
samboornna jyamiti

samantharamaaya swamedhayaa allengil athinte itharamaarggangalonnumillathu euclidean jyaamithiyude svayamsidha samvidaanathe adisthaanamaakkiyulla oru jyaamithiyaan kevala jyamiti. parambaraagathamaayi, yooklidinte aadyathe naalu swamedhayaa ulla sidhaanthangal maathrame upayogikkunnulloo, ennaal euclidean jyaamithiyude adisthaanamaayi iva ippol aparyapthamaayi kanakkakkappedunnathinaal, samanthara swasidhathayillaatha hilbertinte swasidhangal polulla mattu samvidaanangal upayogikkunnu. 1832-ll janos bolyay aanu eee padam avatharippichathu. samanthara swamedhayaa ithu nishpakshamayathinaal ithine chilappol neutral jyamiti ennu vilikkunnu.
swathukkal.
kevala jyamiti oru durbalamaaya samvidaanamaanennu sangalppikkaam, pakshe anganeyalla. vaasthavathil, yooklidinte "moolakangalil", aadyathe 28 nirddeshangalum nirddeshangalum 31 samantharamaaya swamedhayaa upayogikkunnathu ozhivaakkunnu, athinaal avaa kevala jyamithiyil saadhuthayullavayaanu. samboornna jyamithiyil baahya konukalude sidhaantham (oru trikonatthinte baahyakonam vidoora konukalekkaal valuthaanu), athupole thanne trikonathile konukalude alavukalude aakethuka paramavadhi 180° aanennu prasthaavikkunna sakeri-legender sidhaantham ennivayum theliyikkaan kazhiyum.
thannirikkunna variyil illaatha oru pooyintiloode oru rekhaykku samaantharamaayi oru rekhayude nirmmaanamaan nirddesham 31. thelivinaayi nirddesham 27 (alternate interior angle sidhaantham) maathram upayogikkunnathinaal, ithu kevala jyamithiyil saadhuvaaya oru nirmmaanamaan. kooduthal krithyamaayi paranjaal, "l" enna variyil illaatha eathenkilum variyum "p" enna binduvum nalkiyaal, "p" vazhi "kuranjathu" oru variyenkilum "l" nuu samaantharamaayirikkum. oru parichithamaaya nirmmaanam upayogichu ithu theliyikkaan kazhiyum oru variyum "l" ll illaatha oru pooyintum "p" nalkiyaal, lambamaya "m" "p" ll ninnu "l" lake iduka, thudarnnu "p" vazhi "n" muthal "m" vare lambamaayi oru nirmmikkuka. alternate interior angle sidhaantham anusarichu, "l" ennathu "n" nuu samaantharamaanu. (alternate interior angle sidhaantham parayunnath "a", "b" ennee varikal oru transversal "t" kondu murichaal, oru jodi kongruvant alternate interior konukal undengil, "a", "b" enniva samaantharamaanu.) melpparanja nirmmanavum ithara interior angle sidhaanthavum samanthara swamedhayaa aasrayikkunnilla, athinaal avaa kevala jyamithiyil saadhuthayullavayaanu.
kevala jyamithiyil ore rekhaykku lambamaayi rund varikalkk murichukadakkaan kazhiyillennum (ithu rund varikaleyum samanthara rekhakalude nirvachanam anusarichu samantharamaakkunnu) theliyikkappedunnu, ithu oru sakeri chathushkonatthinte ucha konukal thirascheenamaayirikkillennum golaakrithiyilulla jyamiti oru samboornna jyamithiyallennum theliyikkunnu.
mattu jyamithikalumaayulla bandham.
kevala jyaamithiyude sidhaanthangal hyperbolic jyaamithiyilum athupole euclidean jyaamithiyilum nilanilkkunnu, ithu euclidean ithara jyaamithiyaan.
samboornna jyamiti deerghavrithaakruthiyumaayi poruthappedunnilla aaa sidhaanthathil, samanthara rekhakalonnumilla, pakshe samanthara rekhakal nilanilkkunnath kevala jyaamithiyude sidhaanthamaanu. ennirunnaalum, svayamsidhamaaya samvidaanathil maattam varuthaan kazhiyum, angane parishkaricha sisttam nirvachichathupole kevala jyamithiyil samanthara rekhakalillaatha golaakrithiyilullathum deerghavrithaakrithiyilullathu jyaamithikal ulppedum.
kramappeduthiya jyaamithiyude vipuleekaranamaanu kevala jyamiti, athinaal, kramappeduthiya jyaamithiyile ellah sidhaanthangalum kevala jyamithiyil nilanilkkunnu. neremarichu parayunnath shariyalla. yooklidinte moonnaamatheyum naalaamatheyum sidhaanthangal anumaanikkaatha affine jyamithiyumaayi thaarathamyappeduthumbol, yooklidinte aadya naalu sidhaanthangal (allengil avaykku thulyamaayava) kevala jyamiti anumaanikkunnu.
(3: "eathenkilum kendravum dooraparidhiyum ulla oru vrithathe vivarikkan.",
4: "ellah samakonukalum parasparam thulyamaanu.") 
kramappeduthiya jyamiti kevalavum affine jyamitiyum thammilulla oru pothu aditharayaanu.
onpathu sidhaanthangalum kevala jyaamithiyude pathinonnu nirddeshangalum upayogichu prathyeka aapekshikathayude jyamiti vikasippicheduthittundu. rachayithakkalaaya edwin bii. vilson, gilbert enn. rund frameukal referenceumaayi bandhapetta parivarthanamaayi hyperbolic bhramanam avatharippikkumbol levis kevala jyamithikkappuram munnott pookunnu.
hilbert vimanangal.
hilbertinte insidence, edaness, kongruvans axium enniva thripthippeduthunna oru thalathe hilbert thalam ennu vilikkunnu. samboornna jyaamithiyude maathrkakalaanu hilbert thalangal.
apoornnatha.
svayamsidhamaaya oru samvidaanathe asthiramaakkaathe oraalkk adhika swathanthramaaya svayamsidhathakal cherkkaan kazhiyum enna ardhathil samboornna jyamiti oru apoornnamaaya svayamsidhamaaya samvidaanamaanu. samaanthararekhakalekkurich vyathyastha sidhaanthangal cherkkunnathiloode oraalkk kevala jyamiti vipuleekarikkanum euclidean allengil hyperbolic jyaamithikk kaaranamakunna poruthappedaatthathum ennaal sthirathayullathumaaya sidhaantha samvidaanangal nedaanum kazhiyum. athinaal kevala jyaamithiyude oro sidhaanthavum hyperbolic jyaamithiyudeyum euclidean jyaamithiyudeyum sidhaanthamaanu. ennaal, neremarichu parayunnath shariyalla.