Thaum uas tig mus rau lub dav hlau: tus mob thiab cov khoom

Parallel mus rau lub dav hlau no yog ib lub tswvyim ua ntej nyob rau hauv lub Euclidean geometry rau ntau tshaj li ob txhiab lub xyoos dhau los.

Main yam ntxwv ntawm classical geometry

Cov me nyuam yug ntawm no scientific kev qhuab qhia txuam nrog nto moo tej hauj lwm ntawm ancient Greek philosopher Euclid, uas tau sau nyob rau hauv lub thib peb xyoo pua BC, phau ntawv "Hais". Faib mus rau hauv kaum peb phau ntawv, "hais" yog lub siab tshaj plaws kev kawm tau ntawm tag nrho cov ancient lej thiab qhia ntxiv txog cov yuav tsum muaj tenets txuam nrog rau thaj chaw ntawm lub dav hlau nuj nqis.

Classical mob ntawm thaum uas tig mus dav hlau twb formulated raws li nram no: ob tug dav hlau tej zaum yuav hu ua thaum uas tig mus yog hais tias lawv txhua tus yuav tau tsis muaj ntau cov ntsiab lus. Qhov no nyeem Euclidean fifth postulate zog.

Thaj chaw ntawm thaum uas tig mus dav hlau

Lub Euclidean geometry ntawm raug rho tawm, feem ntau yog tsib:

• Cov cuab yeej yog tus thawj (thiab thaum uas tig mus rau lub dav hlau piav lawv uniqueness). Los ntawm ib tug taw tes, uas lies sab nraum ntawm no xyov dav hlau, peb yuav kos ib tug thiab tsuas yog ib qho uas tig mus dav hlau

• Qhov thib ob tej khoom (kuj hu ua zog triplicate). Nyob rau hauv lub rooj plaub uas ob tug dav hlau yog mus tib seem nrog hwm rau peb, nruab nrab ntawm lawv tus kheej, lawv kuj tib.

• Peb tej khoom (nyob rau hauv lwm yam lus, nws yog hu ua ib tug kab sib tshuam thaum uas tig mus rau lub dav hlau). Yog hais tias coj nyias txoj kab ncaj nraim crosses ib tug ntawm cov mus tib seem dav hlau, nws yuav hla thiab lwm.

• Plaub khoom (khoom teejtug uas yog cov kab ncaj nraim carved rau hauv dav hlau thaum uas tig mus txhua lwm yam). Thaum ob tug uas tig mus dav hlau tshuam qhov peb (los ntawm tej lub), thiab lawv txoj kab ntawm kev tshuam ua thaum uas tig mus

• Fifth khoom (cov cuab yeej uas piav txog ntau yam feem ntawm thaum uas tig mus ncaj kab, uas pw ntawm lub dav hlau thaum uas tig mus txhua lwm yam). Cov theem ntawm thaum uas tig mus kab, uas yog muab ntim rau ntawm ob thaum uas tig mus dav hlau tas sib npaug zos.

Parallel mus rau lub dav hlau nyob rau hauv uas tsis yog-Euclidean geometry

Xws li ib tug mus kom ze yog nyob rau hauv kev lub geometry ntawm Lobachevsky thiab Riemann. Yog hais tias Euclidean geometry yog DVR ntawm lub tiaj rau tej qhov chaw, ces Lobachevsky nyob rau hauv tsis zoo nkhaus rau tej qhov chaw (nkhaus tsuas muab tso rau), thaum Riemann nws pom nws realization nyob rau hauv zoo nkhaus rau tej qhov chaw (nyob rau hauv lwm yam lus - qhov chaw). Muaj yog ib tug heev stereotypical saib hais tias Lobachevsky thaum uas tig mus rau lub dav hlau (thiab kuj kab) sib tshuam. Txawm li cas los, qhov no yog tsis muaj tseeb. Tseeb cov me nyuam yug ntawm hyperbolic geometry twb txuam nrog ib tug pov thawj ntawm Euclid tus fifth postulate thiab hloov views on nws, tab sis tus heev txhais ntawm thaum uas tig mus dav hlau thiab cov kab ncaj nraim txhais tau tias lawv yuav tsis hla tsis Lobachevsky tsis Riemann, nyob rau hauv txawm rau tej qhov chaw lawv siv. Ib tug kev hloov ntawm lub plawv thiab hais yog raws li nram no. Nyob rau hauv qhov chaw ntawm lub postulate tias tsuas yog ib tug uas tig mus dav hlau yuav kos los ntawm ib tug taw tes tsis tau rau ib tug muab dav hlau, tuaj lwm formulation: los ntawm ib tug taw tes hais tias tsis dag rau no xyov dav hlau muaj peev xwm noj ob, yam tsawg kawg, ncaj, uas yog nyob rau hauv ib tug dav hlau nrog rau qhov no thiab tsis hla nws.