
polytope

prathamik bhoomitimadhye, bahupadi hii sapat bajoo ("chehare") asaleli bhaumitik vastoo aahe. bahupadi mhanje trimiteey bahupadiche konatyahi sankhyechya parimananmadhye samanyikaran. bahupadi-aayami bahupadi kinwa-bahupadi mhanoon konatyahi samanya sankhyechya parimananmadhye astitvat asoo shakatat. udaharanarth, dvi-aayami bahubhuj haah dvi-bahupadi aahe aanee tri-aayami bahudhruv haah tri-bahupadi aahe. yaa sandarbhat, "sapat bajoo" mhanje-bahupadichya bajunna-bahupadi asatat jyat-bahupadi samaan asoo shakatat.
kahee siddhant simahin aperotope aanee tessellation, golakar bahu-vibhajak aanee sanch-saiddhantik amoort bahu-prushthansah vakr bahu-pattyanche vighatan kinwa jhukane yasarkhya vastooncha samavesh karnyachya kalpanela aanakhi samanya karatat.
teenpeksha jaast parimananche bahupadi 1853 purvi ludwig schlafly yanni pahilyanda shodhale hote, jyanni asha aakrutila polyschem mhatale hote. german shabd 'polytop' haah ganitshastradnya reinhold hoppey yanni tayar keela hota aanee elysia bool stott yanni ingraji ganitshastradnyanna 'polytope' mhanoon olakh karoon deela hota.
vyakhyesathiche marg.
aajkal, 'polytope' hii sandnya eka vyapak sandnya aahe jyamadhye vastooncha eka vistrut varg samavisht aahe aanee ganiteey sahityat vividh vyakhya aadhaltat. yapaiki anek vyakhya ekmekanchya samatulya naheet, parinami vastoonchya vegvegalya parasparsambandhit sanchanna "polytope" mhanatat. samaan gunadharm asalelya itar vastooncha samavesh karnyasathi teey bahirmukh bahupadiche samanyikaran karnyasathi vegvegalya paddhatinche pratinidhitva karatat.
mool drushtikon vyapakpane ludwig schlafly, thorold gosset aanee itarandvare anusaran keela jaato, jyachi suruvat chaar kinwa adhik parimananmadhye, anukrame done aanee teen parimananmadhye bahubhuj aanee bahubhuj kalpanechya sadrushyane hote.
polyheadrache euler vaishishtya uchch-aayami polytopmadhye samanya karnyachya prayatnanmule topologycha vikas jhaala aanee polytopeshy samaan asalelya vighatana kinwa see. doubleu.-complexver upchar kele gele. yaa drushtikonat, polytope heey kahee dilelya anek patanche tessellation kinwa vighatan manale jaau shakate. yaa drushtikonache eka udaharan polytopechi vyakhya binduncha sanch mhanoon karate jeey eka sadhe vighatan swikarte. yaa vyakhyet, polytope haah maryaditpane anek sopya goshtincha sangam aahe, jyamadhye atirikt gunadharm aahe qee, rikt chhed nasalelya konatyahi done sopya goshtinsathi, tyanche chhedanbindu heey doghanche shirobindu, kadaa kinwa uchch aayami mukh aahe. tathapi, hii vyakhya antargat sanrachanansah taranchya bahupadila parvanagi dett nahee aanee mhanunach ganitachya kahee kshetranparyant maryadit aahe.
taara bahubhuj aanee itar asamanya bandhkamanchya shodhamule tyachya aateel bhagakade durlaksh karoon, eka bahubhuj prushthbhag mhanoon bahubhuj rachana nirman jhaali. yaa prakash bahirmukh bahupadimadhye "pii"-spacemadhil bahupadi ("pii"-1)-golakachya jhukanyachya samatulya asatat, tarr itar lambvartulakar, sapat kinwa toroidal ("pii"-1)-prushthbhaganche jhukalele asoo shakatat-lambvartulakar farshi aanee toroidal polyhedron pahaa. bahudhruviy prushthbhag mhanje jyache chehare bahubhuj asatat, 4-bahupadi mhanje atidhruviy prushthbhag jyache pailu (peshi) bahudhruviy asatat, ityadi.
khalchya parimanatil uchch bahupadi tayar karanyachi kalpana dekhil kadhikadhi parimanat khalchya dishene vadhavali jaate, jyamadhye (kadaa) bindu jodine vedhalele 1-bahupadi aanee 0-bahupadi mhanoon bindu kinwa shirobindu mhanoon pahile jaate. haah drushtikon udaharanarth amoort bahupadichya siddhantat vaparla jaato.
ganitachya kahee kshetranmadhye, 'polytope' aanee 'polyhedron' yaa sandnya vegalya arthane vaparlya jaatatah 'polyhedron' hii konatyahi parimanatil samanya vastoo aahe (yaa lekhat 'polytope' mhanoon sandarbhit) aanee 'polytope' mhanje baddh polyhedron. hii shabdavali samanyatah bahirmukh asalelya bahupadi aanee bahupadiparyant maryadit asate. yaa paribhashemadhye, bahirmukh bahudhruv heey maryadit sankhyechya ardhsthanakanche chhedanbindu aahe aanee tyachya bajunni paribhashit kele jaate tarr bahirmukh bahudhruv heey maryadit sankhyechya bindunche bahirmukh prushthbhag asate aanee tyachya shirobindundware paribhashit kele jaate.
qamee sankhyechya parimananmadhil bahupadila manak nave asatatah
ghatak.
polytopmadhye shirobindu, kadaa, chehare, peshi ityadi vividh aayamanche ghatak asatat. vegvegalya lekhakanmadhye hyansathichi shabdavali purnapane susangat nahee. udaharanarth, kahee lekhak ("aine"-1)-aayami ghatakacha sandarbh denyasathi "faces" vapartat tarr itar lock visheshtah 2-faces darshavinyasathi "faces" vapartat. jeey" parimanancha ghatak darshavinyasathi lekhak "jeey"-faces kinwa "jeey"-faces vaparu shakatat. kahijan dongarrangancha sandarbh denyasathi "dhaar" vapartat, tarr etch. ess. emm. coxitor ("aine" −1)-aayami ghatak darshavinyasathi "seela" vaparato. 
yaa lekhat swikarlelya atty khaleel taktyat dilya aaheth
eka "aine"-aayami bahubhajak anek ("aine"-1)-aayami "pailunni" vedhalela asato. heey pailu swatach bahupadi aahet, jyanche pailu mool bahupadiche ("aine"-2)-aayami "cudd" aahet. pratyek parvatrang done bajunche chhedanbindu mhanoon udbhavate (parantu done bajunche chhedanbindu parvatrang asane aavashyak nahee). kadya punha ekada bahupadi aahet jyanche pailu mool bahupadichya ("aine"-3)-aayami seemanna janm detat, aanee asech. yaa bandhane asalelya up-bahupadila chehare kinwa visheshtah 'jeey'-aayami chehare kinwa 'jeey'-chehare asey sambodhale jaau shakate. 0-aayami chehnyala 'shirobindu' mhanatat aanee tyat ekach bindu asato. eka-aayami chehnyala 'dhaar' mhanatat aanee tyat reshecha vibhag asato. dvi-aayami mukhamadhye bahubhuj asate aanee tri-aayami mukhamadhye, jyala kadhikadhi 'peshi' mhanatat, bahu-hedron asato.
polytopeche mahattvache varg.
bahirmukh bahupadi.
polytope haah "bahirmukh" asoo shakato. uttal bahupadi heey sarvat sope prakarche bahupadi aahet aanee bahupadi sankalpanechya anek vegvegalya samanyikaranansathi aadhar tayar karatat. bahirmukh bahupadichi vyakhya kadhikadhi ardhya jaganchya sanchache chhedanbindu mhanoon kelee jaate. hii vyakhya polytopla naan maryadit kinwa naan maryadit asanyachi parvanagi dete. polytopechi vyakhya asha prakare kelee jaate, udaa.jee., resheey programingmadhye. jarr maryadit trijyecha chendu asel tarr polytope 'baddh' asato. jarr polytopmadhye kiman eka shirobindu asel tarr tyala 'tokdar' mhatale jaate. pratyek bandist rikt nasalela polyphe toke nirdeshit keela jaato. naun-point polytopeche udaharan mhanje sanch sootra _ 1. jarr teey maryadit sankhyechya vastoonchya sandarbhat paribhashit kele gele asel tarr polytope "maryadit" aahe, udaa.jee., ardh-samatalanchya maryadit sankhyeche chhedanbindu mhanoon.
jarr tyachya serve shirobindunmadhye purnank nirdeshank asateel tarr teey eka avibhajya bahupadi aahe.
uttal bahupadicha eka vishisht varg mhanje "prativartak" bahupadi asatat. jarr kahee avibhajya matricks sutrasathi _ 3, sootra _ 4, jethe sootra _ 5 sarvancha vector darshavite aanee asamanata ghatak-nihay asel tarr avibhajya sootra _ 2 prativartak aahe. yaa vyakhyevarun asey disun yete qee sootra _ 2 heey reflexive aahe jarr aanee fakt jarr serve sootra _ 8 sathee sootra _ 7 asel. dusya shabdant, sootra _ 2 chaa a, purnank jaali bindunchya drushtine, sootra _ 2 chya a pasoon, fakt simareshevar midavalelya jaali bindundware vegala aahe. samatulya, sootra _ 2 heey reflexive aahe jarr aanee keval jarr tyache duheri polytope sootra _ 12 heey avibhajya polytope asel.
niyamit polytope.
niyamit bahupadimadhye serve bahupadipeksha sarvadhik pramanat samamiti asate. niyamit polytopecha samamiti gut tyachya zendyanvar sankramansheelpane karya karato; mhanoon, niyamit polytopeche duheri polytope dekhil niyamit asate.
niyamit polytopeche teen mukhya varg aahet jeey konatyahi parimanat aadhaltatah
done, teen aanee chaar parimananmadhye niyamit aakruti samavisht aahet jyat paach putt samamiti aahet aanee tyapaiki kahee abhisarte taare aahet aanee done parimananmadhye "aine"-putt samamitiche anant pramanat niyamit bahubhuj aahet, donhi bahirmukh aanee ("aine" ≤5 sathee) taara. parantu uchch parimananmadhye itar konatehi niyamit bahupadi naheet.
teen parimananmadhye bahirmukh platonic ghanpadarthanmadhye paach putt samamiteey dodecahedron aanee icosahedron yancha samavesh hoto aanee paach putt samamitisah chaar taare asalele capler-pointsot polyhedradekhil asatat, jyamule ekoon nauu niyamit polyhedradekhil hotat.
chaar parimananmadhye niyamit 4-bahupadimadhye chaar putt samamitisah eka atirikt bahirmukh ghan aanee paach putt samamitisah done samavisht asatat. serve paach putt samamiti asalele, serve sola niyamit 4-bahupadi dett, daha taara schlafly-ches 4-bahupadi aahet.
starr polytope.
naun-convex polytope heey swayam-vichhedan karnare asoo shakate; polytopechya yaa vargat starr polytopecha samavesh hoto. kahee niyamit bahupadi taare asatat.
malmatta.
euler vaishishtyapurn.
sootra _ 13 parimananmadhil eka (bharalela) bahirmukh bahupadi "p" eka binduparyant sankuchit asalyane, tyachya seema ± 14 chey euler vaishishtyapurn sootra paryayane berijdware dile jaateh
heey polyhedrasathi yularache sootra samanya karate.
antargat coone.
gramm-euler pramey tyachpramane uttal bahupadisathi uchch-aayami bahupadisathi antargat konanchya sootra _ 18 chya paryayi berries samanya karateh
polytopeche samanyikaran.
anant bahupadi.
serve bahuvidh maryadit nasatat. jethe polytope heey anek patanche tyle kinwa vighatan mhanoon samajale jaate, hii kalpana anant anek patanparyant vistarit kelee jaau shakate. samtal jhukane, jaaga bharane (madhmashya) aanee hyperbolic jhukane yaa arthane polytope aahet aanee tyanna kadhikadhi aperotope mhanatat kaaran tyanchyakade anant peshi asatat.
yapaiki, niyamit tirkas bahubhuj aanee niyamit epirogon, chauras tyle, ghan madhmash ityadidvare darshavilelya jhukanyanchya anant malikansah niyamit rupe aahet.
amoort bahupadi.
amoort polytopecha siddhant polytopanna tyanchya nivval sanyukt gunadharmancha vichar karoon, tyanna asalelya jagepasoon vegale karnyacha prayatna karato. yamule 11-peshisarkhya antardnyani antarbhut jaagechi vyakhya karane katheen asalelya vastooncha samavesh karnyasathi shabdachi vyakhya vadhavta yete.
amoort bahubhajak haah ghatak kinwa sadasyancha anshatah krambaddha sanch asato, joou kahee niyamanche palan karato. hii purnapane bijganitiy rachana aahe aanee susangat ganiteey chaukatit vividh bhaumitik varganche samet karane katheen banvinare kahee mudde talhanyasathi haah siddhant vikasit karnyat aala hota. bhaumitik bahupadi hii sambandhit amoort bahupadichya kahee vastavik jaaget eka jaaniv asalyache mhatale jaate.
jatil polytope.
polytopschya samaan sanrachana jatil hilbert space sootra _ 20 madhye astitvat aahet jithe "aine" vastavik parimane "aine" kalpanik parimanansah asatat. niyamit jatil bahupadi adhik yogya prakare sanrachana mhanoon manale jaatat.
dwait.
pratyek "aine"-bahupadichi duheri rachana asate, jee ghatakanmadhil jodani kinwa ghatana tikavoon thevat, tyachya shirobinduchi paraspar bajoo, kadanchya kadaa aanee asha prakare samanyatah ("aine" − "jeey")-aayami ghatkansathi ("jeey" = 1 teey "aine" − 1) tyachya ("jeey" − 1)-aayami ghatakanchi paraspar adlabadal karoon prapt hote.
amoort polytopsathi, heey fakt sanch kramvari ulat karate. heey ulat niyamit polytopsathi schlafly chinhanmadhye disun yete, jithe duheri polytopeche chinh moolchya agadi ulat asate. udaharanarth, {4,3,3} heey {3,3,4} chey duheri aahe.
bhaumitik bahupadichya babatit, dvigunanasathi kahee bhaumitik niyam aavashyak aahet, udaharanarth duheri bahupadisathi varnan kelele niyam pahaa. paristhitinusar, duheri aakruti dusari bhaumitik bahubhajak asoo shakate kinwa nasoo shakate.
jarr duheri ulat kele tarr mool polytope parat midvila jaato. asha prakare, bahupadi duheri jodyanmadhye astitvat aahet.
swayam-duheri polytope.
jarr polytopmadhye bajunpramanech shirobindu, kadanchya kadanchya kadanchya aanee ashach prakare jodanya asateel, tarr duheri aakruti mool aakrutisarkhich asel aanee polytope swayam-duheri asel.
kahee samanya swa-duheri bahupadimadhye khaleel goshtincha samavesh hotoh
itihas.
bahubhuj aanee bahubhuj heey pracheen kalapasoon dnyat aahet.
1827 madhye jevha august ferdinand mobiusne shodh lavala qee done aarshatil ghan padarth tyapaiki ekala chauthya ganiteey parimanadvare firvoon adhiropit kele jaau shakatat, tevha uchch parimanancha prarambhik sanket midala. 1850chya dashkaparyant arther kelee aanee herman grassman yansarkhya itar mojkya ganitshastradnyannihi uchch parimanancha vichar keela hota.
yaa uchch jaganmadhye bahubhuj aanee bahubhujanchya anurupatecha vichar karnare ludwig schlafly heey pahile hote. tyanni 1852 madhye sahaa bahirmukh niyamit 4-bahupadiche varnan kele parantu tyanche karya tyanchya mrutyunantar sahaa varshanni 1901 paryant prakashit jhaale nahee. 1854 paryant, bernhard remanchya 'habilitationscrift' neey uchch parimananchi bhoomiti drudhpane sthapit kelee hotee aanee asha prakare 'aine'-aayami polytopechi sankalpana swikararh banavali gelee. pudhil dashakanmadhye, agadi tyachya aayushyatahi, schlafliche polytope anek vela punha shodhale gele.
1882 madhye german bhashet lihitana, hopene bahubhuj aanee bahubhuj yaa adhik samanya sankalpanecha sandarbh denyasathi 'haah shabd tayar keela. kalantarane, tarkshastradnya george bool yanchi mulgi elysia bool stott hiney ingraji bhashet 'polytope' haah ingraji bhashet aanala.
1895 madhye, thorold gossetne keval schlaflichya niyamit polytopecha punha shodh lavala nahee tarr uchch parimananmadhye ardh-rekheey polytope aanee space-bharane tessellation kalpanancha dekhil tapas keela. hyperbolic spesasarkhya gair-euclidean jaganmadhye dekhil polytopscha abhyas keela jaau lagla.
1948 madhye etch. saha eka mahattvacha tappa gathala gela. ess. emm. coxterche pustak 'regular polytope', jeey aajparyantchya kamacha saransh dete aanee tyachya swatachya naveen nishkarshanchi bharr ghalate.
darmyan, french ganitshastradnya heinry poinkare yanni polytopechi topological kalpana peacewice apghatan mhanoon vikasit kelee hotee (udaa.jee. see. doubleu.-complexes) heey anek patanche asate. branco grunbamne 1967 madhye 'bahirmukh bahupadi' varr tyache prabhavshali karya prakashit kele.
1952 madhye geophrey collin sheferdney yaa kalpanela jatil jaaget jatil bahupadi mhanoon samanyikrut kele, jithe pratyek vastavik pariman tyachyashi sambandhit eka kalpanik asate. coxterne haah siddhant pudhe vikasit keela.
jatil bahupadi, gair-abhisaran, dwaitata aanee itar ghatanandware upasthit kelelya sankalpanatmak samasyanmule grunbaum aanee itaranni shirobindu, kadaa, chehare ityadinshi sambandhit amoort sanyukt gunadharmancha adhik samanya abhyas keela. eka sambandhit kalpana mhanje aaptan sankul, jyat vividh ghatakanchya aaptan kinwa ekmekanshi asalelya sambandhancha abhyas keela gela. yaa ghadamodinmule akheris asha ghatakanche anshatah krambaddha sanch kinwa sthitee mhanoon amoort bahupadichya siddhantakade nele. pieter mcmullan aanee agon schulte yanni 2002 madhye tyanche 'abstract regular polytope' heey pustak prakashit kele.
uttal aanee unconvex yaa eksaman bahupadichi ganana chaar kinwa adhik parimananmadhye karane hii eka pralambit samasya aahe. 1965 madhye johnn konave aanee michael gaay yanni sanganakacha vapar karoon bahirmukh eksaman 4-bahupadichi purnapane ganana kelee hotee; 1997 paryant uchch parimananmadhye hii samasya ajunahi khulee hotee. 2008 paryant naun-bahirmukh eksaman bahupadichi sampoorn ganana chaar aanee tyahoon adhik parimananmadhye dnyat nahee.
aadhunik kalat, sanganak graphics, optimization, shodh engine, vishvashastra, quantum mechanics aanee itar asankhya kshetranmadhye polytope aanee sambandhit sankalpananche anek mahattvache upayog aadhalale aahet. 2013 madhye saiddhantik bhautikshastrachya vishisht ganananmadhye eka sulabh rachana mhanoon amplituhedroncha shodh lagla.
anuprayog.
optimization kshetrat, resheey programming resheey karyanchya maxima aanee minimacha abhyas karate; heey maxima aanee minima heey "aine"-dimensional polytopechya simevar aadhaltat. rekheey programingmadhye, polytope samanyikrut barricentric nirdeshank aanee slack variableschya vaparat aadhaltat.
twister siddhantat, saiddhantik bhautikshastrachi eka shakha, amplituhedron navachya polytopecha vapar upaparmanvik kananna takkar detana tyanche vikhurlele amplitude mojanyasathi keela jaato. rachana hii purnapane saiddhantik aahe aanee konatihi dnyat bhautik abhivyakti nahee, parantu kahee ganana mothya pramanat sulabh karate asey mhatale jaate.