Izindlela zeJiyomethri namahlozi

Ngesikhathi ngibhala lesi sihloko, ngakhumbula ingoma endala kakhulu kaJan Pietrzak, ayicula ngaphambi komsebenzi wakhe wokubhuqa ku-cabaret Pod Egidą, eqashelwa e-Polish People's Republic njenge-valve yokuphepha; umuntu angahleka ngokweqiniso izixakaxaka zesistimu. Kuleli culo umbhali uncome ukubamba iqhaza kwezepolitiki yeSocialist, egcona labo abafuna ukuvukela umbuso nokucisha umsakazo ephephandabeni. "Kungcono ukubuyela esikoleni," uPetshak owayeneminyaka engu-XNUMX ubudala wacula ngendlela exakayo.

Ngibuyela esikoleni ngifunda. Ngifunda kabusha (hhayi okokuqala) incwadi kaShchepan Yelensky (1881-1949) "Lylavati". Kubafundi abambalwa, igama ngokwalo lisho okuthile. Leli igama lendodakazi yesazi sezibalo esingumHindu esidumile eyaziwa ngokuthi uBhaskara (1114-1185), ogama lakhe lingu-Akaria, noma isihlakaniphi esabiza incwadi yakhe ye-algebra ngalelo gama. ULilavati kamuva waba isazi sezibalo nesazi sefilosofi esaziwayo. Ngokweminye imithombo, nguyena ozibhalele le ncwadi.

U-Szczepan Yelensky unikeze isihloko esifanayo encwadini yakhe yezibalo (uhlelo lokuqala, 1926). Kungase kube nzima ukubiza le ncwadi ngomsebenzi wezibalo - ibiyiqoqo lezindida, futhi ibhalwe kabusha kusukela emithonjeni yesiFulentshi (amalungelo okushicilela ngomqondo wesimanje abengekho). Kunoma yikuphi, iminyaka eminingi bekuwukuphela kwencwadi yesiPolish ethandwayo ngezibalo - kamuva incwadi yesibili kaJelensky, Amaswidi kaPythagoras, yengezwa kuyo. Ngakho-ke abantu abasha abathanda izibalo (okuyilokho kanye engangiyikho) kwakungekho ongakhetha kukho ...

ngakolunye uhlangothi, "i-Lilavati" kwakudingeka yaziwe cishe ngekhanda ... Ah, kwakukhona izikhathi ... Inzuzo yabo enkulu yayiwukuthi ngangi ... ngisemusha ngaleso sikhathi. Namuhla, ngokombono wesazi sezibalo esifunde kahle, ngibheka uLilavati ngendlela ehluke ngokuphelele - mhlawumbe njengomqaphi emagodini endlela eya eShpiglasova Pshelench. Akekho noyedwa olahlekelwa ubuhle bakhe ... Ngokwesitayela sakhe sobuntu, u-Shchepan Yelensky, obiza lokho okubizwa ngemibono yezwe empilweni yakhe yomuntu siqu, ubhala esanduleleni:

Ngaphandle kokuthinta incazelo yezici zezwe, ngizosho ukuthi ngisho nangemva kweminyaka engamashumi ayisishiyagalolunye, amazwi kaYelensky mayelana nezibalo awazange alahlekelwe ukubaluleka kwawo. Izibalo zikufundisa ukucabanga. Kuyiqiniso. Singakufundisa yini ukucabanga ngendlela ehlukile, elula futhi enhle kakhulu? Kungenzeka. Kumane ... asikwazi. Ngichazela abafundi bami abangafuni ukwenza izibalo ukuthi nalokhu kuwukuhlolwa kobuhlakani babo. Uma ungakwazi ukufunda ithiyori yezibalo elula ngempela, khona-ke... mhlawumbe amakhono akho engqondo mabi kakhulu kunalokho esingathanda sobabili...?

Izimpawu esihlabathini

Futhi nansi indaba yokuqala "Lylavati" - indaba echazwe isazi sefilosofi saseFrance uJoseph de Maistre (1753-1821).

Itilosi elaliphuma emkhunjini ophahlazekile lajikijelwa amagagasi ogwini olungenalutho, elalilubheka njengelingahlalwa muntu. Kungazelelwe, esihlabathini esisogwini, wabona umkhondo wesithombe sejometri esidwetshwe phambi kothile. Kungaleso sikhathi lapho ebona khona ukuthi isiqhingi asikho dengwane!

Ecaphuna u-De Mestri, u-Yelensky uyabhala: umfanekiso wejometribekungaba inkulumo eyisimungulu kumuntu oneshwa, ophukelwe umkhumbi, ngokuqondana, kodwa wamkhombisa ngokubheka isilinganiso kanye nenombolo, futhi lokhu kwamemezela indoda ekhanyisiwe. Kakhulu ngomlando.

Qaphela ukuthi itilosi lizobangela ukusabela okufanayo, isibonelo, ngokudweba uhlamvu K, ... kanye nanoma yimiphi eminye iminonjana yobukhona bomuntu. Lapha i-geometry ihlelwe kahle.

Nokho, isazi sezinkanyezi uCamille Flammarion (1847-1925) sasikisela ukuba izimpucuko zibingelelane zikude zisebenzisa i-geometry. Wabona kulokhu kuwukuphela komzamo olungile nongenzeka wokuxhumana. Ake sibonise amaMartians anjalo onxantathu bePythagorean ... bazosiphendula ngeThales, sizobaphendula ngamaphethini e-Vieta, umbuthano wabo uzongena kunxantathu, ngakho-ke ubungane baqala ...

Ababhali abanjengoJules Verne noStanislav Lem babuyela kulo mbono. Futhi ngo-1972, amathayili anamaphethini wejometri (hhayi kuphela) afakwa ebhodini le-Pioneer probe, esawela umkhathi, manje cishe amayunithi angu-140 avela kithi (1 I ibanga eliphakathi komhlaba ukusuka eMhlabeni) . Ilanga, okungukuthi, cishe amakhilomitha ayizigidi eziyi-149). Ithayela laklanywa, ngokwengxenye, isazi sezinkanyezi uFrank Drake, umdali womthetho oyimpikiswano ngenani lempucuko yangaphandle komhlaba.

I-geometry iyamangalisa. Sonke siyawazi umbono ojwayelekile ngomsuka wale sayensi. Thina (thina bantu) sisanda kuqala ukukala umhlaba (futhi kamuva umhlaba) ngezinjongo eziwusizo kakhulu. Ukunquma amabanga, ukudweba imigqa eqondile, ukumaka ama-engeli angakwesokudla nokubala amavolumu kancane kancane kwaba yisidingo. Ngakho-ke yonke into iJiyomethri (“Isilinganiso somhlaba”), yingakho zonke izibalo ...

Nokho, isikhathi eside lesi sithombe esicacile somlando wesayensi wasifiphalisa. Ngoba ukube izibalo bezidingeka ngezinjongo zokusebenza kuphela, besingeke sizibandakanye ekuqinisekiseni amathiyori alula. “Uyabona ukuthi lokhu kufanele kube yiqiniso nhlobo,” omunye angasho ngemva kokuhlola ukuthi konxantathu abambalwa abangakwesokudla isamba sezikwele ze-hypotenuses silingana nesikwele se-hypotenuse. Kungani kwenziwa lokho?

I-Plum pie kufanele ibe mnandi, uhlelo lwekhompiyutha kufanele lusebenze, umshini kufanele usebenze. Uma ngibala umthamo womgqomo izikhathi ezingamashumi amathathu futhi yonke into ihlelekile, kungani-ke?

Phakathi naleso sikhathi, kwafika kumaGreki asendulo ukuthi kwakudingeka kutholwe ubufakazi obusemthethweni.

Ngakho-ke, izibalo ziqala ngoThales (625-547 BC). Kucatshangwa ukuthi kwakunguMilethu owaqala ukuzibuza ukuthi kungani. Akwanele kubantu abahlakaniphile ukuthi kukhona abakubonile, ukuthi baqiniseka ngokuthile. Babona isidingo sobufakazi, ukulandelana okunengqondo kwezimpikiswano kusukela ekuqageleni kuye ekuqaguleni.

Nabo babefuna okwengeziwe. Cishe kwakunguThales owaqala wazama ukuchaza izenzakalo ezingokomzimba ngendlela engokwemvelo, ngaphandle kokungenelela kwaphezulu. Ifilosofi yaseYurophu yaqala ngefilosofi yemvelo - ngalokho osekusele ngemuva kwe-physics (yingakho igama: i-metaphysics). Kodwa izisekelo ze-ontology yaseYurophu kanye nefilosofi yemvelo zabekwa amaPythagoreans (Pythagoras, c. 580-c. 500 BC).

Wasungula isikole sakhe eCrotone eningizimu yeNhlonhlo ye-Apennine - namuhla singasibiza ngokuthi ihlelo. Isayensi (ngomqondo wamanje wegama), imfihlakalo, inkolo kanye nenganekwane konke kuxhumene eduze. UThomas Mann wethule kahle kakhulu izifundo zezibalo endaweni yokuzivocavoca yaseJalimane enovelini ethi Doctor Faustus. Ihunyushwe nguMaria Kuretskaya noWitold Virpsha, lesi siqeshana sifundeka kanje:

Encwadini ethakazelisayo kaCharles van Doren ethi, Umlando Wolwazi Kusukela Ekuqalekeni Komlando Kuze Kufike Osukwini Lwamanje, ngithole umbono othakazelisa kakhulu. Kwesinye sezahluko, umbhali uchaza ukubaluleka kwesikole sePythagorean. Sona kanye isihloko sesahluko sangihlaba umxhwele. Ifundeka kanje: "Ukusungulwa Kwezibalo: AmaPythagoreans".

Sivame ukuxoxa ngokuthi izithiyori zezibalo ziyatholwa yini (isib. izindawo ezingaziwa) noma ziyasungulwa (isib. imishini eyayingekho ngaphambilini). Abanye ongoti bezibalo abanobuhlakani bazibona bengabacwaningi, abanye njengabaqambi noma abaklami, abavamile ukubala.

Kodwa umbhali wale ncwadi ubhala ngokusungulwa kwezibalo jikelele.

Ukusuka ekwenzeni ihaba kuye ekudukeni

Ngemva kwale ngxenye ende yesethulo, ngizodlulela ekuqaleni. iJiyomethriukuchaza ukuthi ukuthembela ngokweqile ku-geometry kungamdukisa kanjani usosayensi. UJohannes Kepler waziwa ku-physics kanye ne-astronomy njengomtholi wemithetho emithathu yokunyakaza kwezindikimba zasemkhathini. Okokuqala, iplanethi ngayinye esesimisweni sonozungezilanga ihamba izungeza ilanga emzileni oyi-elliptical, kwenye yezindawo ezigxile kuzo okuyilanga. Okwesibili, ngezikhathi ezivamile umsebe oholayo weplanethi, okhishwe eLangeni, udonsa amasimu alinganayo. Okwesithathu, isilinganiso sesikwele senkathi yokuguquguquka kweplanethi ezungeza iLanga kuya kukhiyubhu ye-eksisi ephakathi nendawo enkulu ye-orbit yayo (okungukuthi, ibanga eliphakathi ukusuka eLangeni) ayishintshile kuwo wonke amaplanethi esimiso sonozungezilanga.

Mhlawumbe lokhu kwakuwumthetho wesithathu - wawudinga idatha eningi kanye nezibalo ukuze usungulwe, okwenza u-Kepler aqhubeke nokusesha amaphethini ekuhambeni kanye nokuma kwamaplanethi. Umlando “wokutholwa” kwakhe okusha ufundisa kakhulu. Kusukela ezikhathini zasendulo, asizange sithande i-polyhedra evamile kuphela, kodwa futhi nezimpikiswano ezibonisa ukuthi kunezinhlanu kuphela zazo emkhathini. I-polyhedron enezinhlangothi ezintathu ibizwa ngokuthi evamile uma ubuso bayo bufana namapholigoni avamile futhi i-vertex ngayinye inenani elilinganayo lemiphetho. Njengomfanekiso, ikona ngalinye le-polyhedron evamile kufanele "libukeke ngendlela efanayo". I-polyhedron edume kakhulu yi-cube. Wonke umuntu usebone iqakala elijwayelekile.

I-tetrahedron evamile ayaziwa kakhulu, futhi esikoleni ibizwa ngokuthi iphiramidi evamile engunxantathu. Ibukeka njengephiramidi. I-polyhedra emithathu esele ayaziwa kakhulu. I-octahedron yakheka lapho sixhuma izikhungo zemiphetho ye-cube. I-dodecahedron ne-icosahedron sezivele zifana namabhola. Ezenziwe ngesikhumba esithambile, ayengakhululeka ukumba. Ukucabanga ukuthi azikho i-polyhedra evamile ngaphandle kwezingqimba ezinhlanu zikaPlato muhle kakhulu. Okokuqala, siyaqaphela ukuthi uma umzimba uvamile, inani elifanayo (yeka u-q) lamapholygoni afanayo avamile kufanele ahlangane ku-vertex ngayinye, lokhu makube ama-engeli angu-p. Manje sidinga ukukhumbula ukuthi iyini i-engeli kupholigoni evamile. Uma othile engakhumbuli esikoleni, sikukhumbuza ukuthi ungathola kanjani iphethini efanele. Sathatha uhambo ekhoneni. Ku-vertex ngayinye sijika nge-engeli efanayo a. Uma sizungeza ipholigoni futhi sibuyela endaweni yokuqala, senze p ukuphenduka okunjalo, futhi sekukonke siguqule amadigri angu-360.

Kodwa u-α ungu-180 degrees' umphelelisi we-engeli esifuna ukuyihlanganisa, futhi kunjalo

Sithole ifomula ye-engeli (isazi sezibalo singasho ukuthi: izilinganiso ze-engeli) yepholigoni evamile. Ake sihlole: kunxantathu p = 3, akekho u-a

Thanda lokhu. Lapho p = 4 (isikwele), ke

amadigri alungile futhi.

Yini esiyitholayo nge-pentagon? Pho kwenzekani uma kukhona amapholigoni angu-q, u-p ngamunye unama-engeli afanayo

 amadigri ehla ku-vertex eyodwa? Ukube bekusendizeni, bekuyokwakheka i-engeli

amadigri futhi awakwazi ukuba ngaphezu kwamadigri angama-360 - ngenxa yokuthi amapholygoni ayagqagqana.

Nokho, njengoba la mapholigoni ehlangana esikhaleni, i-engeli kufanele ibe ngaphansi kwe-engeli egcwele.

Futhi nakhu ukungalingani okuvela kukho konke:

Hlukanisa ngo-180, phindaphinda zombili izingxenye ngo-p, ngokulandelana (p-2) (q-2) < 4. Yini elandelayo? Masiqaphele ukuthi u-p no-q kumelwe kube izinombolo ezingokwemvelo nokuthi p > 2 (kungani? Futhi uyini u-p?) kanye no-q > 2. Azikho izindlela eziningi zokwenza umkhiqizo wezinombolo zemvelo ezimbili ube ngaphansi kuka-4. ngizowabala wonke.kuthebula 1.

Angithumeli imidwebo, wonke umuntu angabona lezi zibalo ku-inthanethi... Ku-inthanethi... ngeke ngenqabe ukwehla kwamazwi - mhlawumbe kuyathakazelisa kubafundi abasha. Ngo-1970 ngakhuluma emhlanganweni wokufundisa. Isihloko besinzima. Nganginesikhathi esincane sokulungiselela, ngangihlala kusihlwa. Isihloko esikhulu besifundwa kuphela endaweni. Indawo yayinethezekile, inomoya wokusebenza, kahle, yavalwa ngehora lesikhombisa. Khona-ke umakoti (manje ongumkami) ngokwakhe wacela ukungibhalela kabusha sonke lesi sihloko: cishe amakhasi aphrintiwe ayishumi nambili. Ngayikopisha (cha, hhayi ngepeni le-quill, sasinawo ngisho namapeni), isifundo saba yimpumelelo. Namuhla ngizamile ukuthola le ncwadi, eseyindala kakade. Ngikhumbula igama lombhali kuphela... Ukusesha ku-inthanethi kwathatha isikhathi eside... imizuzu eyishumi nanhlanu egcwele. Ngicabanga ngakho ngokumomotheka nokuzisola okuncane okungenasizathu.

Siphindela emuva I-Keplera ne-geometry. Ngokusobala, uPlato wabikezela ukuba khona kwefomu lesihlanu elivamile ngenxa yokuthi wayentula okuthile okuhlanganisayo, okuhlanganisa umhlaba wonke. Mhlawumbe yingakho eyalele umfundi (uTheajtet) ukuthi ambheke. Njengoba kwakunjalo, kwaba njalo, ngesisekelo lapho kwatholakala khona i-dodecahedron. Lesi simo sengqondo sikaPlato sibizwa ngokuthi yi-pantheism. Bonke ososayensi, kuze kufike kuNewton, banqotshwa yikho ngezinga elikhulu noma elincane. Kusukela ngekhulu leshumi nesishiyagalombili elinengqondo kakhulu, ithonya layo liye lehla kakhulu, nakuba akufanele sibe namahloni ngokuthi sonke sinqotshwa yiyo ngandlela thile.

Emcabangweni kaKepler wokwakha isimiso sonozungezilanga, yonke into yayilungile, idatha yokuhlola yahambisana nenkolelo-mbono, inkolelo-mbono yayihambisana ngokunengqondo, inhle kakhulu ... kodwa ingamanga ngokuphelele. Ngesikhathi sakhe, kwakwaziwa amaplanethi ayisithupha kuphela: iMercury, iVenus, uMhlaba, iMars, iJupiter neSaturn. Kungani kunamaplanethi ayisithupha kuphela? Kubuza uKepler. Futhi yikuphi ukujwayela okunquma ibanga labo ukusuka eLangeni? Wayecabanga ukuthi konke kuxhunyiwe, lokho i-geometry kanye ne-cosmogony zihlobene eduze. Kusukela emibhalweni yamaGreki asendulo, wayazi ukuthi kwakukhona i-polyhedra emihlanu kuphela evamile. Wabona ukuthi kunezikhala ezinhlanu phakathi kwemizila eyisithupha. Ngakho mhlawumbe ngayinye yalezi zikhala zamahhala ihambisana ne-polyhedron ethile evamile?

Ngemva kweminyaka eminingana yokubuka kanye nomsebenzi wetiyetha, wadala le thiyori elandelayo, ngosizo lapho wabala ngokunembile ubukhulu bemizila, ayethule encwadini ethi "Mysterium Cosmographicum", eyanyatheliswa ngo-1596: Cabanga njengendawo enkulu, ububanzi bayo ububanzi be-orbit ye-Mercury ekuhambeni kwayo kwaminyaka yonke ukuzungeza ilanga. Bese ucabanga ukuthi kulo mkhakha kukhona i-octahedron evamile, i-sphere, i-icosahedron, phezu kwayo futhi i-sphere, i-dodecahedron phezu kwayo, enye i-sphere, phezu kwayo i-tetrahedron, futhi i-sphere, i-cube. futhi, ekugcineni, kule cube ibhola lichazwe.

UKepler waphetha ngokuthi ububanzi balezi zimbulunga ezilandelanayo babuwububanzi bemizila yezinye iplanethi: iMercury, iVenus, uMhlaba, iMars, iJupiter, neSaturn. Lo mbono wawubonakala unembe kakhulu. Ngeshwa, lokhu kuhambisane nedatha yokuhlola. Futhi yibuphi ubufakazi obungcono bokunemba kwethiyori yezibalo kunokuxhumana kwayo nedatha yokuhlola noma idatha yokubona, ikakhulukazi "ethathwe ezulwini"? Ngifingqa lezi zibalo kuThebula lesi-2. Wenzani-ke u-Kepler? Ngazama futhi ngazama kwaze kwasebenza, okungukuthi, lapho ukucushwa (ukuhleleka kwama-sphere) kanye nezibalo eziwumphumela ziqondana nedatha yokubuka. Nazi izibalo zesimanje ze-Kepler nezibalo:

Umuntu anganqotshwa ukuthakaseleka kwethiyori futhi akholelwe ukuthi izilinganiso ezisesibhakabhakeni azinembile, hhayi izibalo ezenziwe ekuthuleni kwe-workshop. Ngeshwa, namuhla siyazi ukuthi kukhona okungenani amaplanethi ayisishiyagalolunye nokuthi konke ukuqondana kwemiphumela kumane kuzenzekele. Isihawu. Bekukuhle kakhulu...