
4-dimensional euclidean bahiraakaashathe bhramanagal

ganithashaasthrathil, naalu dimensional euclidean spacil oru nischitha binduvinekkurichulla bhramanangalude groupine soochippikkunnathu (4) ennaanu. ithu oordar 4-nte prathyeka oorthoogoonal grooppaanenna vasthuthayil ninnaan eee peru vannath.
eee lekhanathil "bhramanam" ennaal "bhramana sthaanachalana" ennaanu ardhamaakkunnathu. savisheshathaykaayi, sandarbham soochippichatho vyakthamaayi soochippichatho ozhike bhramana konukal eee vibhaagathil undennu anumaanikkappedunnu.
bhramanathinushesham thalatthile oro vektarum maattamillaatha oru thalamaanu "sthira thalam". oru "invariyant playin" ennathu oru playinaanu, athinaayi playinile oro vektarum, bhramanathaal baadhichekkaamenkilum, bhramanathinu sesham playinil nilanilkkunnu.
4dii bhramanangalude jyamiti.
naalu dimensional bhramanagal rund tharatthilunda lalithamaaya bramanangalum eratta bramanangalum.
lalithamaaya bhramanagal.
oru bhramana kendrathekkurichulla oru lalithamaaya bhramanam oru muzhuvan thalatheyum (axis-thalam) sthiramaayi vidunnu. poornnamaayum oorthoogoonal aaya oro thalavum oru nischitha pooyintil murichukadakkunnu. atharam oro pooyintinum inn indues cheytha 2 dii bhramanathinte kendramaanu. eee 2dii bhramanangalkkellam ore bhramana konanullathu.
akshathalathil ninnulla pakuthi varikal sthaanabhramsham varuthiyittilla; oorthoogoonalil ninnulla pakuthi varikal sthaanabhramsham varuthunnu; matellaa pakuthi varikalum athinekkaal kuranja koniloode sthaanabhramsham varuthunnu.
eratta bhramanagal.
4-space (uthbhavam nischayikkunnathu) yude oro bhramanathinum, kuranjathu oru jody oorthoogoonal 2-playinukalenkilum undu, avaa oronnum maattamillaathathum 4-space ullathum aanu. athinaal eee vimaanangalil eathenkilum onnil pravartthikkunnathu aaa thalatthinte oru saadhaarana bhramanam srishtikkunnu. mikkavarum ellaavarkkum (trimana upavibhagam ozhikeyulla ellah 6-dimensional bramanangalum), thalatthileyum thalatthileyum bhramana konukal-randum poojyamallaathavayaanennu karuthappedunnu-vyathyasthamaan. asamamaya bhramana konukalum samthrupthikaravum eathaandu savisheshamaayi nirnnayikkappedunnathu. 4-space ooriyantad aanennu karuthuka, thudarnnu 2-playinukalude ooriyanteshanukal eee ooriyanteshanumaayi poruthappedunna rund vazhikaliloode thiranjedukkaam. bhramana konukal asamamaanengil (), chilappol "eratta bhramanam" ennu vilikkunnu.
eratta bhramanathinte saahacharyathil, oreyoru jody invariyant playinukalum uthbhavasthaanathu ninnulla pakuthi varikalum yathaakramam sthaanabhrashtamaakkukayum uthbhavasthaanathu ninnulla pakuthi varikal akathallaathatho allengil karsanamaayi edayilulla konukaliloode sthaanabhrashtamaakkukayum cheiyunnu.
isoclinic bhramanagal.
eratta bhramanathinte bhramana konukal thulyamaanengil, randinupakaram ananthamaaya niravadhi invariyant playinukal undu, athil ninnulla ellah pakuthi varikalum ore koniloode sthaanabhramsham varuthunnu. atharam bhramanangale isoclinic allengil equiangular bhramanagal allengil clifford sthaanachalanangal ennu vilikkunnu. shradhikkuka ellah thalangalum isoclinic bhramanangalkk keezhil maattamillaathavayalla; oru pakuthi varayal vyaapichukidakkunna thalangalum anubandha sthaanabhramsham sambhavicha pakuthi variyum maathram maattamillaathavayaanu.
4-dimensional spacinaayi oru nischitha ooriyanteshan thiranjedutthuvennu karuthuka, isoclinic 4 dii bhramanangale rund vibhagangalaayi thirikkam. ithu kaanunnathin, oru isoclinic bhramanam pariganikkuka, koodaathe oru varianted playinil (soochippikkunnathu) parasparam lambamaya ararekhakalude oru ooriyanteshan-sthiramaaya oordar settu edukkuka, athinaal oru varianted playinil vyaapikkukayum cheyyuka. ippol bhramanakonam maathrame vyakthamaakkiyittullu ennu karuthuka. appol, bhramana indriyangale aashrayich, samathalangalilum bhramana konilum pothuve naalu isoclinic bhramanangalund.
bhramanangalkk muthal muthal vare positteev aayi kanakkaakkappedunnu enna kanvenshan nammal undaakkunnu. appol namukk naalu bhramanangalund, oppam. avaa parasparam vipareethamaanu; athupole thanne avayum. 0 num idayil ullidatholam kaalam, eee naalu bramanangalum vyathyasthamayirikkum.
samanamaaya adayaalangalulla isoclinic bhramanangale "idathu-isoclinic" ennum vipareeta adayaalangalullava "valath-isoclinic" ennum soochippikkunnu. idathu, valath isoclinic bhramanangale yathaakramam idathu, valath gunanam yoonit kwaatterniyanukal prathinidheekarikkunnu; chuvadeyulla khandika "kwaatterniyanukalumaayulla bandham" kaanuka.
allengil ozhike naalu bhramanagal jody thirichulla vyathyasthamaan. angle aidantitti rotteshanumaayi poruthappedunnu; aidantitti maatriksinte negatteevu nalkunna central inversionumaayi yojikkunnu. (4)-nte eee rund gadakangal oresamayam idathu-valath-isoclinic aanu.
mukalil nirvachicha idathu, valath isoclini eathu prathyeka isoclinic bhramanam thiranjedutthu ennathine aasrayichirikkunnu. ennirunnaalum, swantham achuthandukalulla mattoru isoclinic bhramanam thiranjedukkumbol, oraalkk ellaayppozhum, oru bhramana-prathiphalanatthiloode parivartthanam cheiyunnathinupakaram oru bhramanathiloode parivartthanam cheyyaan kazhiyunna kramam thiranjedukkaam (athaayath, kramappeduthiya adisthaanam,,,, enna athe nischitha ooriyanteshan thiranjeduppumaayi poruthappedunnu). athinaal, oraal oru ooriyanteshan thiranjeduthukazhinjaal (athaayath, saarvathrikamaayi valamkaiyaayi soochippikkunna achuthandukalude oru samvidaanam), oraalkk oru prathyeka isoclinic bhramanathinte idathu allengil valath swabhaavam nirnnayikkaan kazhiyum.
angane group ghadana (4).
athinaal (4) oru noonn commutative compact 6-dimensional lye grouppaanu.
bhramana kendrathiloodeyulla oro thalavum oru commutative subgroupinte isomorphic aksha-thalamaanu (2). eee upavibhagangalellaam parasparam samyojippichirikkunnu (4).
poornnamaayum oorthoogoonal thalangalude oro jodiyum oru commutative subgroupinte (4) isomorphic aaya inavariyant thalangalude jodiyaanu.
eee grouppukal (4) nte paramavadhi toriyaanu, avayellaam (4) ll paraspara samyojithamaanu. clifford torres kaanuka.
ellah idathu-isoclinic bramanangalum yoonit kwaatterniyanukalude gunana groupinu isomorphic aaya (4) enna noonn commutative upavibhagam roopappeduthunnu. ellah raitt-isoclinic bramanangalum athupole thanne (4) isomorphic enna oru upavibhagam roopappeduthunnu. randum angane (4) nte paramavadhi upavibhagangalaanu.
oro idathu-isoclinic bhramanavum oro valath-isoclinic bhramanathodoppam sanjarikkunnu. ithu soochippikkunnathu saadhaarana upagrooppukalulla oru nerittulla uthppannam nilavilundennum; bandhapetta rund ghadaka grouppukalum nerittulla uthppannathinte mattu gadakamaaya i-nuu isomorphic aanu.iii. isomorphic. (ithu anganeyalla (4) allengil athinte oru upagroop, kaaranam avaa verthirikkappedunnil identityum central inversion oronnum aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aandu aaa)
oro 4dii bhramanavum rund tharatthil idathu, valath-isoclinic bhramanangalude ulppannamaanu. avayellaam koodichernnu central inversion, i vare nirnnayikkappedunnu.iii. randum kendra vipareethathaal gunikkumbol avayude uthppannam veendum varum.
ithu soochippikkunnathu athu (4)-athinte athulyamaaya eratta cover------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ aidantitti rotation central inversion oordar 2-nte oru grouppaanu, ithu angane (4)-nuu kendramaanu, koodaathe randum. oru grouppinte kendram aaa grouppinte oru saadhaarana upavibhagamaanu. c2-nte ghadakam group athinaal (4) ennathu (3) × athinaal (3)-nuu isomorphic aanu. 3l/c2,3r/c2 ennivayude ghadaka grouppukal oronnum angane (3) nuu isomorphic aanu. athupole, athinaal (4) bye 3l, athinaal (4) bye 3r ennivayude ghadaka grouppukal oronnum angane (3) nuu isomorphic aanu.
athinaal (4) nte topology lye groupinu thulyamaanu, athaayath space formula _ 1, ivide formula _ 2 ennathu alavukal 3-nte yadhaartha projective space aanu, formula _ 3 ennathu 3-golamaanu. ennirunnaalum, oru nuna group enna nilayil, athinaal (4) nuna grouppukalude nerittulla uthppannamalla, athinaal ithu nuna grouppukalkku isomorphic allah ennathu sradheyamaanu.
saadhaaranayaayi bhramana grouppukalkkidayil (4) enna prathyeka swath.
otta-dimensional rotteshan grouppukalil central inversion adangiyittilla, avaa lalithamaaya grouppukalaanu.
eratta-dimensional rotteshan grouppukalil central inversion adangiyirikkunnu, koodaathe groupinu avayude kendramundu. n ≤6 ennathinu, athinaal (n) eathaandu lalithamaanu, athinaal athinte madhyabhaagathu (n)/c2 enna ghadaka group oru lalithamaaya grouppaanu.
athinaal (4) vyathyasthamaan idathu, valath-isoclinic bhramanangale parasparam parivartthanam cheiyunna (4) enna moolakatthinte oru samyojanavum illa. prathiphalanangal oru idathu-isoclinic bhramanathe samyojanathiloode valath-isoclinic bhramanamakki maattunnu, thirichum. ithu soochippikkunnathu, nischitha binduvullah "ellah" isometrikaludeyum group oo (4) nuu keezhil vyathyasthamaaya upagrooppukal parasparam samyojippichirikkunnathu oo (4) nte saadhaarana upagrooppukalaakaan kazhiyilla. 5d bhramana groupil (5) umm ellah uyarnna bhramana grouppukalilum o (4) vare isomorphic upagrooppukal adangiyirikkunnu. athupole (4), ellah thulya-dimensional bhramana grouppukalilum isoclinic bhramanagal adangiyirikkunnu. ennaal (4) ll ninnu vyathyasthamayi, angane (6) lum ellah uyarnna eratta-dimensional bhramana grouppukalilum ore koniloodeyulla eathenkilum rund isoclinic bhramanagal samyojithamaanu. ellah isoclinic bhramanangaludeyum koottam oru saadhaarana upagroop ennu vechaal (2) enna upagroop polumalla.
4d bhramanangalude beejaganitam.
athinaal (4) saadhaaranayaayi thirichariyappedunnathu oru 4d vekttar spacinte ooriyanteshan-prejerving isometric leanier maappingukalude grouppumaayi, yadhaartha samkhyakalkku mukalilulla aantharika ulppannavumaayi.
atharamoru sthalathe oorthoonoormal adisthaanavumaayi bandhappett (4) nirnnayaka + 1 ulla yadhaartha 4-aamm oordar oorthoogoonal maatriksukalude groupaayi prathinidheekarikkunnu.
isoclinic vighadanam.
athinte matricks nalkunna 4D bhramanam thaazhepparayunna reethiyil idathu-isoclinic, valath-isoclinic bhramanamaayi vighadikkunnu
anuvadikkuka.
ekapaksheeyamaaya oorthoonoormal adisthaanavumaayi bandhappett athinte matricks aayirikkuka.
ithil ninnu "associate matricks" ennu vilikkappedunnava kanakkaakkuka."
 4dii rotteshan matricks aanenkil maathram 16dii vektaraayi yoonit euclidean maanadandamulla rank onnaanith. eee saahacharyathil yadhaartha samkhyakal nilavilundu,
oppam
krithyamaayi rund settukal undu, atharam. avar parasparam ethirvashathaanu.
appol bhramana matricks thulyamaanu
vaan elphinkhof (1897) aanu eee soothravaakyam.
eee vighadanathile aadya ghadakam idathu-isoclinic bhramanatheyum randaamathe ghadakam valath-isoclinic bhramanatheyum prathinidheekarikkunnu. gadakangal nirnnayikkunnath negatteevu 4-aamm oordar aidantitti matricks, athaayath.iii. kendra vipareeta.
kwaattarniyanukalumaayulla bandham.
cartician kordinettukalulla 4-dimensional speysile oru pooyintine oru kwaternian prathinidheekarikkam.
oru idathu-isoclinic bhramanathe prathinidheekarikkunnath oru yoonit kwaternian vazhi idathu-gunanam aanu. matricks-vector bhashayil ithu
athupole, oru valath-isoclinic bhramanathe prathinidheekarikkunnath matricks-vector roopathilulla oru yoonit kwaternian vazhi valath-gunanam aanu.
muun vibhaagathil (#isoclinic vighadanam) oru pothuvaaya 4d bhramanam idathu, valath-isoclinic gadakangalaayi engane vibhajikkappettirikkunnu kaanikkunnu.
kwaternian bhashayil vaan elphincophinte soothravaakyam vaayikkunnu
allengil pratheekathmaka roopathil,
jarmman ganithashaasthranjanaaya felix clain parayunnathanusarichu eee soothravaakyam 1854-ll kaelick ithinakam ariyamaayirunnu.
kwaternian gunanam associative aanu. athinaal,
idathu-isoclinic, valath-isoclinic bhramanagal sanjarikkunnuvennu ithu kaanikkunnu.
4dii bhramana maatriksukalude eejan moolyangal.
oru 4dii rotteshan maatriksinte naalu eejan moolyangal saadhaaranayaayi yoonit maagnittyoodinte sankeernna samkhyakalude rund samyojitha jodikalaayi sambhavikkunnu. oru eejan value yadhaarthamaanengil, athu ±1 aayirikkanam, kaaranam oru bhramanam oru vektarinte vyaapthiye maattamillathe vidunnu. aaa eejan moolyathinte samyojanavum ekathvamaanu, ithu oru nischitha thalathe nirvachikkunna oru jodi egenvectorukal nalkunnu, athinaal bhramanam lalithamaanu. quaternian notteshanil, a properor (i.iii.yoonit kwaatterniyanukalude yadhaartha bhagangal vyapthiyil thulyavum ore chinnavumullathaanenkil maathram (4) ennathile noonn-inverting) bhramanam sariyaaya lalithamaaya bhramanamaan. randum poojyamaanengil, bhramanathinte ellah eejan moolyangalum ikyamaanu, bhramanam shoonya bhramanamaan. athinte yadhaartha bhaagangalum thulyamallaathathum aanenkil ellah eejan moolyangalum sankeernamaanu, bhramanam eratta bhramanamaan.
3dii bhramanangalkkulla euler-rodrigues formula.
nammude saadhaarana 3D space soukaryapradamaayi coordinate sisttam uxyz ulla 4d spacinte 0xyz coordinate sisttamulla subspace aayi kanakkakkunnu. athinte bhramana group athinaal (3) metriksukal ulkkollunna (4) nte upavibhagavumaayi thirichariyunnu.
muun upavibhaagathile vaan elphinkhophinte soothravaakyathil, moonnu alavukalilekkulla eee niyanthranam kwaternian praathinidhyathilekku nayikkunnu
3dii bhramana matricks pinneedu 3dii bhramanangalkkulla euler-rodrigues formulayayi maarunnu.
ithu 3D bhramanathinte prathinidheekaranam athinte euler-rodrigukal paaraameettarukal
anubandha kwaternian formula, evide, allengil, vipuleekritha roopathil
haamilttan-kayley formula ennaanu ithu ariyappedunnathu.
hoopp kordinettukal.
golaakrithiyilulla kordinettukal upayogichu 3dii sthalathe bhramanagal ganithashaasthraparamaayi kooduthal tractable akkunnu. 3d-yile eathu bhramanavum oru nischitha bhramana achuthandum aaa achuthandinu lambamaya oru asthira thalavum kondu savisheshamaakkaam. saamanyatha nashtappedaathe, namukk-playin inavariyant playnaayum-axis fixed axis aayum edukkaam. radial doorangale bhramanam baadhikkaathathinaal, yoonit golatthil (2-golam) athinte swaadheenam kondu sthira akshatheyum asthira thalatheyum paraamarsikkunna golaakrithiyilulla kordinettukal upayogichu namukk oru bhramanathe savisheshamaakkaam
kaaranam, pooyintukal 2 golathilaanu. - axisinu chuttumulla oru konil karangunna oru pooyint lalithamaayi vyakthamaakkunnath. 4dii bhramanagal kaikaaryam cheiyunnathil hyperferrical kordinettukal upayogapradamaanengilum, 4dick ithilum upayogapradamaaya oru coordinate sisttam hoopp kordinettukal nalkunnu, avaa 3-golatthil oru sthaanam vyakthamaakkunna moonnu koneeya kordinettukalude oru koottamaanu. udaaharanathina
kaaranam, pooyintukal 3 golathilaanu.
4dii spacil, uthbhavathekkurichulla oro bhramanathinum rund invariyant thalangalundu, avaa parasparam poornnamaayum oorthoogoonal aanu, avaa uthbhavasthaanath murichukadakkunnu, koodaathe rund swatantra konukalaal karangunnu. saamanyatha nashtappedaathe, namukk yathaakramam-umm-playinukalum eee invariyant playinukalaayi thiranjedukkaam. konukaliloode oru pooyintinte 4d yill oru bhramanam nadathukayum pinneedu hoopp kordinettukalil ingane prakadippikkukayum cheiyunnu.
4dii bhramanangalude drishyavalkkaranam.
3dii speysile oro bhramanathinum bhramanathaal maattamillaatha oru nischitha akshamundu. bhramanathinte akshavum aaa akshathinu chuttumulla bhramana konum vyakthamaakkiyaan bhramanam poornnamaayum vyakthamaakkunnath. saamanyatha nashtappedaathe, eee aksham oru cartician coordinate sisttathinte-axis aayi thiranjedukkaam, ithu bhramanathinte lalithamaaya drishyavalkkaranam anuvadikkunnu.
3dii spacil, golaakrithiyilulla kordinettukale 2-golatthinte oru parametric expressionaayi kaanavunnathaanu. sthirathaykkaayi avar-akshathinu lambamaya 2-golatthile vrithangale vivarikkunnu, eee vrithangale golatthile oru binduvinte sanjaarapathangalaayi kaanam. golatthile oru pooyint,-aaksisine chuttipattiyulla oru bhramanathinu keezhil, koonn vyathyaasappedumbol oru paatha pinthudarum. bhramanapathathe samayathile oru bhramana paaraamettariyaayi kaanam, avide bhramanakon samayathil rekheeyamaanu, oru "koneeya vegatha" aanu.
3dii kesinu samaanamaayi, 4dii speysile oro bhramanathinum kuranjathu rund invariyant axis-playinukalundaakum, avaa bhramanathaal invariyant aayi avasheshikkukayum poornnamaayum oorthoogoonal (i.iii. avaa oru pooyintil murichukadakkunnu). akshathalangalum avaykku chuttumulla bhramana konukalum vyakthamaakkiyaan bhramanam poornnamaayum vyakthamaakkunnath. saamanyatha nashtappedaathe, eee akshathalangal oru cartician coordinate sisttathinte-aandu-thalangalayi thiranjedukkaam, ithu bhramanathinte lalithamaaya drishyavalkkaranam anuvadikkunnu.
4dii sthalathu, hoopp konukal 3-golathe paaraameettar cheiyunnu. sthirathaykkaayi avar aandu aandu playinukalile clifford torresinte prathyeka kesai kanakkaakkappedunna oru torressine vivarikkunnu. eee torikal 3dii-spacil kaanapedunna saadhaarana torikalalla. avaa ippozhum 2dii uparithalangalaanenkilum, avaa 3-golatthil ulchernnirikkunnu. 3-golathe euclidean 3dii-space muzhuvan stereographic aayi project cheyyaan kazhiyum, eee torikale appol viplavathinte saadhaarana toriyaayi kaanunnu. - umm-playinukalum inverentumaayi oru bhramanathinu vidheyamaakunnathiloode vyakthamakkiya oru pooyint torressil nilanilkkumennu kaanaan kazhiyum. oru binduvinte sanjaarapatham samayathinte oru pravartthanamaayi ezhuthaanum thaazhepparayunna chithrangalilennapole athinte anubandha torressil stereographic aayi project cheyyaanum kazhiyum. eee kanakkukalil, prarambha pooyint, i aayi kanakkakkunnu.iii. clifford torressil. figgle. 1, rund lalithamaaya bhramana pathakal karuppil kaanikkunnu, athesamayam idathu, valath isoclinic pathakal yathaakramam chuvapp, neela nirangalil kaanikkunnu. figgle. 2, oru pothu bhramanam, athil kaanikkunnu, amshakathil. 3, kaanikkunna oru pothuvaaya bhramanam.
4dii rotteshan maatriksukal srishtikkunnu.
rodrigukalude bhramana soothravaakyathil ninnum kayley soothravaakyathil ninnum naalu dimensional bhramanagal uruthirinjekkaam. oru 4x4 scue-symmetric matricks ayirikkatte. vakra-samamiti matricks advitheeyamaayi vighadippikkaan kazhiyum
rund vakra-samamithikalayi maattukayum gunangale thripthippeduthukayum cheiyunnu, koodaathe, evide, evideyaanu ivayude eejan moolyangal. thudarnnu, 4 dii rotteshan maatriksukal scue-symmetric maatriksukalil ninnum rodrigukalude rotteshan formulayiloodeyum kayley formulayiloodeyum labikkum.
eejan moolyangalude koottathodukoodiya 4x4 poojyamallaatha vakra-samamitiyilulla oru matricks ayirikkatte
appol vighadippikkaam
evide, avaa vakra-samamitiyilulla maatriksukal gunangale thripthippeduthunnu
koodaathe, vakra-samamitiyilulla maatriksukal savisheshamaayi eee reethiyil labikkum

