
hyperbolic plain loe ekareeti vampulu

hyperbolic jyaamitiloo, ekareeti hyperbolic tailing (ledhaa regular, quasiregular ledhaa semirigular hyperbolic tailing) anedi hyperbolic plain yokka anchu nundi anchu varaku nimpadam, idhi mukhaluga regular bahubhujalu kaligi untundi mariyu vertex-transitive (daani sheershaalapai transitive, isogonal, i.ii. edainaa vertexnu itara vaatipai isometry mapping untundi). idhi anni sheershaalu ekaruupamgaa unnayani, mariyu tailing adhika sthaayi bhramana mariyu anuvaada samaruupatanu kaligi undani anusaristundi.
ekareeti vampulanu vaati sheersha aakruteekarana dwara gurtinchavacchu, prati sheersham chuttoo unna bahubhujaala vaipula sankhyanu soochinche sankhyala kramam. udaaharanaku, 7.7 anedi prati sheersham chuttoo 3 heptagonlanu kaligi unna heptagonal tilingnu suuchistumdi. anni bahubhujalu oke parimaanamlo unnanduna idhi kooda kramam, kabatti dheeniki shfley chihnam {7,3} kooda ivvavacchu.
ekareeti vampulu kramam tappakunda undavacchu (mukham mariyu anchu-sankramana ayithe), pakshika-kramam (anchu-sankramana ayithe conny mukham-sankramana kakapothe) ledhaa pakshika-kramam (anchu ledhaa mukham-sankramana kakapothe). kudi tribhujaalu ("p" q "2) kosam, rendu saadhaarana vampulu unnaayi, ivi shfley chihnam {" p "," q "} mariyu {" q "," p "} dwara suuchimchabadataayi.
waithoff nirmaanam.
squarge tribhujaalu ("p" "q" "r") aadhaaramgaa anantamaina sankhyalo ekareeti vampulu unnaayi, ikkada + + <1, ikkada "p", "q", "r" anevi praathamika domine tribhujam yokka moodu paayintla vadda pratibimbam samaruupata yokka prati arderlu-samaruupata samuuham anedi hyperbolic tribhujam samuuham.
prati samaruupata kutumbamlo 7 ekareeti vampulu untaayi, ivi waithoff chihnam ledhaa cauxeter-dainkin rekhaachitram dwara nirvachinchabaddaayi, 7 anedi 3 kriyaaseela addaala kalayikalanu suuchistumdi. 8vadhi oka pratyaamnaaya charyanu suuchistumdi, anni addaalu churukuga unnanduna atyunnata roopam nundi pratyaamnaaya sheershaalanu tolagistundi.
r" = 2 unna kutumbaalu saadhaarana hyperbolic tillingsnu kaligi untaayi, veetini [7,3], [8,3], [9,3], vanti cauxeter samuuham nirvachistundi... [5,4], [6,4],...
r" = 3 ledhaa anthakante ekkuva unna hyperbolic kutumbaalu ("p" q "r") dwara ivvabadataayi mariyu (4 3 3), (5 3 3), (6 3 3) kaligi untaayi... (4 4 3), (5 4 3),... (4 4)...
hyperbolic tribhujaalu ("p" q "r") compact ekareeti hyperbolic tilling lanu nirvachistaayi. parimitiloo edainaa "p", "q" ledhaa "r" nuu φ dwara bharti cheyavachu, idhi paracompact hyperbolic tribhujaanni nirvachistundi mariyu oke aadarsha binduvuku kalusukune anantamaina mukhaalato (apyrogons anie pilustaaru) ekareeti vampulanu srushtistundi, ledhaa anantamaina vertex figure adhe aadarsha binduvu nundi aneka anchulu mallutaayi.
tribhujaalu kaani praathamika domainla nundi marinta samaroopa kutumbaalanu nirminchavacchu.
ekareeti vampula yokka enchukunna kutumbaalu krinda chuupabaddaayi (hyperbolic plain kosam paayinkaare disk modalnu upayoginchi). vaatilo moodu-(7 3 2), (5 4 2), mariyu (4 3 3)-mariyu maray itara sankhyalu, vaati nirvachinche sankhyalalo deeninainaa chinna puurnaamkamtoe bharti chesthe, phalitamgaa vachhe namuunaa hyperbolic kante euclidean ledhaa golakaramga untundi; dheeniki viruddhamgaa, itara hyperbolic namuunaalanu roopondinchadaaniki yea sankhyalanu ayina penchavacchu (anantaniki kooda).
prati ekareeti tailing dwandwa ekareeti tilingnu utpatti chestundi, vaatilo chaalaa varaku krinda ivvabaddaayi.
kudi tribhuja domainlu.
anantamaina aneka ("p" q "2) tribhuja samuuha kutumbaalu unnaayi. eee vyaasam "p", "q" = 8 varaku kramabaddhamaina tilingnu chooputundi, mariyu 12 kutumbaalalo ekareeti tilingnu chooputundih (7 3 2), (8 3 2), (5 4 2), (6 4 2), (7 4 2), (8 4 2), (5 5 2), (6 5 2), (6 6 2), (7 7 2), (8 6 2), mariyu (8 8 2).
saadhaarana hyperbolic tillings.
hyperbolic tilling yokka saralamaina samiti regular tillings {"p", "q"}, ivi regular polyheadra mariyu euclidean tillingsto oka maatrukalo untaayi. saadhaarana tailing {"p", "q"} pattika yokka vikarna aksham antataa dwandwa tailing {"q", "p"} kaligi untundi. sweeya-dwandwa vampulu {2,2}, {3,3}, {4,4}, {5,5} modalainavi. table yokka vikarnam krindiki vellandi.
(7 3 2).
(7 3 2) tribhujam samuuham, coxeter samuuham [7,3], kakshya (* 732) eee ekareeti vampulanu kaligi untundih
(8 3 2).
(8 3 2) tribhuja samuuham, coxeter samuuham [8,3], arbifold (* 832) eee ekareeti vampulanu kaligi untundih
(5 4 2).
(5 4 2) tribhuja samuuham, coxeter samuuham [5,4], kakshya (* 542) eee ekareeti vampulanu kaligi untundih
(6 4 2).
(6 4 2) tribhuja samuuham, coxeter samuuham [6,4], arbifold (* 642) eee ekareeti vampulanu kaligi untaayi. anni muulakaalu samanamga unnanduna, prati ekareeti dwandwa tailing okati pratibimbita samaruupata yokka praathamika domaine suchistundihi * 3333, * 662, * 3232, * 443, * 222222, * 3222, mariyu * 642. alaage, anni 7 ekareeti tilingnu pratyamnaayamgaa cheyavachu, mariyu vatiki dvandvaalu kooda untaayi.
(7 4 2).
(7 4 2) tribhujam samuuham, coxeter samuuham [7,4], kakshya (* 742) eee ekareeti vampulanu kaligi untundih
(8 4 2).
(8 4 2) tribhuja samuuham, coxeter samuuham [8,4], arbifold (* 842) eee ekareeti vampulanu kaligi untaayi. anni muulakaalu samanamga unnanduna, prati ekareeti dwandwa tailing okati pratibimbita samaruupata yokka praathamika domaine suchistundihi * 4444, * 882, * 4242, * 444, * 222222, * 4222, mariyu * 842. alaage, anni 7 ekareeti tilingnu pratyamnaayamgaa cheyavachu, mariyu vatiki dvandvaalu kooda untaayi.
(5 5 2).
(5 5 2) tribhuja samuuham, coxeter samuuham [5,5], kakshya (* 552) eee ekareeti vampulanu kaligi untundih
(6 5 2).
(6 5 2) tribhuja samuuham, coxeter samuuham [6,5], arbifold (* 652) eee ekareeti vampulanu kaligi untundih
(6-2).
(6 6 2) tribhuja samuuham, coxeter samuuham [6,6], kakshya (* 662) eee ekareeti vampulanu kaligi untundih
(8-6 2).
(8,62) tribhuja samuuham, coxeter samuuham [8,6], arbifold (* 862) eee ekareeti vampulanu kaligi untaayi.
(7-2).
(7 7 2) tribhujam samuuham, coxeter samuuham [7,7], kakshya (* 772) eee ekareeti vampulanu kaligi untundih
(8-2).
(8-82) tribhuja samuuham, coxeter samuuham [8,8], kakshya (* 882) eee ekareeti vampulanu kaligi untundih
saadhaarana tribhuja domainlu.
anantamaina aneka saadhaarana tribhuja samuuha kutumbaalu ("p" q "r") unnaayi. eee vyaasam 9 kutumbaalalo ekareeti vampulanu chooputundih (4 3 3), (4 4 3), (4 4 4), (5 3 3), (5 4 3), (5 4 4), (6 3 3), (6 4 3), mariyu (6 4 4).
(4-3).
(4 3 3) tribhuja samuuham, coxeter samuuham [(4,3,3)], arbifold (* 433) eee ekareeti vampulanu kaligi untundi. praathamika tribhujamlo samakonalu lekunda, waithoff nirmaanaalu koddigaa bhinnamgaa untaayi. udaaharanaku (4,3,3) tribhujam kutumbamlo, "snoob" roopam oka sheersham chuttoo aaru bahubhujaalanu kaligi untundi mariyu daani dwandwa pentagonla kante shatkonalanu kaligi untundi. saadhaaranamgaa tribhujamlo ("p", "q", "r") oka snoob tailing yokka sheersham p. 3.q.3.r.3, eee sandarbhamlo 4.3.3.3.3.3 kinda undhi.
(4 4 3).
(4 4 3) tribhuja samuuham, coxeter samuuham [(4,4,3)], arbifold (* 443) eee ekareeti vampulanu kaligi untundi. 
(4 4).
(4 4) tribhuja samuuham, coxeter samuuham [(4,4,4)], arbifold (* 444) eee ekareeti vampulanu kaligi untaayi. 
(5 3 3).
(5 3 3) tribhuja samuuham, coxeter samuuham [(5,3,3)], arbifold (* 533) eee ekareeti vampulanu kaligi untaayi.
(5 4 3).
(5 4 3) tribhuja samuuham, coxeter samuuham [(5,4,3)], arbifold (* 543) eee ekareeti vampulanu kaligi untundi.
(5 4 4).
(5 4 4) tribhuja samuuham, coxeter samuuham [(5,4,4)], arbifold (* 544) eee ekareeti vampulanu kaligi untaayi.
(6 3 3).
(6 3 3) tribhuja samuuham, coxeter samuuham [(6,3,3)], arbifold (* 633) eee ekareeti vampulanu kaligi untundi.
(6 4 3).
(6 4 3) tribhuja samuuham, coxeter samuuham [(6,4,3)], arbifold (* 643) eee ekareeti vampulanu kaligi untundi.
(6 4 4).
(6 4 4) tribhuja samuuham, coxeter samuuham [(6,4,4)], arbifold (* 644) eee ekareeti vampulanu kaligi untundi.
parimita tribhujaakaara praathamika domainlatho vampula saaraamsam.
praathamika domainlatho ("p" "q" "r") anni ekareeti hyperbolic tilling pattika kosam, ikkada 2 ≤ "p", "q", "r" ≤ 8.
chaturbhuja domainlu.
(3 2 2).
hyperbolic plainloo quadrilatoral fundamental domainlu kooda unnaayi, * 3222 arbifold ([φ, 3, φ] coxeter sangnaamaanam) athi chinna kutumbamgaa undhi. chaturbhuja domainlalo ekareeti tailing kosam 9 taram sthaanaalu unnaayi. sheersham sankhyanu praathamika domine nundi 3 sandarbhaalu (1) muula (2) madhya anchu, mariyu (3) kendramgaa velikiteeyavacchu. paayintlanu utpatti chesetappudu, arder-2 muulalaku prakkane muulalu unnappudu, aaa moolallo degenerate {2} diagon mukhalu untaayi, conny vaatini vismarinchavacchu. oka sheersham chitramlo sama-vaipu mukhalu matrame unte snoob mariyu pratyaamnaaya ekareeti vampulu kooda utpatti cheyabadataayi (chuupabadavu).
chaturbhuja domainla yokka coxeter rekhaachitraalanu 6 anchulalo 2 nii anantamgaa label chesina ledhaa chukkala rekhalugaa unna ksheeninchina tetrahedron graphga pariganistaaru. rendu samaantara addaalalo kaneesam okadaani yokka taarkika avasaram kriyasilamga undatam anedi ekareeti kesulanu 9 kii parimitam chestundi, mariyu itara ringed namuunaalu chellavu.
aadarsha tribhuja domainlu.
anantamaina shrenulatho sahaa anantamaina aneka tribhuja samuuha kutumbaalu unnaayi. eee vyaasam 9 kutumbaalalo ekareeti vampulanu chooputundih (φ3), (φ42), (φ3), (φ4), (φ4), (φ4), (φ3), (φ4), (φ4), (φ4), (φ4), mariyu (φ4).
(φ32).
aadarsha (φ32) tribhujam samuuham, coxeter samuuham [φ, 3], kakshya (* φ32) eee ekareeti vampulanu kaligi untundih
(φ42).
aadarsha (φ42) tribhujam samuuham, coxeter samuuham [φ, 4], kakshya (* φ42) eee ekareeti vampulanu kaligi untundih
(φ52).
aadarsha (φ52) tribhujam samuuham, coxeter samuuham [φ, 5], kakshya (* φ 52) eee ekareeti vampulanu kaligi untundih
(φ φ2).
aadarsha (φ φ2) tribhujam samuuham, coxeter samuuham [φ, φ], kakshya (* φ φ2) eee ekareeti vampulanu kaligi untundih
(φ3).
aadarsha (φ3) tribhujam samuuham, coxeter samuuham [(φ, 3,3)], kakshya (* φ33) eee ekareeti vampulanu kaligi untundi. 
(φ43).
aadarsha (φ43) tribhujam samuuham, coxeter samuuham [(φ, 4,3)], kakshya (* φ43) eee ekareeti vampulanu kaligi untundih
(φ4).
aadarsha (φ4) tribhujam samuuham, coxeter samuuham [(φ, 4,4)], kakshya (* φ44) eee ekareeti vampulanu kaligi untundi. 
(φ φ3).
aadarsha (φ φ 3) tribhuja samuuham, coxeter samuuham [(φ, φ, 3)], kakshya (* φ φ 3) eee ekareeti vampulanu kaligi untundi. 
(φ φ4).
aadarsha (φ φ 4) tribhuja samuuham, coxeter samuuham [(φ, φ, 4)], kakshya (* φ φ 4) eee ekareeti vampulanu kaligi untundi. 
(φ φ).
aadarsha (φ φ φ) tribhuja samuuham, coxeter samuuham [(φ, φ, φ)], kakshya (* φ φ φ) eee ekareeti vampulanu kaligi untundi. 
anantamaina tribhujaakaara praathamika domainlatho vampula saaraamsam.
praathamika domainlatho ("p" "q" "r") anni ekareeti hyperbolic tilling pattika kosam, ikkada 2 ≤ "p", "q", "r" ≤ 8, mariyu okati ledhaa anthakante ekkuva φ.

