thathwangal undaayirikkanamennum vaasthavathil, avayude aperiodysitic thelivukalonnumillaathu aperiodic tailukalude settukal nilavilundennum urappaakkunnu.
oru alavil aperiodic tailukalonnum undaakilla ennathu sradheyamaanu variyile eathenkilum tailukal oru samboornna tiling roopeekarikkaan upayogikkaanaakillenn kaanikkukayo allengil oru periodic tiling roopeekarikkaan upayogikkukayo cheiyunnathu oru lalithamaaya vyaayaamamaanu. protottailukalude aperiodysitic rando athiladhikamo alavukal aavasyamaanu.

