apoloniy gasket

ganitat, apollonian gasket kinwa apollonian jaale heey teen vartulanpasun suru houn, pratyek vartul itar done vartulanna sparshika deuun aanee adhik vartulanmadhye salag bharoon, pratyek vartul aanakhi teen vartulanna sparshika deuun tayar kelele eka fractal aahe. greek ganitshastradnya, pergachya apollonius yachya navavarun yala heey naav denyat aale aahe.
bandhkam.
apollonian gasketche bandhkam teen vartul sootra _ 1, sootra _ 2 aanee sootra _ 3 (aakrutimadhye kaale) pasoon suru hote, jeey pratyek itar donchya sparshika aahet, parantu jyat tiheri sparshikecha ekach bindu nahee. hii vartule ekmekansathi vegvegalya aakarachi asoo shakatat aanee doghanna doghanna aat kinwa tinhi ekmekanna baher asanyachi parvanagi aahe. apolloniusne shodhalyapramane, aanakhi done vartule aahet, sootra 4 aanee sootra 5 (laal), jee tinhi mool vartulanna sparshika aahet-tyanna "apoloniy vartule" mhanatat. hii paach vartule ekmekanpasun sahaa vakr trikoni pradeshanni vegali kelee aahet, pratyek vartul teen jodichya dishene asalelya vartulanchya kamanine vedhalele aahe. yaa sahaa vakr trikonanpaiki pratyekamadhye eka, tyachya teen bajunna sparshika asaleli aanakhi sahaa vartule jodoon bandhkam suru aahe. hyachya badalyat aanakhi 18 vakr trikon tayar hotat aanee teey punha sparshika vartulanni, aanee anantane bharoon bandhkam suru rahate.
asha prakare tappyatappyane suru thevalelya bandhkamamadhye, 7 vya tappyatil sutravar sootra 6 naveen vartule jodali jaatat, jyamule 7 vya tappyanantar ekoon sootra 8 vartule midatat. maryadet, vartulancha haah sanch eka apollonian gasket aahe. tyat, sparshikechya vartulanchya pratyek jodimadhye jodyatil donhi vartulanna sparshikechi vartulanchi anant pups sakhali asate.
pratyek naveen vartulacha aakar decarteschya prameyadvare nirdharit keela jaato, jyamadhye asey mhatale aahe qee, konatyahi chaar paraspar sparshika vartulansathi, vartulanche trijya sootra _ 10 samikaran palate.
sootra _ 11
yaa samikaranat nakaratmak trijya asalele uttar asoo shakate; yacha arth assa qee eka vartul (nakaratmak trijya asaleli eka) itar teen vartulanna vedhate.
yaa bandhkamachya suruvatichya vartulanpaiki eka kinwa done kinwa yaa bandhkamacha parinam asaleli vartule, eka saral reshet rupantarit houu shakatat, jee anant trijya asaleli vartul manali jaau shakate. jevha done resha asatat, tevha tyaa samantar asane aavashyak aahe aanee anantatechya eka binduvar sparshika manalya jaatat. jevha gasketmadhye sootra 12-akshavar done resha aanee tyachya varr eka ekak aanee sootra 13-akshavar kendrit asalelya donhi reshanna ekak vyasache sparshika asalele vartul asate, tevha sootra 12-akshala sparshika asaleli vartule hii sankhya siddhantat mahattvachi, fords vartule asatat.
apollonian gasketche aakarman sumare 1.3057 aahe. kaaran tyache pariman suspashtapane bhinn asale taree, tyache pariman agadi tantotant swa-samaan nasale taree, teey eka fractal manale jaau shakate.
samamiti.
pratalache mobius parivartan vartulanche aakar aanee sparshika jatan karate aanee mhanoon apollonian gasketchi rachana jatan karate.
apollonian gasketmadhil paraspar sparshikechya vartulanchya konatyahi done tiheri vartulanna mobius parivartanadvare ekmekanmadhye mapp kele jaau shakate aanee konatyahi done apollonian gasketna mobius parivartanadvare ekmekanmadhye mapp kele jaau shakate. visheshtah, konatyahi apollonian gasketmadhil konatyahi done sparshikechya vartulansathi, sparshikechya binduvar kendrit asalelya vartulatil vyutkram (mobius parivartanache eka vishesh prakaran) yaa done vartulanche done samantar reshanmadhye rupantar karel aanee urvarit gasketche done samantar reshanmadhil gasketchya vishesh swarupat rupantar karel. yaa vyutkramanchya rachanancha vapar sparshikechya konatyahi done bindunche ekmekanmadhye rupantar karnyasathi keela jaau shakato. mobius parivartan heey hyperbolic pratalache samamiti dekhil aahet, mhanoon hyperbolic bhoomitimadhye serve apollonian gasket sarvasam aahet. eka arthane, tyamule (hyperbolic) isometriparyant fakt ekach apollonian gasket aahe.
apollonian gasket haah clenian gut mhanoon olakhalya olakhalya mobius parivartananchya gatacha maryada sanch aahe.
mobius parivartanaivaji euclidean samamiti parivartanansathi, sarvasadharanpane, apollonian gasketla tyachya teen vartulanchya nirmiti sanchachya samamiti varshane midatil. tathapi, vartulanchi kahee tippat suruvatichya tippatpeksha jaast samamitisah apollonian gasket tayar karoo shakatat; jevha tyach gasketmadhye tayar tayar vartulancha vegala aanee adhik samamiteey sanch asato tevha heey ghadate. visheshtah samamiteey prakarananmadhye done samantar reshanmadhil (anant dihedral samamitisah) apollonian gasket, sambahu trikonatil teen sarvasam vartulanni (trikonachya samamitisah) tayar keleli apollonian gasket aanee trijya 1 chya done vartulanni tayar keleli apollonian gasket yancha samavesh hoto, jyachya bhovati trijya 2 chya vartulane (paravartak samamitichya done reshansah) vedhalele asate.
avibhajya apollonian vartul packing.
jarr apollonian gasketmadhil konatyahi chaar paraspar sparshika vartulanmadhye purnank vakrata (tyanchya trijyecha ulat) asel tarr gasketmadhil serve vartulanmadhye purnank vakrata asel.
kaaran apollonian gasketmadhil vakrateshi sambandhit samikaran, avibhajya asso vaa naso, heey aahe
sootra _ 15
tyanantar, naveen markov kramank shodhatana jase vietta jampingdware eka vyakti eka chaupat vakratepasoon vakratepasoon jaau shakate.
yaa avibhajya apollonian gasketpacky pahilya kahee khaleel taktyat soochibaddha aahet. table gasketmadhil sarvat mothya vartulanchya vakratechi yadee dete. pratyek gasketche purnapane varnan karnyasathi fakt pahilya teen vakrata (tablemadhye darshavilelya pachpaiki) aavashyak aahet-itar serve vakrata yaa teenmadhun midavata yetat.
avibhajya apollonian vartul packingschi ganana karane.
vakrata sootra _ 16 heey sootra _ 17 asalyas mool chaupat (kahee avibhajya vartul packingmadhye sarvat lahan) aahet. sootra _ 18 asatana teey adim asatat. matricks samikaranadvare chull sootra _ 19 chaa naveen sanch paribhashit karatat.
sootra _ 20
sootra _ 16 hii eka ashee pranali dete jithe sootra _ 22 achookpane jevha decartes samikaran purn karate. yavyatirikt, sootra _ 24 achookpane jevha formula _ 16 adim asate aanee sootra _ 26 achookpane jevha mool chaupat asate.
yaa sambandhancha vapar dilelya nakaratmak vakya sutrasah serve adim mool chaupat shodhanyasathi keela jaau shakato. teey sootra _ 28 aanee sootra _ 29 pasoon anusaran karate jeey sootra _ 30 aahe aanee mhanunach teey sootra _ 31. mhanoon, konatehi mool chaupat sootra _ 32 purn karel. sootra _ 33, sootra _ 34 aanee sootra _ 35 chya serve sambhavya mulyanvar punaravrutti karoon serve adim mool chaupat shodhu shakato. khaleel python code haah algoridam darshavito, varr soochibaddha kelelya adim mool chaupat tayar karato.
aayat ganeet
dafe gett _ premitive _ bends (aine):
 jarr n = = 0 asel tarah
 utpann 0,0,1,1
 parat yene
 emm. sathee shreneet (ganeet.ceal (aine/ganeet.ess. kyu. aar. tii. (3)):
 s = m * * 2 + n * * 2
 dii 1 sathee shreni (kamal (2 * mea, 1), ganeet.majala (ganeet.ess. kyu. aar. tii. (ess) + 1):
 d2, urvarit = devode (s, d1)
 jarr urvarit = = 0 aanee ganeet.jee. see. dii. (aine, aine, dii1 = = 1:
 utpann-aine, aine + aine, heey + aine, aine + heey + aine-2 * emm
aine sathee shreneet (15):
 gett _ premitive _ bends (aine) madheel vakyansathih
 chhapun (vakane)
avibhajya apollonian vartul packing samamiti.
vartulanchya vakratenusar gaskets udbhavu udbhavu dihedral samamitiche anek prakar aahet.
samamiti nahee.
pahilya pachmadhye jarr konatyahi vakratechi punaravrutti jhaali nahee, tarr gasketmadhye konatihi samarupata nasate, jee samarupata gut "see" 1 dware darshavili jaate; vakratedvare varnan keleli gasket (−10,18,23,27) heey eka udaharan aahe.
"dii" 1 samamiti.
jevha gasketmadhil sarvat mothya paach vartulanpaiki done vartulanmadhye samaan vakrata asel, tevha tyaa gasketmadhye "dii" 1 samamiti asel, jee konatyahi firati samamitishivay, seema vartulachya vyasachya paravartanshi sambandhit asel.
"dii" 2 samamiti.
pahilya pachmadhye done bhinn vakrata punha kelyas, gasketmadhye d2 samamiti asel; asha samamitimadhye 180° chya done putt firki samamitisah, seema vartulachya vyasasah done paravartan (ekmekanna lumb) asatat. vakratene (− 1,2,3) varnan kelele gasket heey heey samamiti asalele ekmev apollonian gasket (scaling factorparyant) aahe.
"dii" 3 samamiti.
d" 3 samamiti asalele purnank gasket naheet.
jarr sarvat lahan sakaratmak vakrata asalelya teen vartulanmadhye samaan vakrata asel, tarr gasketmadhye "d" 3 samamiti asel, jee 120° chya teen-putt firki samamitisah, seema vartulachya vyasasah (120° antaravar) teen paravartananshi sambandhit asel. yaa prakaranat teen aateel vartulanchya seema vartulachya vakrateche gunottar 2−3 aahe. heey gunottar tarksangat naslyamule, konatyahi avibhajya apollonian vartul packingmadhye hii 'dii' 3 samamiti nasate, jaree anek packing jawal yetat.
jawaljawal-"dii" 3 samamiti.
davikadil aakruti hii eka avibhajya apollonian gasket aahe jyamadhye "dii" 3 samamiti asalyache disate. teech aakruti ujavikade darshavili jaate, labels aateel vartulanchi vakrata darshavitat, heey darshavitat qee gasketmadhye pratyakshat itar anek avibhajya apollonian gasketsathi samanya asaleli fakt "dii" 1 samamiti asate.
khaleel taktyat yapaiki adhik 'jawaljawal'-'dii' 3 avibhajya apollonian gasketchi yadee aahe. yaa anukramat kahee manoranjak gunadharm aahet aanee taktyat vakrateche gunank, tasech mageel sanchatoon vartaman sanchat jaanyasathi aavashyak asalelya gunakanchi yadee aahe. aaa" diskchya vakratechi paripoorn moolye punaravrutti sambandhanche palan karatat, jyatoon gunak + 2 ± 3.732050807 madhye abhisaran karato.
anukramik vakrata.
konatyahi purnank "n"> 0 sathee, khaleel vakratedvare paribhashit eka apollonian gasket astitvat aaheh (− "n", "n" + 1 ", n" ("n" + 1), "n" ("n" + 1) + 1). udaharanarth, (− 2,3,6,7), (− 3,4,12,13), (−8,9,72,73) aanee (−9,10,90,91) yanni paribhashit kelele gasket heey serve yaa namunyache anusaran karatat. kaaran 'aine' + 1 neey paribhashit kelele pratyek aateel vartul vartul gasketmadhil seema vartul (-'aine' neey paribhashit kelele) banoo shakate, yaa gasketla nested kele jaau shakate. heey ujavikadil aakrutimadhye darshavile aahe, jyamadhye 2 teey 20 paryant paryant 'aine' saha yaa anukramik gasket aahet.

