Geometric nzira uye matanda
Ndichiri kunyora chinyorwa ichi, ndakayeuka rwiyo rwekare zvikuru naJan Pietrzak, rwaakaimba pamberi pebasa rake rekunyomba mu cabaret Pod Egidą, inozivikanwa muPolish People's Republic sevhavhavhavhavha; munhu aigona kuseka nokutendeseka pamagakava ehurongwa. Murwiyo urwu, munyori akakurudzira kuti vanhu vapinde mune zvematongerwo enyika, vachinyomba avo vanoda kuita zvematongerwo enyika uye kudzima nhepfenyuro mubepanhau. "Zviri nani kudzokera kuchikoro kuverenga," Petshak aive nemakore XNUMX akaimba zvinokatyamadza.
Ndave kudzokera kuchikoro ndichiverenga. Ndiri kuverenga zvakare (kwete kekutanga) bhuku raShchepan Yelensky (1881-1949) "Lylavati". Kune vashoma vaverengi, shoko pacharo rinotaura chimwe chinhu. Iri izita remwanasikana wenyanzvi yemasvomhu yechiHindu inozivikanwa saBhaskara (1114-1185), anonzi Akaria, kana kuti shasha yakatumidza bhuku rake paalgebra nezita iroro. Lilavati akazova nyanzvi yemasvomhu uye muzivi ane mukurumbira. Maererano nemamwe mabhuku, ndiye akanyora bhuku racho pachake.
Szczepan Yelensky akapa zita rimwechete kubhuku rake remasvomhu (yekutanga edition, 1926). Zvinogona kutonetsa kudaidza bhuku iri ibasa remasvomhu - raive rakawanda remapuzzle, uye rakanyanya kunyorwa kubva kuFrench zvinyorwa (copyrights mupfungwa yemazuva ano yakanga isipo). Chero zvazvingava, kwemakore akawanda ndiro chete bhuku rakakurumbira rechiPolish pamusoro pemasvomhu - gare gare bhuku rechipiri raJelensky, Pythagoras's Sweets, rakawedzerwa kwariri. Saka vechidiki vanofarira masvomhu (ndizvo chaizvo zvandaimbove) vaive vasina chekusarudza kubva ...
Ukuwo, "Lilavati" aifanira kuzivikanwa nemusoro ... Ah, pane dzimwe nguva ... Kubatsira kwavo kukuru kwaiva kuti ndaiva ... jaya ipapo. Nhasi, kubva pakuona kwenyanzvi yemasvomhu akadzidza, ndinotarisa Lilavati nenzira yakasiyana zvachose - zvichida semukwikwidzi pamakona enzira inoenda kuShpiglasova Pshelench. Hapana kana imwe kana imwe inorasikirwa nerunako rwayo ... Mumaitiro ake, Shchepan Yelensky, uyo anotaura anonzi mafungiro enyika muhupenyu hwake pachake, anonyora munhanganyaya:
Pasina kubata pamusoro pekutsanangurwa kwemaitiro enyika, ndichataura kuti kunyange mushure memakore makumi mapfumbamwe, mazwi aYelensky pamusoro pemasvomhu haana kurasikirwa nekukosha kwawo. Masvomhu inokudzidzisa kufunga. Ichokwadi. Tinogona kukudzidzisa kufunga zvakasiyana, zviri nyore uye zvinoyevedza here? Zvingava. Zvinongodaro ... isu hatigone. Ndinotsanangurira vadzidzi vangu vasingade kuita masvomhu kuti uyu muyedzo wenjere dzavo. Kana iwe usingakwanise kudzidza dzidziso yakapusa yemasvomhu, zvino ... pamwe kugona kwako kwepfungwa kwakaipa kupfuura isu tose tinoda ...?
Zviratidzo mujecha
Uye heino nyaya yekutanga mu "Lylavati" - nyaya inotsanangurwa nemuzivi wechiFrench Joseph de Maistre (1753-1821).
Mumwe mutyairi wengarava akanga aparara akakandwa nemafungu pamhenderekedzo yakanga isina chinhu, yaaifunga kuti yakanga isingagarwi. Nenguva isipi, mujecha remahombekombe, akaona mucherechedzo wechimiro chejometri chakadhonzwa pamberi peumwe munhu. Ipapo ndipo paakaona kuti chitsuwa chacho hachina kusiyiwa!
Achitora mashoko kubva kuna Mestri, Yelensky anonyora kuti: geometric nhambakungadai kwakave kutaura kwechimumumu kumunhu ane rombo rakanaka, kupunzika kwechikepe, zvakangoitika, asi akamuratidza pachiyero nenhamba, uye izvi zvakazivisa murume akavhenekerwa. Zvakawanda zvenhoroondo.
Ziva kuti mutyairi wengarava anokonzeresa kuita kwakafanana, semuenzaniso, nekudhirowa vara K, ... uye chero mamwe maitiro ehuvepo hwemunhu. Pano iyo geometry inogadziriswa.
Zvisinei, nyanzvi yoruzivo rwemitumbi yomudenga Camille Flammarion (1847-1925) yakakarakadza kuti vanhu vakabudirira vanokwazisana vari kure vachishandisa geometry. Akaona mune izvi kuedza chete kwakarurama uye kunobvira pakukurukurirana. Ngatiratidzei maMartians akadaro maPythagorean triangles ... ivo vachatipindura neThales, isu tichavapindura neVieta mapatani, denderedzwa ravo richapinda mumatatu, saka hushamwari hwakatanga ...
Vanyori vakadai saJules Verne naStanislav Lem vakadzokera kune iyi pfungwa. Uye muna 1972, mataira ane geometric (uye kwete chete) mapatani akaiswa muchikepe chePioneer probe, iyo ichiri kuyambuka expanses yechadenga, zvino ingangoita 140 mayuniti emuchadenga kubva kwatiri (1 Ini ndiri avhareji chinhambwe chePasi kubva pasi) . Zuva, kureva, anenge mamiriyoni 149 km). Iyo tile yakagadzirwa, muchidimbu, nenyanzvi yenyeredzi Frank Drake, musiki wekutonga kukakavara pamusoro pehuwandu hwemagariro ekunze.
Geometry inoshamisa. Tese tinoziva maonero akazara pamusoro pekwakabva sainzi iyi. Isu (isu vanhu) tangotanga kuyera ivhu (uye gare gare ivhu) nekuda kwezvinangwa zvekushandisa zvakanyanya. Kuona madaro, kudhirowa mitsetse yakatwasuka, kutara makona akarurama uye kuverenga mavhoriyamu zvishoma nezvishoma zvakava chinhu chinodiwa. Saka chinhu chose geometry ("Kuyera kwepasi"), saka masvomhu ese ...
Zvisinei, kwenguva yakati mufananidzo uyu wakajeka wenhoroondo yesainzi wakatiputira. Nokuti dai masvomhu aidikanwa nokuda kwezvinangwa zvekushanda chete, tingadai tisina kubatikana mukuratidza dzidziso dzakapfava. “Unoona kuti ichi ngachive chokwadi zvachose,” mumwe aizotaura mushure mekutarisa kuti mumakona matatu ekurudyi huwandu hwemasikweya e hypotenuses hwakaenzana ne sikweya ye hypotenuse. Sei kuita zvechivanhu zvakadaro?
Plum pie inofanira kuva inonaka, purogiramu yekombiyuta inofanira kushanda, muchina unofanira kushanda. Kana ndikaverenga huwandu hwebhodhoro kakapetwa makumi matatu uye zvese zviri muhurongwa, saka nei zvimwe?
Munguva iyi, zvakaitika kune vaGiriki vekare kuti humwe uchapupu hwepamutemo hwaifanira kuwanikwa.
Saka, masvomhu anotanga naThales (625-547 BC). Zvinofungidzirwa kuti Mireto ndiye akatanga kushamisika kuti nei. Hazvina kukwana kune vanhu vakachenjera kuti vaona chimwe chinhu, kuti vagutsikana nechimwe chinhu. Vakaona kudikanwa kwoufakazi, nhevedzano ine mufungo yenharo kubva pakufungidzira kusvika kurondedzero.
Vaidawo zvimwe. Zvichida aiva Thales akatanga kuedza kutsanangura zviitiko zvemuviri nenzira yezvisikwa, pasina kupindira kwaMwari. Uzivi hweEurope hwakatanga nehuzivi hwechisikigo - nezvatova kumashure kwefizikisi (naizvozvo zita: metaphysics). Asi hwaro hweEuropean ontology nehuzivi hwepanyama hwakaiswa nevaPythagoras (Pythagoras, munenge muna 580-c. 500 BC).
Akavamba chikoro chake muCrotone kumaodzanyemba kweApennine Peninsula - nhasi tingachidaidza kuti sekete. Sainzi (mupfungwa yazvino yezwi), kusanzwisisa, chitendero uye fungidziro zvese zvakabatana. Thomas Mann airatidza zvinoyevedza zvidzidzo zvemasvomhu mune imwe gymnasium yekuGermany mune novel Doctor Faustus. Chakashandurwa naMaria Kuretskaya naWitold Virpsha, chikamu ichi chinoti:
Mubhuku rinofadza raCharles van Doren, The History of Knowledge kubva muMambakwedza eNhoroondo kusvikira kuZuva Ranhasi, ndakawana murangariro unofadza zvikuru. Mune chimwe chezvitsauko, munyori anotsanangura kukosha kwechikoro chePythagorean. Musoro chaiwo wechitsauko chacho wakandibaya mwoyo. Inoti: "Kugadzirwa kweMathematics: The Pythagoreans".
Tinowanzo kurukura kuti dzidziso dzemasvomhu dziri kuwanikwa here (semuenzaniso nyika dzisingazivikanwe) kana kugadzirwa (semuenzaniso michina yakanga isipo kare). Vamwe vanogadzira masvomhu vanozviona sevatsvaguri, vamwe sevagadziri kana vagadziri, vasingawanzo kuverenga.
Asi munyori webhuku rino anonyora nezvekutanga kwemasvomhu zvachose.
Kubva pakuwedzeredza kusvika pakunyengera
Mushure meichi chikamu chesumo chakareba, ndichaenderera mberi nekutanga. geometrykutsanangura kuti kunyanya kuvimba nejeometry kunogona kutsausa musayendisiti. Johannes Kepler anozivikanwa mufizikisi uye nyeredzi semuwani wemitemo mitatu yekufamba kwemitumbi yedenga. Kutanga, nyika imwe neimwe iri muchadenga inofamba ichitenderera zuva mudenderedzwa redenderedzwa, pane imwe yefoci yayo izuva. Kechipiri, panguva dzenguva dzose mwaranzi inotungamira yepasi, yakatorwa kubva kuZuva, inodhirowa minda yakaenzana. Chetatu, chiyero chesikweya yenguva yekuchinja kwenyika yakatenderedza Zuva kusvika kune cube yehafu-hombe axis yekutenderera kwayo (kureva, chiyero chepakati kubva kuZuva) chinogara chiripo kune ese mapuraneti ari muzuva.
Zvichida uyu waive mutemo wechitatu - waida data rakawanda uye kuverenga kuti uzvisimbise, izvo zvakaita kuti Kepler arambe achitsvaga mapatani mukufamba uye chinzvimbo chemapuraneti. Nhoroondo ye "kuwanikwa" kwake kutsva inodzidzisa zvikuru. Kubva makarekare, takayemura kwete chete polyhedra yenguva dzose, asiwo nharo dzinoratidza kuti pane zvishanu chete munzvimbo. Itatu-dimensional polyhedron inodaidzwa kunzi chenguva dzose kana zviso zvayo zvakafanana mapolygons uye vertex imwe neimwe ine nhamba yakafanana yemapendero. Sekuenzanisira, kona yega yega yepolyhedron yenguva dzose inofanirwa "kutaridzika zvakafanana". Iyo inonyanya kuzivikanwa polyhedron icube. Munhu wese akaona chigumbo chamazuva ose.
Tetrahedron yenguva dzose hainyanyi kuzivikanwa, uye muchikoro inonzi piramidhi yenguva dzose yetriangular. Inotaridzika sepiramidhi. Akasara matatu enguva dzose polyhedra anozivikanwa zvishoma. Octahedron inoumbwa patinobatanidza nzvimbo dzemicheto yecube. Iyo dodecahedron uye icosahedron yatoita semabhora. Yakagadzirwa kubva kumatehwe akapfava, angave akasununguka kuchera. Mafungiro ekuti hakuna akajairwa polyhedra kunze kwemashanu ePlatonic solids yakanaka kwazvo. Chekutanga, tinoona kuti kana mutumbi uri wenguva dzose, saka nhamba imwe chete (rega q) yemapolygoni akafanana enguva dzose anofanira kuchinjika pane imwe neimwe vertex, ngaive ma-angles. Zvino isu tinofanirwa kurangarira kuti chii chiri kona muporigoni yenguva dzose. Kana mumwe munhu asingayeuki kubva kuchikoro, tinokuyeuchidza kuti ungawana sei maitiro akakodzera. Takafamba rwendo rwekukona. Pa vertex imwe neimwe tinotenderera nekona imwe chete a. Patinopoterera polygon uye todzokera kunzvimbo yekutanga, tinenge taita p kutendeuka kwakadaro, uye pamwe chete takashandura 360 degrees.
Asi α ndeye 180 madhigirii 'inoenderana nekona yatinoda kuverengera, uye naizvozvo
Tawana fomula yekona (nyanzvi yemasvomhu ingati: zviyero zvekona) yeporigoni yakajairika. Ngatitarisei: mugonyo p = 3, hapana a
Sezvizvi. Kana p = 4 (square), ipapo
madhigirii akanaka zvakare.
Chii chatinowana kune pentagon? Saka chii chinoitika kana paine q polygons, imwe neimwe p iine makona akafanana
madhigirii achidzika pane imwe vertex? Dai yaive mundege, ipapo kona yaizoumba
madhigirii uye haagone kupfuura 360 madhigirii - nekuti ipapo mapolygoni anopindirana.
Zvakadaro, sezvo mapolygoni aya achisangana muchadenga, kona inofanira kunge iri shoma pane yakazara kona.
Uye heino kusaenzana kubva kune zvese zvinotevera:
Igovane ne180, wedzera zvikamu zviviri ne p, kuronga (p-2) (q-2) < 4. Chii chinotevera? Ngatizivei kuti p naq dzinofanira kunge dziri nhamba dzechisikigo uye kuti p > 2 (nei? Uye chii chinonzi p?) uyewo q > 2. Hapana nzira dzakawanda dzekuita kuti chigadzirwa chenhamba mbiri dzakasikwa dzisasvike 4. Isu Ndichanyora dzese mutafura 1.
Handisi kutumira mifananidzo, munhu wose anogona kuona nhamba idzi paInternet ... PaInternet ... ini handingarambi chirevo chemashoko - zvichida chinonakidza kune vaverengi vaduku. Muna 1970 ndakataura pane mumwe musangano. Nyaya yacho yanga yakaoma. Ndakanga ndine nguva shoma yekugadzirira, ndakagara manheru. Chinyorwa chikuru chakaverengwa-chete panzvimbo. Nzvimbo yacho yaive yakasununguka, ine mhepo yekushanda, zvakanaka, yakavhara kumanomwe. Ipapo mwenga (iye zvino mudzimai wangu) pachake akazvipa kuti andinyorerezve nyaya yacho yose: anenge gumi nemaviri akadhindwa mapeji. Ndakaikopa (kwete, kwete nechinyoreso, taitova nezvinyoreso), hurukuro yacho yakabudirira. Nhasi ndaedza kutsvaga bhuku iri, iro ratosakara. Ndinoyeuka chete zita remunyori ... Kutsvaga paInternet kwakatora nguva yakareba ... maminitsi gumi nemashanu akazara. Ndinofunga nezvazvo nekuseka uye nekuzvidemba kusina chikonzero.
Tinodzokera ku Keplera uye geometrii. Sezviri pachena, Plato akafanotaura kuvapo kwechimiro chechishanu chenguva dzose nemhaka yokuti akanga asina chimwe chinhu chinobatanidza, chinofukidza nyika yose. Zvichida ndosaka akarayira mudzidzi (Theajtet) kuti amutsvake. Sezvazvaive, saka zvaive, pahwaro hweiyo dodecahedron yakawanikwa. Tinodaidza mafungiro aya ePlato pantheism. Masayendisiti ose, kusvikira kuNewton, akakurirwa nawo kumwero mukuru kana kuti muduku. Kubva muzana remakore regumi nemasere rine musoro, simba rayo rakaderera zvikuru, kunyange zvazvo tisingafaniri kunyara pamusoro pekuti isu tose tinokundwa nayo neimwe nzira.
Mune pfungwa yaKepler yekuvaka solar system, zvinhu zvose zvaive zvakarurama, data yekuedza yakabatana nedzidziso, dzidziso yacho yakanga yakanyatsoenderana, yakanaka kwazvo ... asi inhema zvachose. Munguva yake, mapuraneti matanhatu chete aizivikanwa: Mercury, Venus, Earth, Mars, Jupiter uye Saturn. Nei kuine mapuraneti matanhatu chete? Kepler akabvunza. Uye ndezvipi zvenguva dzose zvinosarudza kureba kwavo kubva kuZuva? Akafunga kuti zvese zvakabatana, izvo geometry uye cosmogony vane hukama hwepedyo kune mumwe nemumwe. Kubva muzvinyorwa zvevaGiriki vekare, aiziva kuti paingova nepolyhedra shanu dzenguva dzose. Akaona kuti pakati penzira nhanhatu paive nepakati shanu. Saka pamwe imwe neimwe yenzvimbo idzi dzemahara inoenderana neimwe yenguva dzose polyhedron?
Mushure memakore akawanda ekucherechedza uye basa rezvidzidzo, akasika dzidziso inotevera, nerubatsiro rwaakaverenga zvakanyatsorurama zviyero zvenzira, iyo yaakapa mubhuku rinonzi "Mysterium Cosmographicum", rakabudiswa muna 1596: Fungidzira giant sphere. dhayamita yaro idhayamita rekutenderera kweMercury mukufamba kwayo kwegore richitenderera zuva. Zvino fungidzira kuti panzvimbo iyi pane octahedron yenguva dzose, pairi denderedzwa, pairi icosahedron, pairi zvakare denderedzwa, pairi dodecahedron, pairi imwe nzvimbo, pairi tetrahedron, zvakare zvakare denderedzwa, cube. uye, pakupedzisira, pane iyi cube bhora rinotsanangurwa.
Kepler akagumisa kuti dhayamita dzemabhora akatevedzana aya aiva madhayamita ekutenderera kwemamwe mapuraneti: Mercury, Venus, Earth, Mars, Jupiter, uye Saturn. Dzidziso yacho yairatidzika kuva yakarurama zvikuru. Nehurombo, izvi zvakabatana nedata rekuyedza. Uye ndehupi humbowo huri nani hwekururama kwedzidziso yemasvomhu kupfuura kunyorerana kwayo nedata rekuyedza kana data rekuona, kunyanya "kutorwa kubva kudenga"? Ndinopfupikisa zviverengero izvi muTable 2. Saka Kepler akaitei? Ndakaedza uye ndakaedza kusvikira yaita, ndiko kuti, apo kurongeka (kurongeka kwezvikamu) uye maverengero anoguma akabatana nedheta rekucherechedza. Heano emazuva ano Kepler manhamba uye kuverenga:
Mumwe anogona kukundwa nekunakidzwa kwedzidziso uye kutenda kuti zviyero mudenga hazvina kururama, uye kwete maverengero akaitwa mukunyarara kwemusangano. Nehurombo, nhasi tinoziva kuti kune angangoita mapfumbamwe mapuraneti uye kuti kusawirirana kwese kwemhedzisiro kunongoitika. Tsitsi. Zvainge zvakanakisa...
