Moswavlis biblioTeka j e m a L k I k n a z e
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moswavlis biblioTeka j e m a l k i k n a Z e j e m a l k i k n a Z e j e m a l k i k n a Z e j e m a l k i k n a Z e j e m a l k i k n a Z e azris jaWvi moswavleTa inteleqtis ganviTarebisaTvis gamomcemloba `inteleqti~ Tbilisi 2001
pirveli nawili daamtkica saqarTvelos ganaTlebis saministrom damxmare saswavlo masalad maTematikaSi zogadsaganmanaTleblo skolis V-XI klasebis moswavleTaTvis 2 winamdebare mecnierul-popularuli wigni-almanaxi Taviseburi damxmare wignia, romelic maTematikis sxvadasxva sakiTxebs moicavs da ZiriTadad zogadsaganmanaT- leblo skolis moswavleTaTvis aris Sedgenili. maTematikis garda masSi sxva saintereso sakiTxebzecaa saubari da saxaliso maTematikis Janrs ganekuTvneba. wignSi mravladaa warmodgenili savarjiSoebi, kiTxvebi, TamaS-gasarTobi Tavsa- texebi da xelovnebis zogierTi nimuSi, romlebic xels Seuwyoben ymawvilebis dainteresebas, SemoqmedebiT saqmianobas, damoukidebeli muSaobisa da garkveuli ti pis inteleqtis ganviTarebas. am mizniT wignSi Setanilia sxvadasxva testebi, romlebic ganaviTareben farul kanonzomierebaTa aRmoCenis, analitikuri da induqciuri azrovnebis unar-Cvevebs. nimuSisaTvis ganxilulia zogadi codnis Sesabamisi inteleqtis koeficientis (ik) dasadgeni ramdenime testi. igi daexmareba maswav- leblebs saklaso da klasgareSe muSaobaSi. wignSi ganxiluli sakiTxebi misawvdomia ara marto zogadsaganmanaTleblo skolis moswavleTaTvis, aramed am sakiTxebiT dainteresebul mkiTxvelTa farTo wrisaTvis. © jemal kiknaZe, 2001. © gamomcemloba `inteleqti~, 2001. ISBN 99928-67-93-0 didi galaktionis msgavsad yovel bavSvSi aRmzrdeli `momavalis did saxes~ xedavs da minda es wigni maT mivuZRvna. `qals patara dahyavs xeliT, bavSvi zrunviT zrdili, momavalis didi saxe _ Svili _ Svili _ Svili ~.
galaktioni redaqtori: arkadi qurCiSvili, Tsu-s ufrosi maswavlebeli. recenzentebi: romanoz danelia, gamomcemloba `inteleqtis~ direqtoris moadgile, Jurnal `fizika da maTematika skolaSi~ redaqtoris moadgile, gela kvirikaSvili, Tsu-s T. gegelias saxelobis respublikuri fizika-maTematikis saSualo skola-internatis fizikis maswavlebeli, guram oTinaSvili, Tbilisis 55-e saSualo skolis maTematikis maswavlebeli. 3 S e s a v a l i mravalwlianma pedagogiurma moRvaweobam da Tanamedrove meToduri literaturis gacnobam damarwmuna, rom swavlebis tradiciuli meTodebi moiTxovs gadasinjvas da sqolastikisagan ganTavisuflebas. sqolastika aris skolis muSaobisa da cxovrebis wesis sistema, romelsac skolis gareT ar gaaCnia araviTari faseuloba, radgan skolas ar SeuZlia moamzados bavSvebi cxovrebisaTvis. zogjer tradiciuli skola iyenebs mxolod axsnis da naTqvamis sailustraciod cdebsac, magram aseTi axsna iZleva mxolod zedapirul da formalur codnas, romelic ver mkvidrdeba adamianis gonebaSi. aseTi codna im norC ylorts waagavs, xes rom Camosxepen da miwaSi CaarWoben. cxadia, igi male iRupeba. swored aseT zedapirul, ufesvebo codnas iZleva tradiciuli skola, romelic Tavis naklovanebebs niRbavs uSinaarso, xmamaRali, wyaliviT morakrake mravalsityvaobiT. ra Tqma unda kargi mexsiereba SesaniSnavi Tvisebaa, magram mas xSirad SecdomaSi Sehyavs pedagogi da mSobeli. saubedurod gamocdebzec mxolod imas amowmeben, Tu ra daimaxsovra moswavlem. dazepireba ar niSnavs codnaso _ uTqvams frang filosofos, humanist miSel de montens. igi uaryofda sqolastikosTa mTavar meTods `CavasxaT~ adamianSi codna, rogorc ZabrSi. saubedurod, xSirad tradiciuli swavleba emyareba mexsierebas, romlis meqanikur gamoyenebas dauZlurebamde da gamofitvamde mihyavs axalgazrdoba. sinamdvileSi aseTi codnis usargebloba sul ufro da ufro aSkara xdeba. axla yvelgan (iSviaTad skolaSi) bevrs laparakoben WeSmarit, Rrma kulturisa da inteleqtis mniSvnelobaze da moazrovne Taobis aRzrdaze. cxadia, am urTules pirobebSi, rom icxovro, saWiroa iazrovno. swored amitom gagaxseneT didi frangi filosofosis da maTematikosis rene dekartis gamonaTqvami: `Cogito, ergo sum~. bevr rameSi veTanxmebi da praqtikulad viyeneb meoce saukunis gamoCenili frangi pedagogis selesten frenes (1896-1966), originaluri pedagogiuri sistemisa da `pedagogiuri invariantebis~ avtoris mosazrebebs: unda vifiqroT, rom namdvili kulturisaken da inteleqtis ganviTarebisaken mimavali gza eqsperimentul mosinjvaze gadis. codnis SeZena miiRweva eqsperimentis gziT da ara wesebisa da kanonebis SeswavliT (gazepirebiT), rogorc zogjer fiqroben. pirvel rigSi wesebisa da kanonebis Seswavla imasa hgavs, uremi xarebs win davuyenoT. cocxali eqsperimentuli muSaobis princi pia _ `mefolade mxolod RumelTan SeiZleba gaxde~. samwuxarod tradiciuli skola aviTarebs mxolod abstraqtuli azrovnebis unars, rac Sorsaa realuri cxovrebis moTxovnilebisagan. maT visac hi pertrofirebuli (aranormaluri ganviTrebuli) aqvT inteleqtis es forma, SeuZliaT SesaniSnavad imsjelon nebismier `Wkvianur~ Temaze, magram sruliad umweoni aRmoCndebian, Tu saqme Seexeba cxovrebasa da garemosTan Seguebas. arsebobs inteleqtis sxva ti pebic da isini eqsperimentuli mosinjvis procesSi viTardeba. magaliTad: xeliT Sromis unari, SemoqmedebiTi unari, problemaTa gonivruli, praqtikuli gagebis unari, Semecnebisadmi mecnieruli midgomis unari, socilaur-politikuri mimarTulebis mqone inteleqti, romlis mflobelTagan izrdebian da yalibdebian sazogadoebrivi moRvaweni. kacobrioba yovelTvis maRal Sefasebas aZlevda inteleqtis am sxvadasxvagvar formebs; sworedac maTi SemweobiT gvyavs xelovnebis genialuri moRvaweni, gamomgoneblebi Tu mecnierebi. aseTi xalxi ver egueboda swavlebis tradiciul xerxebs da xSirad isini skolaSi CamorCebodnen. Tanamedrove sazogadoebas esaWiroeba gansxvavebuli mimarTulebis kadrebi _ gamomgoneblebi da mkvlevarebi, rac saWiroebs inteleqtis sxvadasxva formis ganviTrebas. cnobilia, rom bavSvis buneba arafriT gansxvavdeba zrdasruli adamianis bunebisagan. gansxvaveba mxolod gamocdilebasa da ganviTrebis doneSia. arc bavSvs da arc ufross ar uyvars, roca rames ubrZaneben. yoveli adamiani asea mowyobili _ nebismieri brZaneba TiTqmis avtomaturad winaaRmdegobis grZnobas iwvevs: adamiani wiTldeba, azri efanteba, da pirveli survili, rac uCndeba, es aris, win aRudges, ar daemorCilos brZanebas. kanonzomieria Semdegi mtkiceba: nebismieri avtoritaruli brZaneba yovelTvis Secdomaa. nebismier brZanebas, rogorc wesi, bavSvi Tu ufrosi pasuxobs uaryofiTad _ `ara~! _ mniSvneloba ara aqvs xmamaRla warmoTqvamen am sityvas, Tu xmadabla. amitom saWiroa maTTan megobruli urTierToba. am qveyanaze aravis ar uyvars fuWi Sroma, robotiviT moqmedeba, aravis ar exaliseba raime samuSaos Sesruleba sxvisi Canafiqris mixedviT. yoveli SromiTi saqmianoba damoukidebeli da SemoqmedebiTi unda iyos. erTi da igive samuSao erTisaTvis mZime monobaa, meoresaTvis Tavisufali SemoqmdebiTi Sroma, romelic mas yovelTvis did siamovnebas aniWebs. codnis aTvisebis yvelze efeqturi saSualebaa ara ubralo dakvirveba, axsna da demonstracia, rogorc mimdinareobs tradiciul skolaSi, aramed Semecnebis bunebrivi, mecnieruli da universaluri meTodi, romelic ciklurad viTardeba: faqtebis erTobliobis analizisa da problemis dasmiT 4 hipoTezisken; hi poTezidan Teoriuli Sedegebisaken; Sedegebidan maTi interpretaciiT eqsperimentuli Semowmebisaken da praqtikuli gamoyenebisaken: maSasadame, mkvlevarma (moswavlem) axali movlena rom Seiswavlos, is am movlenis Sesaxeb agrovebs da asistemebs cnobil faqtebs; dagrovebuli faqtebis safuZvelze Sesaswavli movlenis `meqnizmis~ Sesaxeb agebs models (hi poTezur models), romelic aZlevs ara marto dagrovebuli faqtebis axsnis, aramed axali, manamde ucnobi movlenis winaswarmetyvelebis saSualebas; bolos, hi poTezasa da miRebul Sedegebs eqsperimentulad amowmebs. am meTodis fuZemdeblad galileo galilei iTvleba. winamdebare wigni, romelic gamocdilebisa da sxvadasxva literaturis gacnobis safuZvelze ymawvilTaTvis Sevadgine, nawilobriv zeviT ganxiluli problemebis gadawyvetas isaxavs miznad. masSi mocemuli bevri saintereso sakiTxi ZiriTadad maTematikas exeba, magram aq sxva sakiTxebzecaa saubari da mecnierul-popularulia. masSi mravladaa warmodgenili (pasuxebis gareSe) savarjiSoebi, kiTxvebi, TamaS-gasarTobi Tavsatexebi da xelovnebis zogierTi nimuSi, romlebic xels Seuwyoben ymawvilebis dainteresebas, SemoqmedebiT saqmianobas, damoukidebeli azrovnebisa da garkveuli ti pis inteleqtis ganviTrebas. swored am mizniTaa wignis bolo paragrafSi mocemuli sxvadasxva testebi (zogierTi fsiqologiuri) da maTi pasuxebi. nimuSisaTvis ganxilulia zogadi codnis inteleqtis koeficientis (ik) dasadgeni testebi. aRsaniSnavia, rom es kiTxvebi, Tavsatexebi da testebi xels uwyoben faruli kanonzomierebis aRmoCenas, analitikuri da induqciuri azrovnebis unar-Cvevebis ganviTrebas. wignis saTauri misi Sinaarsidan gamomdinareobs. moswavlem es wigni ubralod ki ar unda waikiTxos, aramed bevri ram sakuTari xeliT, damoukideblad unda Seasrulos. vfiqrobT, kiTxvebisa da Tavsatexebis pasuxebi calke gamovceT. dainteresebuli mkiTxveli am wignSi ganxilul sakiTxebs (zogjer isini mokled da gacnobis mizniTaa warmodgenili) ufro Rrmad gaecnoba im literaturaSi, romelic wignis boloSia miTiTebuli. uRrmes madlobas movaxseneb: redaqtors, baton arkadi qurCiSvils. recenzentebs, batonebs: romanoz danelias, gela kvirikaSvils da guram oTinaSvils. agreTve i. gogebaSvilis saxelobis pedmecnierebaTa erovnuli institutis mecnier-TanamSromlebs, qalbatonebs: iza xuciSvils, eTer guldamaSvils, da Tsu-s T. gegelias saxelobis respublikuri fizika-maTematikis saSualo skola-internatis maTematikis maswavlebels, baton iuri fiCxaZes, romlebmac guldasmiT waikiTxes xelnaweri da damexmarnen dasabeWdad momzadebaSi. gamomcemloba `inteleqtis~ xelmZRvanelebs, maT yvela TanamSromels, gansakuTrebiT ilia xelaias, romelmac udidesi Sroma gaswia am wignis Seqmnaze da kompiuterul grafikaze. yvela moswavles, maT Soris Cems SviliSvils _ laSa tabataZes, romelTac stimuli momces am wignze muSaobaSi, xelnaweri yuradRebiT waikiTxes, zogierTi savarjiSo damoukideblad amoxsnes _ Seamowmes da damexmarnen koreqturis kiTxvaSi. agreTve madlobas vuxdi yvela mkiTxvels, romelTa saqmian SeniSvnebs madlierebiT miviReb. jemal kiknaZe Tbilisis ivane javaxiSvilis saxelobis saxelmwifo univer- sitetis Tengiz gegelias saxelobis fizika-maTematikis respubli- kuri saSualo skola-internatis fizikis maswavlebeli. gTxovT survilebi da SeniSvnebi gamogzavnoT misamarTze: 380079, Tbilisi, guram rCeuliSvilis quCa, korp. #3, bina 21. 29-21-23. modeli Sedegebi
faqtebi eqsperimenti 1 2
3 5 dedamiwa, dedamiwaze da mis gareT, sivrceSi mze, planetebi, varskvlavebi da sxva, realurad arsebuli yvelaferi erTad samyaros _ bunebas warmoadgens. adamiani samyaros na- wilia, albaT erT-erTi yvelaze didi saocreba, rac ki bunebas Seuqmnia. bunebam mas misca sam- yaros Secnobis saSuale- bebi _ grZnobis orga- noebi da tvini. adamiani samyaroSi yvela informacias grZnobis organoebis saSu- alebiT iRebs, romelic Tavis tvins gadaecema (sur. 1). mxedveloba, smena, ynosva, gemovneba, Sexeba _ ai Cveni `fanjrebi~ garemomcvel samyaroSi. tvini amuSavebs giganturi raodenobis sig- nalebs, romlebic momdinareobs samyarodan da adamianSi iwvevs siTbosa da sicivis, sigluvisa da simqisis, ferisa da sikaSkaSis, xmaurisa da siCu- Sen _ adamiani, rogorc samyaros sarke buneba, Sen da gamokvlevebi buneba, Sen da gamokvlevebi buneba, Sen da gamokvlevebi buneba, Sen da gamokvlevebi buneba, Sen da gamokvlevebi vidre raimes Seswavlas Seudgebode, winaswar saWiroa gaerkve, ra ginda, saxeldobr ris gagebas da Cawvdomas apireb. zogierTi sityvebi: `samyaro~, `buneba~, `fizika~, `maTematika~, `mecniere- ba ~ da sxva mravaljer gsmenia, magram yvelas ar SeuZlia axsnas, Tu ras aRniSnaven isini. saWiroa gaerkve ara marto maT dedaazrSi, aramed nebismieri sityvis arsSic. winaaRmdeg SemTxvevaSi gad- mocemuli azris gagebas da, miT umetes, masSi Rrmad Cawvdomas ver SeZleb. Tavidanve zemoT naxsenebi yvela sityvis ganmartebas ar Sevecdebi, radgan vfiqrob, es igivea vimsjeloT im qveynis Sesaxeb, sadac jer ar vyofilvarT. es TandaTanobiT sxvadasxva saswavlo procesSi moxdeba. daxmarebas es patara wignic gagiwevT. `
~ .
`vazrovneb, maSasadame varsebob~. rene dekarti romelman Seqmna samyaro ZaliTa miT ZlieriTa, zegardmo arsni suliTa yvna zeciT monaberiTa, Cven, kacTa, mogvca qveyana, gvaqvs uTvalavi feriTa, misgan ars yovli xelmwife saxiTa mis mieriTa. S o T a sur. 1
mis, zomebisa da formis, sxeulebamde manZilebis SegrZnebebs. realur samyaroSi, rom viarseboT, an rom gvqondes saSualeba gaviumjobesoT cxovrebis pirobebi, saWiroa rac Sei- Zleba meti vicodeT am samyaroze. aucilebelia g a n v a s x v a v o T x m e l e T i zRvisagan, saWiroa sakvebi mcenareebis da cxoveleb- is gamravleba, maTi mov- la. sacxovreblebis ageba da mtacebeli cxovele- bisagan Tavis dacva. ratom ar SeiZleba am miznebis ganxorcielebisaTvis daveyrdnoT Cveni grZnobe- bis organoebs? marTlac xom ase iqcevian primi- tiuli civilizaciebi. arsebobisaTvis materialuri pirobebis gaum- jobesebis mcdelobisas iZulebulni varT gavi- farTovoT Cveni codna gare samyaroze. risTvisac saWiroa maqsimalurad davZaboT Cveni grZnobis organoebi. magram samwuxarod maT zogjer Sec- domaSi SevyavarT.
6 buneba, Sen da gamokvlevebi daakvirdi sur. 2-ze gamosaxul T-s magvar figurebs, romlebzec gamosaxulia horizonta- luri da vertikaluri monakveTebi. Seadare maTi sigrZeebi erTmaneTs. Semdeg gamoiyene saxazavi da gazome maTi sigrZeebi. a) SemTxvevaSi ver- tikaluri monakveTi ufro grZeli mogeCveneba, vidre horizontaluri. Seadare dakvirvebisa da gazomvebis Sedegebi erTmaneTs. b) SemTxvevaSi monakveTebis sigrZeebi toli mogeCveneba. gazom- vebiT miiReb sawinaaRmdego pasuxs. aq saqme gvaqvs mxedvelobiT (optikur) ilu- ziasTan. amis mizezi isaa, rom Tvalis mobruneba horizontaluri mimarTulebiT ufro advilia, vidre vertikaluri mimarTulebiT. aseve daakvirdi sur. 3-ze da sur. 4-ze mocemuli figurebis AB da AC monakveTebs. dakvirvebiTa da gazomviT Seadare maTi sigrZee- bi. gaakeTe daskvna. ganixile sur. 5-ze gamosaxuli figura jer pirdapiri dakvirvebiT, Semdeg gverdidan ise, rom Tvali rac SeiZleba axlos iyos furclis sib- rtyesTan. Seadare dakvirvebebi erTmaneTs, gaa- keTe daskvna. daamzade bzriala sur. 6-ze gamosaxuli muyaos an fanerisagan gamoWrili diskos gamoye- nebiT. diskoze naCvenebia TeTri da Savi zolebi. bzriala gaanaTe, daabrune da daakvirdi. dain- axav ferebs! sur. 7-ze naCvenebia rkinigza. ori parale- luri relsi gveCveneba, rom Sors erTdebian. es wrfivi perspeqtivis magaliTia. sur. 8-ze sxivebi miemarTebian Sekrebis wertilisaken (perspeqtivis centri), moculo- biTi scenis iluziis SeqmnisaTvis. suraTze ga- mosaxulia erTnairi zomis maRali yuTebi, mag- ram gveCveneba sxvadasxva sididis _ `Sors~ ufro didi. am iluziis mizezi aris gamocdileba: rac ufro Sorsaa dasakvirvebeli sagani, miT ufro mcire zomis gveCveneba igi, amitom marjvena yuTi ufro didi gamoiyureba, vidre sinamdvileSia. realisturi maneriT daxatuli suraTi or ganzomilebiania (sibrtyezea), magram Tu isini daxatulia wrfivi perspeqtivis Sesabamisad, ma- Sin Cven is gveCveneba sivrceSi (samganzomilebi- ani scena). aseTi `moculobiTi gamosaxulebis~ iluz
iluz iluz
iluz iluziebi
iebi iebi
iebi iebi
sur. 2 sur. 3
sur. 4 A B A C a b moric kornelis esxeri. `belvederi~*.
liTografia **. 1958.
* belvederi (it. Belvedere _ mSvenieri xedi) _ koSki, saidanac gadaiSleba garemos xedi; fanCaturi (talavari) maRlobze. ** liTografia (berZ. lithos qva da graph õ vwer) _ beWdva specialuri brtyeli qvis zedapiridan, romelzedac winaswar xataven.
7 buneba, Sen da gamokvlevebi sur. 7 sur. 8
sur. 6 sur. 5
O perspeqtivis centri salvador dali. `jvarcma. hiperkuburi sxeuli~ . saukeTeso magaliTebi uamravia. mas xSirad iy- eneben konstruqtorebi, arqiteqtorebi da mxat- vrebi.
am saxis kargi magaliTia XX saukunis erT- erTi yvelaze popularuli adamianis, udidesi espaneli mxatvris salvador dalis suraTi: `jvarcma~ (hiperkuburi sxeuli). 1954 w. (sur. 9). saocari movlena: mTvare amosvlisas an Cas- vlisas (horizontTan) gveCveneba ufro didi, vidre maRla caSia aweuli. sinamdvileSi gazom- vebis Tanaxmad mTvaris xiluli diametri hori- zontTan ufro pataraa, vinaidan am dros mTva- re damkvirveblidan ufro Sorsaa. fsiqologebi Tvlian, rom es faqti ganpirobebulia adamianis tvinisa da mxedvelobis TvisebebiT. damkvirvebe- li midrekilia warmoidginos mTvare ufro Sors, rodesac igi axlos aris dedamiwis zedapirTan (horizontTan), vidre maSin, rodesac maRlaa caSi. e. i. is Seucnoblad rogorRac moaTavsebs mTva- res horizontze. radgan mTvaris kuTxuri zomebi ucvleli rCeba, igi mogveCveneba SesamCnevad didi horizontTan, sadac mTvare mogveCveneba ufro daSorebuli (sur. 10). iluziebsa da Secdomebs adgili aqvs sxva grZnobis organoebiT SegrZnebisas: temperatu- ris, gemos, bgerisa da moZraobis dros. magaliTad: aiRe sami WurWeli _ cxeli, Tbili da civi wyliT. marcxena xeli moaTavse civ wyalSi, marjvena _ cxelSi. cota xnis Semdeg orive xeli gadaitane Tbil wylian WurWelSi. ra SeigrZeni? sur. 9.
8 buneba, Sen da gamokvlevebi davuSvaT saaTma Seasrula 6 dartyma 5 wamSi. ramden wamSi Seasrulebs is 12 dartymas? intui- cia gvkarnaxobs: 10 wamSi. magram araa swori.
dartyma _ TerTmetiT. maSasadame swori pasuxia: 11 wami da ara 10 wami. e.i. intuiciam gviRalata! mocemuli gaqvT erTnair perimetrebiani (perimetri aris gverdebis sigrZeTa jami) ori marTkuTxedi. erTnairi farTobebi eqnebaT Tu ara orives? albaT ifiqreb, rom erTnairi. mag- ram rogorc gamoTvlebi gviCvenebs sazogadod farTobebis toloba ar aris aucilebeli (See- cade damoukideblad daamtkico). bunebrivad ibadeba kiTxva, erTnair peri- metrebian romel marTkuTxeds aqvs udidesi far- Tobi? irkveva, rom kvadrats! (samkuTxedebis SemTxvevaSi ki tolgverda _ wesier samkuTxeds). oci metri siRrmis WaSi bayayi Cavarda da cdilobs amosvlas. yovel saaTSi 5 metrze amo- dis, magram me-5 metrze ise iRleba, rom Tavs ver ikavebs, fexi ucurdeba da 4 metriT ukanve cocdeba. rogor fiqrob, ramden saaTSi amova bayayi Widan? pirobis Tanaxmad bayayi saaTSi erTi metriT iwevs zeviT, radgan Wis siRrme 20 metria, al- baT ifiqreb, rom bayayi Widan amova 20 saaTSi. magram intuicia aqac Segacdens. Tu kargad dafiqrdebi mixvdebi, rom bayayi Widan amosvlas moandomebs 16 saaTs. intuiciiT Cven vcdebiT mravali sxva situ- aciis drosac. magaliTad adamiani, romelic garkveul manZilze imyofeba vaSlis xidan da xedavs, rom erTi vaSli es-es aris unda Camo- vardes. mas unda, rom Tofis tyvia moaxvedros vardnil vaSls. romel wertilSi unda dau- miznos man, rom tyvia moxvdes vaSls? albaT ifiqreb vaSlis qveviT, romeliRac wer- i n t u i n t u
i n t u i n t u
i n t ui c i a i c i a
i c i a i c i a
i c i a `mTvariani~ optikuri iluzia: or Sav diskos aqvs erTnairi zomebi. (rokisa da kaufmanis suraTis mixedviT). sur. 10. albaT marcxena xeliT wyali geCveneba cxeli, xolo marjvenaTi igive wyali civi! oTaxSi sxvadasxva sagans erTnairi tempera- tura aqvs. magram, Tu xeliT Seexebi rkinisa da xis sagnebs, rkina ufro civi mogveCveneba, vidre xe. e.i. grZnobis organoebi zogjer gvatyueben, xolo rac erTxel mainc mogvatyuebs, sando araa. de- kartiseuli eWvis pirveli wyaroa grZnobadi codna.
maSasadame SegrZnebebi yovelTvis san- do ar aris! tilSi, radgan raRac droSi tyvia miaRwevs im adgilamde, sadac wydeba vaSli, imave droSi vaS- li Tavisufali vardniT, raRac manZilze daiw- evs qveviT. magram mtkicdeba, rom gasrola unda moxdes mowyvetis momentSi da msrolelma unda daumiznos pirdapir vaSls da ara mis qveviT. riT SeiZleba Sevewina- aRmdegoT sxvadasxva sa- xis iluziebsa da in- tuiciiT daSvebul Sec- domebs? Cveni yvelaze efeq- turi pasuxia: maTematikis gamo- yenebiT! rodesac aleqsandre makedonelma (Zv. w. 356- 323) ganizraxa msoflios dapyroba, man saberZneTis oikumene* aTenidan gada- itana egviptis erT qa- laqSi, romelsac man Tavisi Cveuli `mokrZale- aleqsandre makedoneli. romauli marmarilos qandake- bis asli. msoflios dampyro- bi mefis saxis nakvTebi albaT gadmocemulia zustad, radgan igi Seqmnilia samefo karis didi moqandakis lisipes mier. didi aleqsandre. (pompeis mozaikis detali). * oikumene (berZ. Oikumene _ dasaxleba) _ dedamiwis adgilebi, romlebic adamianebiTaa dasaxlebuli. ~ 9 buneba, Sen da gamokvlevebi * Tavisi batonobis gansamtkiceblad igi dapyrobil teritoriebze aarsebda gamagrebul qalaqebs _ aleqsandriebs. daarsa daaxloebiT 70 aseTi qalaqi. ** hipoTeza (berZ. hypothesis varaudi) _ raime movlenis asaxsnelad wamoyenebuli mecnieruli varaudi, romlis ueWveloba jer ar aris cdiT damtkicebuli. bis ~ gamo saxeli gadaarq- va da aleqsandria* uwo- da. swored iq, aleqsandri- aSi, evklidem (Zv. w. 300) dawera pirveli RirsSesa- niSnavi maTematikuri cod- nis dokumenti _ Tavisi klasikuri `sawyisebi~. am naSromSi gamoyenebuli iyo damtkiceba. garda amisa evklides ekuTvnis naSrome- bi musikaSi, optikaSi, meqanikaSi, romlebSic Zir- iTadi roli gamoyofili aqvs maTematikas. maTem- atika gamovida rogorc idealuri versia imisa, rac Seadgenda CvenTvis cnobili realuri sam- yaros agebulebas. samyaroSi mimdinare movle- 2347 _
g a m o k v l e v e b i g a m o k v l e v e b i g a m o k v l e v e b i g a m o k v l e v e b i g a m o k v l e v e b i `dazverva~, Semdeg `dalaSqvra~. Cveni mizania dagexmaroT im gzeb- isa da xerxebis _ meTodebis moZebnaSi, romli- Tac midiodnen didi maTematikosebi da bunebis mkvlevarebi, moZraobis mimarTulebas ki Sen TviTon airCev. gaxsovdes qarTuli andaza: `xer- xi sjobia Ronesa, Tu kaci moigonebsa ~. daiwyeb ra romelime movlenis SeswavliT, ma- galiTis amoxsniT, ideiT an faqtiT Sen Seudge- bi namdvil kvleviT muSaobas. movlenebis Ses- wavlisas Caatareb dakvirvebebs da cdebs, daag- roveb faqtebs. maT safuZvelze raRacas ivaraudeb, e. i. Camoayalibeb egreT wodebul hipoTezas**. dagebadeba kiTxvebi: ratom xdeba es? ra gamova Tu gavakeTeb ase? ra SeiZleba gamovides am ma- galiTidan? da ase Semdeg. magaliTis an amocanis amoxsna, movlenis Sesas- wavlad cdis an dakvirvebis Catareba, sxva da sxva SesaZlo variantebis ganxilva, Sedegis miReba an imis Segneba, rom dakavebuli xar raRac Rir- seuliT, Sen siamovnebas moganiWebs (Tu es grZnoba ar dageufleba, maSin sjobs sxva saqmes mohkido xeli). zogierT SemTxvevaSi Sen aRmoaCen, rom winaswar gansazRvruli gzebidan zogierTi miznamde ar migiyvans. aseT SemTxvevaSi unda eZe- bo axali ideebi da mimarTulebebi. cxadia gadaw- yvetilebas iRebs mkvlevari, e. i. Sen. gamokvlevebisas miznis misaRwevad arsebobs sxvadasxva xerxebi. cxadia yvelas migneba da gamoy- eneba Znelia. sasurvelia, Tu erTi da imave prob- lemas sxvadasxva xerxiT gadawyvitav. yvela xerxi misaRebia, magram Sen amoarCie is, romelic ufro mogwons. maTematikuri gamokvlevebis dros yovelTvis asea: Sen mas SenTvis ufro moxerxe- buli da sasiamovno mxridan Sexedav. magram rogorc ar unda miudge, yovelTvis dagWirdeba mondomeba, nebisyofa, warmosaxvis unari, orga- nizebuloba da dro. magram mTavari mainc is aris, rom gqondes amis survili da ganwyoba! ganvixiloT umartivesi magaliTi: ra miiReba, Tu 2347-s gamovaklebT 298-s? pasuxis miReba SeiZleba sxvadasxva xerxiT: 1. qaRaldisa da fanqris gamoyenebiT saklebs qveS miuweren maklebs: Semdeg yvelasaTvis cnobili wesiT bolodan TanmimdevrobiT asruleben gamoklebas: a) 17 _ 8 = 9; b) 13_ 9 = 4; g) 2 _ 2 = 0 da iReben 2049. 2. maklebs daumateben 2-s, miiReben 300 (e.i. S e a v s e b e n m T e l a s e u l a m d e ) . S e m d e g 2347_300=2047. magram, radgan maklebi gaz- ardes 2-iT, cxadia sxvaoba Semcirdeboda 2- iT. e.i. saZiebeli pasuxis misaRebad saWiroa
kl.) cota ucnaurad asrulebs am moqmedebas, magram sworad! ai ase: a)
_
nebs ZiriTadad Seiswavlis mecniereba fizika (berZ.
igi aris erT-erTi Ziri- Tadi mecniereba bunebis Sesaxeb. didi italieli mecnieri galileo ga- lilei (1564-1642) ambob- da: `maTematikuri sim- boloebi aris is damwer- loba, romliTac RmerTma dawera bunebis udidesi wigni ~. gindaT SeicnoT garemomcveli samyaro, buneba? iswavleT maTematika! da nurafers daijerebT damtkicebis gareSe! galileo galilei. evklide.
10 buneba, Sen da gamokvlevebi b) 8 _ 7 = 1; g) 10 _ 1 = 9; d) 9 _ 3 = 6; e) 10 _ 6 = 4; v) 200 _ 200 = 0 da bolos iRebs pasuxs: 2049.
cxadia nebismieri xerxi misaRebia! magram sjobs vecadoT SevarCioT yvelaze mar- tivi. am magaliTSi cxadia, yvelaze martivia kalkulatoris gamoyeneba, magram winaswar sxva xerxebis Seswavlaa saWiro! 6. sainteresoa, arsebobs kidev sxva xerxi? yovelgvari gamokvleva SemoqmedebiTi Sro- maa. aseTi Sroma ki adamians did siamovnebas ani- Webs... kvleviTi moRvaweobis sfero ki aris mec- niereba, romelic gamiznulia bunebis, sazoga- doebis da azrovnebis Sesaxeb axali codnis ga- mosamuSaveblad. dasmul kiTxvebs da amoxsnis process xandax- an amoxsnis magier mivyavarT axal kiTxvebamde. raime axlis aRmoCenisas ar damSvidde. ikiTxe (dafiqrdi) ratom da rogor xdeba igi. magaliTisaTvis daakvirdi da Seiswavle pi- Tagoras (berZeni maTematikosi) cxrili _ gam- ravlebis cxrili: bis gadakveTa). am cxrils Sen maTematikis gakveTi- lebidan icnob da misi gamoyeneba Zalian advi- lia. magram, Tu Sen SegekiTxebian: namravlebis mo- Zebnis garda ra aris saintereso am cxrilSi? Sen SegiZlia bevri saintereso ram aRmoaCino da dasmul kiTxvas upasuxo. marTlac, SearCie romelime mamravli, vTqvaT 9. daakvirdi Sesabamis vertikalur svetSi an horizontalur striqonSi daweril ricxvebs _ namravlebs (e.i. 9-ze gamravlebiT miRebul nam- ravlebs). advilad mixvdebi, rom: a) `yoveli namravli 9-iT metia Tavis wina namravlze ~, b) `Tu 9 mravldeba luw ricxvze, namravli luwia, xolo kentze _ kenti ~, g)
`TiToeuli namravlis cifrTa jami to- lia! da udris 9-s ~. axla wamoiWreba sxva kiTxvebic. ratom aris namravlis cifrTa jami yovelTvis 9-s toli? didi ricxvebis 9-ze gamravlebis drosac asea? risi tolia gamravlebis cxrilSi sxva ricxve- bze namravlis cifrTa jami? da sxva. piTagoras cxrilSi gamoyofilia `kuTxeebi~, daStrixuli da dauStrixavi. SearCie romelime maTgani.
Sekribe masSi Cawerili ricxvebi da miRebu- li jami Seadare Sesabamis samravls (an mamravls) Tavis Tavze samjer gamravlebiT miRebul nam- ravls. magaliTad: 3+6+9+6+3=27 da 3 ×3×3=27, aRmoCnda toli! ase moiqeci sxva `kuTxeebis~ SemTxvevaSic. Se- giZlia Tu ara, dainaxo raime kanonzomiereba? dagexmareba Tu ara es kanonzomiereba iwi- naswarmetyvelo Semdegi? saidan miiReba es kanon- zomiereba?
×3×3=27 5 ×5×5=125 cxadia es cxrili SeiZleba gagrZeldes usas- rulod, magram yvela ricxvs xom ver Cavwer- diT? naturalur ricxvTa simravle: 1, 2, 3, 4... usasruloa! samravli (mamravli) ricxvebi: 1, 2, 3, ... 9, 10, ... moTavsebulia pirveli svetis ujredebSi, xolo mamravli (samravli) ricxvebi: 1, 2, 3, ... 9, 10, ... ki _ zemodan pirveli striqonis ujredebSi. danarCeni ricxvebi namravlebia. magaliTad 3 ×7=7×3=21. amitom namravli 21 am cxrilis or sxvadasxva ujraSi weria (Sesa- bamisi horizontaluri da vertikaluri mwkrive- piTagora
(Zv.w. VI saukune). portreti rafaelis freskidan.
11 buneba, Sen da gamokvlevebi gamokvlevaTa organ gamokvlevaTa organ gamokvlevaTa organ gamokvlevaTa organ gamokvlevaTa organizacia izacia
izacia izacia
izacia gamokvlevebi, rom ufro efeqturi iyos da rac SeiZleba meti sargebloba moitanos saWiroa Sedegebis sistemaSi moyvana. maSin ufro advi- lia axali kanonzomierebis SemCneva. aba, daakvirdi 9-ze gamravlebis Sesabamis svets, aq SeimCneva garkveuli wesrigi (namravlebSi cifrebis gan- lagebis mixedviT). es ricxvebi rom mTel fur- celze iyos mimofantuli, usistemod (rogorc Cvevia zogierT moswavles), maSin raRac wesrigis danaxva da kanonzomierebis aRmoCena Zneli iqne- boda.
bunebrivia gamokvlevebis dros dagebadeba sxva- Tu imuSaveb gulmodgined, bejiTad, gamokv- levebi ganaviTareben Sens warmosaxvas, inteleqts da giCveneben rogor SeiZleba maTematikam migiy- vanos aRmoCenamde. Seecdebi ra amoxsna amocana an Seiswavlo movlena sxvadasxva xerxebiT, mo- Zebni ra SenTvis sxvadasxva gzebs, Sen iswavli, rogor unda ufro ukeT amoxsna ara marto ra mn ra mn
ra mn ra mn
ra mniSvneloba aqvs gamokvlevebs iSvneloba aqvs gamokvlevebs iSvneloba aqvs gamokvlevebs iSvneloba aqvs gamokvlevebs iSvneloba aqvs gamokvlevebs dasxva azrebi. maT Soris mniSvnelovani maSinve unda Caiwero, imisaTvis, rom ar dakargo an ar dagaviwydes. zogjer saWiroa suraTebis (naxaze- bis) Caxatva, Sedegebis gaformebisaTvis cxrilebisa da grafebis daxazva. gamoiyene SenTvis xelmi- sawvdomi yvela xerxi. Canawerebi rom ufro mowes- rigebuli iyos, gamoiyene ujredebiani an mili- metrula qaRaldi. aucileblobis SemTxvevaSi isargeble mikrokalkulatoriT. igi dagexmare- ba swrafad miiRo zusti Sedegebi da rTuli gamokvlevebi gaxado ufro xelmisawvdomi. maTematikuri, aramed cxovrebis mier dasmuli yvela amocana. ganiviTaro zusti da naTeli azrovneba. cxadia es mxolod dasawyisia did da rTul gzaze `mogzaurebisaTvis~. yovelTvis unda gaxsovdes kiTxvebi: ratom xdeba es? rogor moxde- boda ama Tu im pirobebSi? ra iqneboda es rom ase gavakeTo? gisurvebT warmatebebs!
L
kargad daakvirdi holandieli mxatvris m. k. esxeris suraTs `belvederi~. aRwere da imsjele dakvirvebis Sedegebis safuZvelze. 2 L
holandieli mxatvari moric kornelis esxeri ambobs: `wisqvilis borbals var- dnili wyali moZraobaSi moiyvans, Semdeg or koSks Soris daxrili arxiT gaedineba da ramdenime zigzagiT isev CanCqeris saTavesTan miva. mxolod aucilebelia d r o d a d r o w i s q v i l s a o r T q l e b a z e dakarguli wyali davumatoT ~. Tqven rogor fiqrobT? Seni azriT ra aris gamosaxuli am suraTze? 3 L m. k. esxeri. `CanCqeri~ . liTografia. 1961. 12 buneba, Sen da gamokvlevebi 8 L
moifiqre iseTi figurebi, romlebic gamoiwveven optikur iluziebs. cxrianis saidumloeba. dawereT nebismieri samniSna ricxvi, im pirobiT, rom pirveli da bolo cifrebi gansxvavebuli iyos. es ricxvi Sebrunebuli TanmimdevrobiT dawere da dids gamoakeli patara. yovelTvis aRmoCndeba saintereso kanonzomiereba: sxvaobis Sua cifri iqneba 9, agreTve pirveli da bolo cifrTa jamic 9! es imas niSnavs, rom Sen sxvaobas swrafad gamoicnob, Tu gecodineba am sxvaobis pirveli an bolo cifri. Tu exla sxvaobas Cawer Sebrunebuli TanmimdevrobiT da am or ricxvs (sxvaobas da mis Seqceuls) Sekribav, Sen yovelTvis miiReb 1089! cxrianis aRniSnuli `saidumloeba~ kargi gasarTobi iqneba, Tu amasTan erTad gamoiyeneb furcelze winaswar daweril ricxvs 6801, misi SebrunebiT miiReba 1089. moifiqre gasarTobi, rom ufro saxaliso gaxdes da SesTavaze Sen amxanagebs. cxrianis yvela `idumali~ Tviseba aixsneba im ubralo faqtiT, rom igi aris aTobiTi sistemis bolo cifri. romeli elifsia didi: qveda Tu zeda SigniT?
L es wrewiria? 5 L kvadratia Tu ara? 6 L rogor fiqrobT, aris am suraTebze kvadrati? 7 L
9 L
amocanebi 1. davuSvaT, Sen da me gvaqvs erTnairi raodenobis Tanxa. ramdeni fuli unda mogce Sen me, rom Cemze 10 lariT meti gagixdes? 2. boTli Rvino Rirs 5 lari. Rvino 4 lariT Zviria boTlze. ra Rirs boTli? 3. vaWarma raRac saqoneli 7 larad iyida da 8 larad gayida. Semdeg igive saqoneli xelaxla iyida 9 larad da gayida 10 larad. ramdeni lari moigo man? 4. zoomaRaziaSi iyideboda didi da patara Tevzebi. didi orjer ufro Zviri Rirda pataraze. myidvelma iyida 5 didi da 3 patara Tevzi. Tu is amis nacvlad iyidda 3 didsa da 5 patara Tevzs, gadaixdida 2 lariT naklebs. ra Rirda TiToeuli Tevzi?
katas _ 5. ramdeni iyo ZaRli da ramdeni kata? Tavsatexi
13 fibonaCis ricxvebi. ricxviTi mimdevrobebi fibonaCis ricxvebi. ricxviTi mimdevrobebi fibonaCis ricxvebi. ricxviTi mimdevrobebi fibonaCis ricxvebi. ricxviTi mimdevrobebi fibonaCis ricxvebi. ricxviTi mimdevrobebi 1202 wels, italiaSi gamovida naSromi maTematikaSi, saxelwodebiT `wigni abakze~ (berZ. aba- kos _ saangariSo CarCo). misi avtori iyo qalaq pizaSi dabadebuli bonaCis vaJiSvili leonardo, romelsac pizelebi leonardo fibonaCis (daaxl. 1180-1240 w.), mokled fibonaCis eZaxdnen (filius Bonacci _ bonaCis vaJiSvili). am wignSi mocemulia Tavsatexi. igi dRemde iqcevs maTematikosebis yuradRebas, im SesaniSnavi ricxviTi mimdevrobis gamo, romelic Tavsatexis pasuxis Zebnisas miiReba. saocaria, rom es mim- devroba sruliad moulodnel situaciebSic gvxvdeba. Seecade SeniSno kanonzomiereba, romlis Tanax- madac xdeba kurdRlebis gamravleba! SevTanxmdeT aseT aRniSvnebze: kurdRlebis wyvili, romelic badebs axal wyvils, aRvniSnoT aseTi niSniT , xolo wyvili, romelic ar sadRac, yvela mxridan SemoRobil adgilSi viRacam erTi wyvili axaldabadebuli kurdReli gauSva, rom gaego ramdeni wyvili kurdReli daugrovdeboda erTi wlis ganmavlobaSi. kur- dRlebis buneba iseTia, rom erTi Tvis Semdeg wyvili kurdReli yovel TveSi aCens axal wyvils, xolo axaldabadebuli wyvili ki _ ori Tvis Semdeg Tavisi dabadebidan. ramdeni wyvili kurdReli dagrovdeboda Tormeti Tvis ganmavlobaSi erTi wyvili axaldabadebulisagan? amoxsna
Tveebi I ianvari 1 1 2 3 5 8 II Tebervali III marti
IV aprili
V maisi
VI ivnisi
VII ivlisi
VIII agvisto
IX seqtemberi wyvilebis raodenoba s q e m a badebs _ . TiToeuli dambadebeli wyvili erTi Tvis Semdeg aCens Tavis Tavs da axal aradambadebel wyvils . erTi Tvis Semdeg aradambadebeli wyvili gadadis dambade- Tavsatexi 14 fibonaCis ricxvebi. ricxviTi mimdevrobebi belTa TanrigSi, magram ar iZleva axal wyvils. exla yuradRebiT ganixile qvemoT moyvanili sqema (ukeTesi iqneba Tu, Senc daxazav da imsjeleb zemoT moyvanili aRniSvnebis Sesabamisad). aba, scade gaagrZelo sqema. Tu sworad im- sjeleb VII TveSi (ivlisSi) wyvilebis raodeno- ba aRmoCndeba 13, VIII TveSi (agvistoSi) _ 21,
axla amovweroT miRebuli wyvilebis raode- noba erT mwkrivSi: 1, 1, 2, 3,
es aris fibonaCis mimdevrobis ramdenime sawyisi wevri. am mimdevrobis gagrZeleba SeiZleba usas- rulod. aseT ricxviT mimdevrobebs usasrulo mimdevrobebs uwodeben (cxadia arsebobs sasru- li mimdevrobebic). Seecade fibonaCis mimdevroba gaagrZelo damoukideblad. amisaTvis saWiroa rogorme miagno is kanonzomiereba, romlis mixedviTac fibonaCim miiRo igi. aba, daiwye mesame ricxvidan. rogor aris is miRebuli wina ori ricxvisagan? aseve daakvirdi am mimdevrobis nebismier wevrs da mis win mdgom or wevrs (ricxvs). kanonzomiereba miageni? kargia! e. i. fibonaCis mimdevrobis yoveli wevri, dawyebuli mesamedan, udris wina ori wevris jams. maSin Sen miiRebdi aseT gagrZelebas: ... 55, 89, 144, 233, 377, 610, ... axla Tavsatexis kiTxvaze pasuxis gacema ad- vilia. cxadia, Tormeti Tvis ganmavlobaSi wyvilebis raodenoba iqneba 144. maTematikidan SenTvis cnobilia, rom ricxve- bi SeiZleba aRiniSnos asoebiT. mag. a, b, c da a. S. ricxviTi mimdevrobis wevrebi aRvniSnoT ase: pirveli wevri a 1 -iT, meore _ a 2 -iT, mesame _ a 3 - iT da a. S. xolo nebismieri wevri a n -iT _ me- n-e wevri, sadac n naturaluri ricxvia. maSin miviRebT ricxviT midevrobis zogad Canawers: a 1 , a 2 , a 3 , ... a n , ... sadac (n=1, 2, 3, ...). fibonaCis mimdevrobis SemTxvevaSi: a 1 =a 2 =1, a 3 =a 1 +a 2 =1+1=2, a 4 =a 2 +a 3 =1+2=3, . . . . . . . . . . . . . . a n =a n-2 +a n-1 , n=3,4,... an ase: U k+2 =U k+1 +U k , k=1,2,..., U 1 =U 2 =1. rogor moiqcevi Tu SegekiTxebian: ras udris fibonaCis mimdevrobis magaliTad me- 100 wevri
? cxadia, mimdevrobas ganxiluli wesiT gaagrZelebdi, magram me-100 wevris gansazRvramde kalkulatoris gamoyenebiTac ki didi Sroma da dro dagWirdeba. maSin rogor unda moviqceT
gamosavali, ra Tqma unda, arsebobs, magram amisaT- vis saWiroa mimdevrobis zogadi wevris gan- sazRvris, e. w. bines formulis: ⎪⎩ ⎪
⎧ ⎪⎭ ⎪ ⎬ ⎫ ⎟⎟⎠ ⎞ ⎜⎜⎝
⎛ − − ⎟⎟⎠ ⎞ ⎜⎜⎝
⎛ + =
n n 2 5 1 2 5 1 5 1 a codna. magram misi miReba (an misi WeSmarite- bis dasabuTeba maTematikuri induqciis meTo- diT) maTematikis ufro met codnas moiTxovs. amis Sesaxeb qvemoT vimsjelebT. fibonaCis mimdevrobam SemdegSi gamoyeneba hpo- va mecnierebaSi. igi cnobilia ara marto maTe- matikosebisaTvis, aramed bunebis mkvlevarebisTvisac. magaliTad, Tu xe itoteba yovel wels da mesame wels aqvs ori Sto, maSin meoTxe wels Cveulebrivad totebis raodenoba izrdeba 3- mde, mexuTe wels _ 5-mde, meeqvse wels _ 8-mde, da a. S. fibonaCis mimdevrobis Sesabamisad (sur. 1). SeiZleba fibonaCis mimdevrobis pirveli ori wevri ar iyos erTianebi, aramed nebismieri ricx- vebi. magaliTad: 4, 7, 11, 18, 29, 47, 76, 123, … aseT SemTxvevaSi mas uwodeben fibonaCis gan- zogadobul mimdevrobas. 1 L gaamravle fibonaCis mimdevrobis TiToeuli wevri nebismier erTi da igive ricxvze. Seamowme xom ar miiReba isev fibonaCis mimdevroba? daamtkice zogadad aRmoCenili Tviseba. sur. 1
weli
15 fibonaCis ricxvebi. ricxviTi mimdevrobebi 1
dawere nebismieri fibonaCis ganzogadebuli mimdevrobis pirveli 10 wevri, Sen aTive wevris jams swrafad miiReb, Tu bolodan meoTxe wevrs (Tavidan meSvides) 11-ze gaamravleb. (ra- tom?) Seecade daamtkico. es WeSmariteba Sen SegiZlia gamoiyeno gasarTobad: SesTavaze vinmes zemoT ganxiluli wesiT Seadginos ganzogadebuli fibonaCos nebismieri mimdevroba da ipovos pirveli 10 wevris jami ise, rom Sen ar icode. sTxove, giTxras bolodan meoTxe wevri. maSin 11-ze gamravlebiT momentalurad etyvi saZiebel jams! magaliTad: 4+7+11+18+29+47+76+123+199+322=836. 76 ×
2
a) daxaze nebismieri marTkuTxa ABC samkuTxedi (sur. 2) da gaatare kaTetebis paraleluri Suaxazebi. miiReb AA 1 C 1 B 1 B texils, romlis sigrZe kaTetebis sigrZeTa jamis tolia (Seamowme!). analogiuri ageba Seasrule AA 1 C 1 da BB 1 C 1 samkuTxedebSi. miiReb AA 2 C 2 B 2 C 1 A 3 C 3 B 3 B texils, romlis sigrZe kvlav kaTetebis sigrZeTa jamis tolia. gaagrZele aseTi ageba miRebuli oTxi patara samkuTxedisaTvis da a. S. usasrulod (warmodgenaSi). cxadia, Tanda- TanobiT yoveli axali texili mcirediT gansxvavdeba hipotenuzisagan da zRvarSi SeuTav- sdeba mas. am texilebis sigrZe yovelTvis mudmivia da kaTetebis sigrZeTa jamis tolia. maSasadame, AC+CB=AB! e. i. kaTetebis sigrZeTa jami hipotenuzis sigrZis tolia! b) daxaze tolgverda samkuTxedi ABC (sur. 3) da dauSvi, rom misi gverdis sigrZe erTi erTeulis tolia: AB=AC=CB=1 erT. Seasrule wina magaliTSi ganxiluli agebebi da Sen geometriulad daamtkiceb, rom AC+CB=AB! e. i. 2=1! ra aris winaaRmdegobebis mizezi?
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miRebuli mimdevroba dawere pirveli mimdevrobis qveS. Semdeg pirveli da meore mimdev- robis wevrebi wevrobriv Sekribe (e. i. pirvels miumate pirveli, meores _ meore, mesames _ mesame da a. S.). Seamowme miRebuli mimdevrobac, xom ar aris fibonaCis mimdevroba? daamtkice zogadad es Tviseba. 3 L
es davalebebi Seamowme ramdenime ricxvisa da mimdevrobis SemTxvevaSi. ra SeiZleba vTqvaT fibonaCis mimdevrobis Tvisebebis Sesaxeb? 4 L
daamtkice, rom: a) a) a) a) a) fibonaCis ganzogadebuli mimdevrobis pirveli n wevris jami ganisazRvreba formuliT: S n = a n+2 _ a 2 ;
b) b) b) b) b) miiRe analogiuri formula fibonaCis mimdevrobisaTvis. 5 L
daamtkice, rom fibonaCis mimdevrobis pirveli n wevris kvadratebis jami udris a n . a n+1 . miTiTeba: 2 1 1 1 1 ) ( k k k k k k k k a a a a a a a a = − = ⋅ − ⋅ − + − + . 6 L
daamtkice, rom mocemuli P = 4a perimetris mqone yvela marTkuTxedebidan udidesi farTobi aqvs kvadrats. Tavsatexi sur. 2
B C sur. 3
A A 1 A A 2 B 1 B B 3 B 2 C 2 C 1 C 3 A 3 C 16 oqros kveTa. fidias ricxvebi oqros kveTa. fidias ricxvebi oqros kveTa. fidias ricxvebi oqros kveTa. fidias ricxvebi oqros kveTa. fidias ricxvebi fidia (Pheidias) iyo Zveli welTaRricx- vis V s. berZeni moqandake, maRali klasis war- momadgeneli, Zveli berZnuli xelovnebis erT- erTi udidesi ostati. daiba- da aTenSi daaxloebiT Zv. w.
swavlobda moqandake gegiasTan, xolo Semdeg saxelganTqmul skulptorTan ageladTan, rom- lis saxelosnoSic Cawvda brinjaos Camosxmis xelov- nebas. misi qmnilebebidan gan- sakuTrebiT popularulia aT- ena promaqosis (aTena _ me- brZolis) brinjaos Svid metriani kolosa- luri figura, romelic sparselebze gamar- jvebis aRsaniSnavad Zv. w. 460 aRmarTes aTe- nis akropolisze sparselebis mier ganadgurebuli taZrebis nangrevebs Soris (akropolisi _ berZ. akropolis _ antikuri berZnuli qalaqis gamagrebuli nawili, ganlagebulia ZiriTadad amaRlebul adgilze). e. w. zeda qalaqi, romelic warmoadgenda cixesimagres. akropolisi _ qarTulad Sidacixe, zedacixe, zedaqalaqi _ hqondaT Zvel qar- Zveli Taobis maTematikosebs kalkulatori ar hqoniaT. amitom maT didi Sroma uxdebodaT angariSTan dakavSirebiT. Sen ki SegiZlia gamoiyeno kalkulatori da ipovo fibonaCis mimdevrobis ori mezobeli ricxvis Sefardeba: 1:1=1; 2:1=2; 3:2=1,5; 5:3 ≈1,666 ...; 8:5=1,6; 13:8=1,625; 21:13≈1,615384 ... Tu ase gaagrZeleb, Sen zRvarSi miuaxlovdebi usasrulo aTwilads 1,618034 ..., romelsac Φ Φ Φ Φ Φ asoTi aRniSnaven: Φ Φ Φ Φ Φ≈1,618034 ... ≈ 1,618. mas `fidias ricxvs~ an `oqros ricxvs~ uwodeben. albaT bunebrivad dagebadeba kiTxva: fidia vin aris? cxenis Tavi parTenonis aRmosavleTis frontonidan parTeEnoni 17 oqros kveTa. fidias ricxvebi Tul qalaqebsac: mcxeTas, Tbiliss, quTaiss, arqeopoliss, anakofiss, vans da a. S. berZe- ni istorikosi dion kasios kokeiani (155 an 164 - 229) Tavis `romis istoriaSi~ akropolisad ixsenebs mcxeTis Sidacixes (zedacixes), igi mogviTxrobs, rom Zv. w. 65 romaelTa sardali pompeusi iberiaSi Sei- Wra da miaRwia akropolisamde. mcxeTis am nawils armazcixe ewodeba). ZvelberZnuli xuroTmoZRvrebis isto- riaSi gansakuTrebuli adgili uWiravs aTe- nis akropolisze nagebobaTa kompleqss. maT SeqmnaSi wamyvani roli ekuTvniT gamoCenil saxelmwifo moRvawes, brwyinvale orator- s, Zveli elitis mxedarTmTavars periklesa da udides moqandakes fidias, romelic Teq- vsmeti weli emsaxura akropolisis mSeneb- lobas. aTenis akropolisis mTavari nageboba aris parTenoni (berZ. Yüklə 4,69 Mb. Dostları ilə paylaş:
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