Moswavlis biblioTeka j e m a L k I k n a z e
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~ ~ 24 egviptis piramidebi da oqros kveTa saukunis ganmavlobaSi qristemde, ar gaaCndaT garkveuli codna siZvelis Sesaxeb, Tumca icodnen, rom igi Zalze Zveli iyo. maTze STabeW- dilebas axdenda misi uzarmazari zoma _ qvis mTa, kacis xeliT nagebi da saocari xelovneba konstruqciisa. xufus (xeofsis) piramida, xafras (xefrenis) da menkauras (mikeranis) piramidebTan erTad qmnis Zveli egvi ptis nekropolis* kompleqss, romelic Zveli samefos epoqaSi (Zv. w. XXVIII- XXII ss. III-IV dinastiebi) gizaSi iqna agebuli. (sur. 1) am ansamblis Seqmnis dros, adgilis SerCevidan dawyebuli yovelgvari wvrilmani didi sizustiT iqna gaTvaliswinebuli. igi nilosis dasavleT sanapiroze CrdiloeTis ganedis 30 0 paralelzea ganlagebuli, sarwyavi miwebisa da libiis mkvdari udabnos qvian grun- tian sazRvarze. Zveli egvi ptelebis rwmeniT, Camavali mzis mxare sikvdilis sasufeveli iyo. snofrus (IV dinastiis, daaxl. Zv. w. 2613-2498 pirveli faraoni) memkvidreebma Tavi- anTi sasufevlis adgilad dRevandeli giza airCies, radganac es adgili akmayofilebda geodeziur, astronomiul Tu sakulto xasiaTis yvela moTxovnas. micvalebulis `maradiuli saxlis~ _ pira- midis konstruqcia religiuri warmodgenebis Sesabamisi unda yofiliyo. samive piramida sam- yaros mxareebis mimarT sakmaod zustad aris orientirebuli _ maTi samxreT-aRmosavleTis kuTxeebi praqtikulad erT wrfeze mdebareobs. xeofsisa da xefrenis piramidebis cenrtebi Zevs erT wrfeze (xeofsis piramidis fuZis diagonalis Tanxvdenil wrfeze), romelic sur. 2. piramidebis ansamblis agebis sqema gizaSi. sur. 3. xeofsis piramidis samxreTidan CrdiloeTisaken Wrili. 3. mdinare nilosis maqsimaluri done; 4. mdinare nilosis minimaluri done 0 10 20 30 40 50 m * nekropolisi (berZ. nekropolis) _ Zveli aRmosavleTis qveynebsa da antikur samyaroSi: sasaflao, samarxi. a varskvlavisaken
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1 2 3 5 a d 5 25 egviptis piramidebi da oqros kveTa momcro piramidis (mikeranis) Crdilo-dasavleT kuTxis wveroze gadis (sur. 2). Sesasvleli moTavsebulia CrdiloeTis mxridan, saidanac Casvla SeiZleboda miwisqveSa senakSi ise, rom masSi SeRweuliyo gveleSapis Tanavarskvla- vedis alfa varskvlavis sinaTlis sxivi (sur. 3), piramidebis mSeneblobis epoqaSi es varskvlavi iTvleboda samyaros polusad. konstruqciis gaangariSebis dros saWiro iyo saaRmSeneblo wesebis mTeli kompleqsis gaTvaliswineba: piramidis fuZis kvadratis farTobis gamoTvla, gruntis SesaZleblobebis gaangariSeba, rom gaeZlo uzarmazari qvis mTis wnevisaTvis. amis Sesabamisad iqna dadgenili fuZesTan gverdebis daxris kuTxe: xeofsis _
′; xefrenis _ 52 0 20 ′; mikeranis _ 51 0 . piramidis wiboebi wverosTan gadaikevTebian 63 0 26 ′ kuTxiT, romelic miaxloebiT Seesabameba fxvieri sxeulebis _ mocemul SemTxvevaSi silis (RorRis) mTis mier Seqmnil kuTxes. varaudoben, rom xeofsis piramida aago xuroTmoZRvarma xemiunma (faraonis naTesavma) daaxloebiT Zv. w. 2 600. mis agebaze daixarja d a a x l o e b i T 2 590 000 m 3 g r a n i t i s q v a . (2 300 000 qvis blokebi da filebi; saSualod TiToeuli qvis blokis moculoba 1m 3 -is
tolia, masiT 2,5 tona; zogierTi blokis masa 50 tonaa, sul saWiro iyo daaxloebiT 6 000 000 tona masis qva _ e. i. 600 000 vagoni. granits ezidebodnen axlandeli asuanis midamoebidan, romelic daaxloebiT 1000 km manZiliTaa daSorebuli piramididan), misi Tavdapirveli simaRle 146,59 m (amJamad 137 m) iyo, gverdis sigrZe _ 232,05 m (amJamad 230 m). piramida agebulia duRabis gareSe. blokebis gadasatanad mxolod Tokebsa da berketebs iyenebdnen, radgan sxva saSualebebi maSin ar arsebobda. mopirkeTebulia gasaocari sizustiT erTmaneTze morgebuli kirqvis samwaxnagovani filebiT. drom Secvala piramidebis garegnuli saxe, mxolod xefrenis piramidis sul zeda nawilzea SemorCenili mzis mcxunvale mxares sxivebiTYgaxunebuli mopirkeTebis niSnebi. gizas piramidebs klasikurs uwodeben. egvi p- tis Zveli samefos arqiteqturaSi klasikuri piramidis forma erTbaSad ar Camoyalibebula. mas win uswrebda grZelvadiani kvlevebi. pirami- debis fuZemdeblad iTvleba III dinastiis fa- raonis joseris safexurebiani, erT-erTi pirveli piramida (daaxl. Zv. w. 2700, simaRle 60 m, mis aSenebas ukavSireben xuroTmoZRvar imhotempis saxels). IV dinastiis pirveli faraonis snofris akldamebi medumSi da daSurSi warmoadgenen gardamaval etaps gizis kompleqsis klasikuri piramidebis Seqmnis gzaze. piramidebis formebis Camoyalibebis procesSi xuroTmoZRvrebma upiratesoba fuZeSi kvadrats mianiWes, vidre marTkuTxeds (joseris safexurebiani piramida agebulia marTkuT- xedovan fuZeze). qvis blokebis damuSavebisa da dawyobis (duRabis gareSe) teqnikis damu- Savebam mSeneblebs saSualeba misca nageboba ufro swori geometriuli formis aegoT. maT konstruqciebSi miRweulia srulyofis zRvari _ mwyobri proporciebi mTeli moculobis masas awonasworeben. nagebobas uryev sidiades aZleven. egvi ptelebma erT-erTi yvelaze mar- tivi da mkacri geometriuli figuris far- glebSi ulamazes maTematikur amoxsnas miagnes, romlis safuZvelzec gizas mTeli ansambli (piramidebi, taZrebi da faraon xefrenis didi sfinqsi) oqros kveTis proporciebzea age- buli (sur. 2). konstruqciis arcerTi elementi, qvis blokebis zomebidan dawyebuli, Siga saTav- soebisa da faraonis dasakrZalavi sakaniT dam- Tavrebuli zogadi TanaSezomvidan ar gamo- dis. sur. 2-ze naCvenebiaBmonakveTis dayofa oqros kveTis Sesabamisad: AB/AO ′′′′′ = AO′′′′′/ O′′′′′B = Φ Φ Φ
Φ = 1,618... aseve AB/BF = BF/ AF = Φ Φ Φ Φ Φ = 1,618.. sadac AD = DB = 1 erTeuls. amave suraTze naCvenebia: kvadratis diagonali 2 , samkuTxedis hi potenuzebi 3 da
5 . gizaSi piramidebis ansamblis sqemaze am propor- ciebis gamoyenebaa naCvenebi. Zveli egvi pturi Zeglebis Seswavlisa da analizis safuZvelze aRmoCnda, rom yvela isini geometriuladaa agebuli. Tanamedrove arqiteq- torebma, mxatvrebma gazomes ra mTeli rigi Zeglebi, daadgines, rom praqtikulad nebismieri Zveli egvi pturi kompoziciebi mkacr maTe- matikur sqemaSi Tavsdeba. amasTan geometriuli figurebi, romlebSic Caixazeba kompoziciis TiToeuli elementi, ar gamodis umartivesi _ samkuTxedis, kvadratis, wrewirisa da sxva nakvTebis sazRvrebidan. Zveli egvi ptis xelov- nebis harmoniuli erTianoba `haromiis algeb- ridan ~ gamomdinareobs, sadac konstruqciuli da mxatvrul-esTetikuri azrovneba urRvev erTianobas warmoadgenen. radgan Cvenamde ar mouRwevia sistemur Teoriul debulebebs, savaraudoa, rom isini 26 egviptis piramidebi da oqros kveTa TavianT SemoqmedebaSi albaT praqtikuli Sede- gebis ganuxreli logikiT midiodnen. esaa gza dakvirvebebisa da Sedegebis. Catarebulma ana- lizma aCvena, rom Zveli egvi ptis saxviTi xelovnebis Zeglebi, gamonaklisis gareSe, pro- porciebis struqturis Sesabamisadaa agebuli. igi warmoadgens rva sididisagan Sedgenil skalas, romelic miRebulia mecxredan _ sawyisidan, romelic mTeli kompoziciis maqsi- mums Seesabameba. piramidebisaTvis aseT sidided misi fuZis kvadratis gverdis sigrZea miRebuli, qandakebisaTvis _ simaRle, reliefis kompo- ziciisaTvis an moxatvisaTvis _ mTeli velis maqsimaluri zoma, romelSic scena horizon- talurad Tu vertikaluradaa moxatuli. piramidebis konstruqcias safuZvlad udevs oqros kveTis proporciebi (sur.4). warmovid- ginoT, rom misi fuZis Wrilis kvadratis gver- debi dayofilia oqros kveTis Sesabamisad. dayofis wertilebidan gatarebuli diago- nalebi gadakveTisas warmoqmnian or kvadrats, romelTa gverdebic agreTve oqros kveTis Sesabamisad daiyofa. am kvadratebis mopirda- pire wveroebisa da gadakveTis wertilebis SeerTebiT miiReba rva monakveTi. es rva sidide
harmoniis gadmocema) proporciul skalas Sead- gens, sawyisi me-9 monakveTi (M) sidide moce- muli kompoziciis maqsimalur zomas Seesabameba. Catarebulma agebam Cven mogvca mxolod proporciuli skalis gamosaxvis geometriuli forma, sadac sidideebi ganlagebulia zrdis mixedviT _ R, I, E N, O, S, C, A.** maTi ricxviTi mniSvnelobebis dadgena imave agebidan SeiZleba, Tu gamoTvlebis gamartivebis mizniT sawyisi kvadratis gverdis sigrZes 2-is tolad mivi- RebT. es ricxviTi mniSvnelobebi saxelde- buli ar aris, amitom isini iZlevian saSualebas nebismieri sigrZis sazomi erTeuli gamoviyenoT, es iqneba Zveli: idayvi, terfi, muSti, romliTac Zveli egvi ptelebi sargeblobdnen, Tu Tanamed- rove metruli sistema. aRsaniSnavia, rom ar arsebobs araviTari dokumenturi monacemebi, iyenebdnen Tu ara Zveli egvi ptelebi proporciuli sidideebis sistemas, magram maTi monawileoba SeimCneva Zvelegvi ptur yvela ZeglSi dawyebuli piramididan da damTav- rebuli mcire formis nebismieri nawarmoebiT. maTi aRmoCena SeiZleba gazomvebiT, es ki moaxerxes arqiteqtorebma: am mxriv pirvel sur. 4. geometriuli agebebi, romlebic gansazRvraven rva proporciul sidideebs
sur. 5. moxele anxirptis gamosaxuleba. Zveli samefo. *abreviatura (ital. abbreviatura < laT. brevis mokle) _ sityvebis Semokleba. ** es asoiTi aRniSvnebi aRebulia frangi arqiteqtoris furne de koras naSromebidan, romelic Tavis mxriv man aiRo petrus talemarianusisagan, romelic iyenebda formulas, sadac miniSnebuli iyo saxeli ARSENICOM. 27 egviptis piramidebi da oqros kveTa rogSi aRsaniSnavia frangi arqitetoris a. furne de koras naSromi `egvipturi proporciebi da RvTaebrivi harmoniis Tanafardobebi ~. avtorma SeZlo yvela Zvelegvi pturi xelovnebis Zegleb- Si gamonaklisis gareSe rva proporciuli Tanafardobis monawileobis aRmoCena. mjdomare mwerali Zveli egvipte. V dinastia daaxl. Zv. w. 2500. SeRebili kirqva. simaRle 53 sm. luvri, parizi sur. 6. kanonikuri tipis mwerlis qandakebis ageba oqros kveTis proporciebSi. sur. 7. sfinqsi gizaSi.
28 oqros kveTa. fidias ricxvebi radgan agebis safuZvels warmo- adgens oqros kveTis proporcia, romelic iZleva oqros ricxvs _ fidias ricxvs Φ Φ
Φ Φ = 1,618..., igi warmoadgens usasrulo aTwilads _ iracionalur ricxvs, amitom RDH skalis proporciuli sidi- deebic iracionaluri ricxvebiT gamoisaxeba, romlebic agreTve usas- rulo aTwiladebia. Zveli egvi p- telebi icnobdnen iracionalur ricxvebs da maTTvis miTologiur warmodgenebSi ganuzRvrelobisa da usasrulobis cnebebi arsebobda. proporciuli Tanafardobebi Zvelegvi ptur plastikaSi xelov- nurad ar iyo Setanili. es TviT bunebam gansazRvra: adamianis figura mis anatomiur agebulebaSi Seicavs proporciebs, romelic gansazRvru- lia oqros kveTiT da gamoisaxeba o q r o s r i c x v i s ( Φ Φ
Φ Φ = 1,618...) xarisxebiT (sur. 5) da (sur. 6). Cveulebrivad sfinqsis Tavi warmoadgens ama Tu im faraonis konkretul gamosaxulebas, rome- lic Serwymulia lomis tanTan (sinkretuli saxe). gizis sfinqsSi gamoxatulia faraon xefrenis saxe (sur. 7). mxolod zustad damu- Savebuli proporciuli Tanafar- dobebis sistemis saSualebiT iyo SesaZlebeli monoliTuri kldi- dan veberTela monumentis, 57 m sigrZis gizis sfinqsis gamokveTa. proporciebis sitema safuZvlad daedo arqiteqturis, qandakebis, mxatvrobis organul sinTezs da kompoziciis Semadgeneli nawilebis urTerTSerwymas. es ki imaze mianiSnebs, rom Zvelegvi pturi kanonis sitema Camoyalibebulia. zogadad kanoni aris proporciuli Tanafardobis skala, romlis nawi- lebic gamonaklisis gareSe monawileoben yvela Zvelegviptur xelovnebis ZeglebSi, ganTavsebulni arian yvela tipis gamosaxulebebis mkacrad dadgenil sistemaSi da kompoziciis yvela nawilebs Soris Tanazomierebas uzrunvelyofen. sur. 8-ze naCvenebia RDH proporciuli skalis sistemaSi faraon exnatonis, dedofal nefertitisa da princesas gamosaxulebebiani stelis* kompoziciis agebis sqema. aq stelis sigane miRebulia ZiriTad sidided (FB monakveTi), romelic oqros kveTis Sesabamisad aris dayofili: FO = A da OB = A + S monakveTebad da im wrewirebis radiusebs warmoadgenen, romlebSic exnatonisa da nefertitis figurebi gamoisaxeba. sur. 8. faraon exnatonis, dedofal nefertitisa da princesas gamosaxulebiani stela. kompoziciis agebis sqema RDH proporciuli skalis sistemaSi. * stela (berZ. stele ) _ qvis fila, romelzedac warwera an reliefuri gamosaxulebaa. ~ ~ C F B O D O ′
OB = A + S FO = A S Φ = = OF OB OB FB 29 ricxvebis idumali samyaro ricxvebis idumali samyaro ricxvebis idumali samyaro ricxvebis idumali samyaro ricxvebis idumali samyaro `RmerTma Seqmna naturaluri ricxvebi, yvela danarCeni adamianis mogonilia~. leopold kronekeri naturaluri ricxvebi dawere Tanmimdevro- biT, miiReb naturalur ricxvTa mimdevrobas: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... am mimdevrobis wevrebidan zogierTi: 2, 3, 5, 7, 11, 13, ... iyofa mxolod Tavis Tavze da erTze, maSas- adame aqvs mxolod ori gamyofi. sxvebi:
iyofa ara marto Tavis Tavze da erTze, aramed sxva ricxvebzec. e. i. aqvs sxva gamyofebic. ricxvs, romelsac aqvs mxolod ori g a m y o f i , m a r t i v i e w o d e b a . d a n a r C e n ricxvebs ki _ Sedgenili. ricxvi 1 ar iTvleba arc martivad da arc Sedgenilad, radgan mas aqvs mxolod erTi gamyofi _ Tavisi Tavi. ricxvebis aseTi dayofa SemoiRo berZenma maTematikosma piTagoram (Zv. w. VI s.). mart
mart mart
mart martivi r ivi r ivi r
ivi r ivi ricxvebi icxvebi icxvebi
icxvebi icxvebi
ricxvTa mwkrivSi (mimdevrobaSi) kanonzom- ierebis aRmoCena sasiamovno movlenaa maTema- tikosebisaTvis. es kanonzomierebani SeiZleba gamoyenebuli iqnas garkveuli varaudebis _ hipoTezebis agebisaTvis, damtkicebebisa da Teorie- bis SemowmebisaTvis. maTematikosebi martivi ricxvebis gansazRvri- saTvis iyeneben sxvadasxva xerxs, magram dRemde erTaderTi WeSmaritad efeqturi xerxi mogvca berZenma maTematikosma da astronomma era- tosTenma (Zv. w. 276-194). es xerxi 2000 welze meti xnisaa. gairkva, rom martivi ricxvebi ar emorCilebian garkveul kanonzomierebas _ swored esaa maTi Tviseba. eratosTenes meTodiT SeiZleba SevadginoT cxri- li da ganvsazRvroT, mar- tivia Tu ara romelime ricxvi. mas `eratosTenes saceri ~ ewodeba. vnaxoT rogor SeiZleba am sacer- iT
`gavcxriloT~ Sedgeni- li ricxvebi naturaluri ricxvebis mwkrividan: pirve- lad gadavxazoT ricxvi 1. is arc martivia, arc Sedge- nili. Semdeg gadavxazoT 2- is jeradi (luwi ricxvebi), garda 2-isa, amisaTvis gada- vxazoT 2-is Semdeg yoveli meore ricxvi. Semdeg gada- vxazoT 3-is jeradi Sedgenili ricxvebi dawye- buli 3-dan _ yoveli mesame (zogierT SemTx- vevaSi mogvixdeba ricxvebis orjer gadaxazva, esenia 2-isa da 3-is jeradi). amis Semdeg gada- vxazavT 5-is jerad Sedgenil r icxvebs: dawye- buli 5-dan yovel mexuTes. (aqac zogjer me- ored gadaixazeba zogierTi ricxvi) da natu- ralur ricxvTa mwkrivSi 1-dan 25-mde darCeba Semdegi martivi ricxvebi:
amis Semdeg igives gavimeorebT 7-ze, Semdeg 11-ze da a. S. `eratosTenes saceris~ erT-erTi varianti warmodgenilia cxrilSi: aq
ricxvebi rgolSia Casmuli. Sen SegiZlia gaagrZelo es cxrili da 100-mde ipo- vo danarCeni martivi ricx- vebi. cxrilSi 5-is jeradi ricxvebi dalagebulia di- agonalze marjvnidan marcx- niv, xolo 7-is jeradi ricx- vebi _ diagonalze marcx- nidan marjvnisaken. yvelaze didi martivi ricxvi romelia? TandaTanobiT eratosTenes meTodi srulyofili iqna da dReisaTvis arsebobs martivi ricxvebis cxrili 1 dan 100
30 ricxvebis idumali samyaro 000 000-mde. maTze dayrdnobiT gamoiTqva sxva- dasxva hipoTeza (varaudi). zogierTis damtki- ceba aRmoCnda rTuli. zogic dResac daumt- kicebelia. erT-erTi pirveli kiTxva martivi ricxvebis Sesaxeb aseTia: sasrulia Tu ara maTi ricxvi usasrulo naturalur ricxvTa mimdevrobaSi? jer kidev evklidem daamtkica maTi usasrulo- ba. e. i. yvelaze didi martivi ricxvi ar arsebobs. davuSvaT piriqiT, vTqvaT arsebobs yvelaze didi martivi ricxvi da aRvniSnoT P- Ti. gadavamravloT yvela martivi ricxvi erT- maneTze da namravls davumatoT erTi. miviRebT:
×3×5×7×...×P+1. cxadia, es ricxvi ar gaiyofa arcerT mar- tiv ricxvze. maSin an TviTon aris martivi ricxvi, an aqvs P-ze didi martivi gamyofi erTic da meorec ewinaaRmdegeba imas, rom P udidesi martivi ricxvia. es winaaRmdegoba gviCvenebs, rom Cveni daSveba mcdaria. xarisxebi
aris ricxvi, igi ase ikiTxeba: `2 xarisxad
~ da niSnavs 2 Tavis Tavze gamravlebuli samjer. e. i. 2
×2×2 = 8. analogiurad `2 xarisxad 4~ iqneba: 2 4 = 2 ×2×2×2 = 16. ipove 2
maSin 2 2 2 = 2 4 = 2 ×2×2×2 = 16. aseve 2
×2×2×2×2×2×2×2 = 356. uZvelesi droidan moyolebuli maTematiko- sebi cdilobdnen iseTi formulis Sedgenas, romelic mogvcemda yvela martiv ricxvs, ma- gram aseTi formlis Sedgena ar moxerxda. frangma maTematikosma pier fermam (1601- 1665) gamoTqva azri, rom formula f (n) = 2 2n + 1, (1) sadac n naturaluri ricxvebia (1, 2, 3, 4, 5, ...) mogvcemda martiv ricxvebs. marTlac, roca n = 1, 2, 3, 4, fermas formula iZleva martiv ricxvebs. magram roca n = 5 _ ara (1732 w. es daamtkica leonard eilerma 1707-1783).
×6700417, ar aris martivi (e. i. ferma Seacdina arasruli induqciis meTodma. am sa- kiTxze dawvrilebiT qvemoT vimsjelebT). eleqtronuli saangariSo manqanebiT damt- kicda, rom f(10) da f(16)-ic Sedgenili ricx- vebia. ganixile formula: F (5) = n 2 - n + 41. Yüklə 4,69 Mb. Dostları ilə paylaş:
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