Njira za geometric ndi nkhalango

Pamene ndikulemba nkhaniyi, ndinakumbukira nyimbo yakale kwambiri ya Jan Pietrzak, yomwe adayimba pamaso pa ntchito yake yonyansa mu cabaret Pod Egidą, yodziwika ku Polish People's Republic ngati valve yotetezera; munthu akhoza kuseka moona mtima zododometsa za dongosolo. M’nyimbo imeneyi, wolembayo analimbikitsa anthu kuti atengepo mbali pa ndale za chikhalidwe cha anthu, kunyoza anthu amene akufuna kukhala andale komanso kuzimitsa wailesi mu nyuzipepala. "Ndibwino kubwereranso kusukulu," Petshak wazaka XNUMX anaimba modabwitsa.

Ndikubwerera kusukulu ndikuwerenga. Ndikuwerenganso (osati kwa nthawi yoyamba) buku la Shchepan Yelensky (1881-1949) "Lylavati". Kwa owerenga ochepa, mawuwo amanena chinachake. Ili ndi dzina la mwana wamkazi wa katswiri wa masamu wachihindu wotchuka wotchedwa Bhaskara (1114-1185), wotchedwa Akaria, kapena wanzeru amene anatchula bukhu lake la algebra ndi dzina limenelo. Pambuyo pake Lilavati anakhala katswiri wa masamu ndi wafilosofi wotchuka. Malinga ndi magwero ena, ndi iye amene analemba bukulo.

Szczepan Yelensky anapereka mutu womwewo ku buku lake la masamu (kope loyamba, 1926). Zingakhale zovuta kutchula bukhuli ngati ntchito ya masamu - linali lazithunzi zambiri, ndipo linalembedwanso kuchokera ku French magwero (zokopera m'lingaliro lamakono kunalibe). Mulimonsemo, kwa zaka zambiri ndilo buku lokhalo lodziwika bwino la ku Poland pa masamu - pambuyo pake buku lachiwiri la Jelensky, Sweets's Pythagoras, linawonjezeredwa kwa ilo. Chifukwa chake achinyamata omwe ali ndi chidwi ndi masamu (zomwe ndizomwe ndidali kale) analibe chilichonse choti angasankhe ...

mbali inayo, "Lilavati" ankayenera kudziwidwa pafupifupi ndi mtima ... Ah, panali nthawi ... Ubwino wawo waukulu unali wakuti ndinali ... wachinyamata panthawiyo. Lero, kuchokera kwa katswiri wa masamu wophunzitsidwa bwino, ndimayang'ana Lilavati mosiyana kwambiri - mwinamwake ngati wokwera pamwamba pa njira yopita ku Shpiglasova Pshelench. Palibe m'modzi kapena winayo amene amataya chithumwa chake ... M'makhalidwe ake, Shchepan Yelensky, yemwe amati ndi malingaliro adziko pamoyo wake, akulemba m'mawu oyamba:

Popanda kukhudza kufotokozera makhalidwe a dziko, ndikunena kuti ngakhale patapita zaka makumi asanu ndi anayi, mawu a Yelensky okhudza masamu sanataye kufunika kwake. Masamu amakuphunzitsani kuganiza. Ndi zoona. Kodi tingakuphunzitseni kuganiza mosiyana, mophweka komanso mokongola kwambiri? Mwina. Zangokhala^sitingathebe. Ndimafotokozera ophunzira anga omwe safuna kuchita masamu kuti ichinso ndi mayeso a luntha lawo. Ngati simungathe kuphunzira masamu osavuta, ndiye ...

Zizindikiro mumchenga

Ndipo apa pali nkhani yoyamba mu "Lylavati" - nkhani yofotokozedwa ndi wafilosofi wa ku France Joseph de Maistre (1753-1821).

Woyendetsa ngalawa yosweka anaponyedwa ndi mafunde pagombe lopanda kanthu, lomwe ankaliona kuti lopanda anthu. Mwadzidzidzi, mumchenga wa m'mphepete mwa nyanja, adawona chithunzi cha chithunzi chojambulidwa pamaso pa munthu. Apa m’pamene anazindikira kuti chilumbacho sichinasiyidwe!

Pogwira mawu a Mestri, Yelensky analemba kuti: chithunzi cha geometricakadakhala mawu osayankhula kwa tsoka, kusweka kwa ngalawa, mwangozi, koma adamuwonetsa pang'onopang'ono gawo ndi nambala, ndipo izi zidalengeza munthu wowunikiridwa. Kwambiri mbiri.

Zindikirani kuti woyendetsa ngalawa amayambitsa zomwezo, mwachitsanzo, pojambula chilembo K, ... ndi zizindikiro zina za kukhalapo kwa munthu. Apa geometry ndi idealized.

Komabe, katswiri wa zakuthambo Camille Flammarion (1847-1925) ananena kuti anthu otukuka apatsana moni ali kutali pogwiritsa ntchito geometry. Iye adawona mu izi njira yokhayo yolondola komanso yotheka yolumikizirana. Tiyeni tiwone Martians oterowo ma triangles a Pythagorean ... adzatiyankha ndi Thales, tidzawayankha ndi machitidwe a Vieta, bwalo lawo lidzalowa mu katatu, kotero ubwenzi unayamba ...

Olemba monga Jules Verne ndi Stanislav Lem anabwerera ku lingaliro ili. Ndipo mu 1972, matailosi okhala ndi mawonekedwe a geometric (osati okha) adayikidwa paulendo wa Pioneer probe, womwe umadutsabe mlengalenga, tsopano pafupifupi mayunitsi 140 zakuthambo kuchokera kwa ife (1 ine ndi mtunda wapakati wa Dziko Lapansi kuchokera ku Dziko Lapansi) . Dzuwa, mwachitsanzo, pafupifupi 149 miliyoni km). Tileyi idapangidwa, mwa zina, ndi katswiri wa zakuthambo Frank Drake, wopanga malamulo otsutsana pa kuchuluka kwa zitukuko zakunja.

Geometry ndi yodabwitsa. Tonsefe tikudziwa malingaliro ambiri pa chiyambi cha sayansi iyi. Ife (ife anthu) tangoyamba kuyeza malo (ndipo kenaka nthaka) kuti tigwiritse ntchito kwambiri. Kuzindikira mitunda, kujambula mizere yowongoka, kulemba mizere yolondola ndi kuwerengera ma voliyumu pang'onopang'ono kunakhala kofunika. Chifukwa chake chinthu chonsecho geometry ("Kuyeza kwa dziko lapansi"), chifukwa chake masamu onse ...

Komabe, kwa nthawi ndithu chithunzi chomveka bwino cha mbiri ya sayansi chidatisokoneza. Pakuti ngati masamu akadafunikira kuti agwire ntchito basi, sitikadakhala ochita kutsimikizira mfundo zosavuta. "Mukuwona kuti izi ziyenera kukhala zoona," wina anganene atawona kuti pamakona atatu akumanja kuchuluka kwa mabwalo a hypotenuses ndi ofanana ndi masikweya a hypotenuse. N’chifukwa chiyani amachita mwambo wotero?

Plum pie iyenera kukhala yokoma, pulogalamu ya pakompyuta iyenera kugwira ntchito, makinawo ayenera kugwira ntchito. Ngati ndidawerengera kuchuluka kwa mbiya nthawi makumi atatu ndipo zonse zili mu dongosolo, ndiye chifukwa chiyani?

Panthaŵiyi, zinafikira kwa Agiriki akale kuti umboni wina wotsimikizirika unafunikira kupezedwa.

Choncho, masamu akuyamba ndi Thales (625-547 BC). Zikuganiziridwa kuti Mileto ndi amene anayamba kudabwa chifukwa chake. Sikokwanira kwa anthu anzeru kuti aona chinachake, kuti akhutiritsidwe ndi chinachake. Iwo anawona kufunika kwa umboni, kutsatizana kotsatirika kwa zitsutso kuyambira pa kulingalira kufikira ku nthanthi.

Anafunanso zambiri. Mwinamwake anali Thales yemwe poyamba anayesa kufotokoza zochitika zakuthupi mwachirengedwe, popanda kuloŵerera kwaumulungu. Nzeru za ku Ulaya zinayamba ndi filosofi ya chilengedwe - ndi zomwe zili kale kumbuyo kwa physics (choncho dzina: metaphysics). Koma maziko a ontology a ku Ulaya ndi nzeru zachilengedwe anayalidwa ndi Pythagoras (Pythagoras, c. 580-c. 500 BC).

Anayambitsa sukulu yake ku Crotone kum'mwera kwa Peninsula ya Apennine - lero tingatchule kuti gulu lampatuko. Sayansi (m’lingaliro la tsopano la mawuwa), kukhulupirira zinsinsi, chipembedzo ndi zongopeka zonse zimagwirizana kwambiri. Thomas Mann anapereka mokongola kwambiri maphunziro a masamu mu bwalo la masewera olimbitsa thupi la Germany mu buku lakuti Doctor Faustus. Chidutswachi chomasuliridwa ndi Maria Kuretskaya ndi Witold Virpsha chimati:

M’bukhu lochititsa chidwi la Charles van Doren, The History of Knowledge from the Dawn of History to the Present Day, ndinapeza lingaliro lokondweretsa kwambiri. Mu umodzi mwa mitu, wolemba akufotokoza kufunika kwa sukulu ya Pythagorean. Mutu womwewo wa mutuwo unandikhudza mtima kwambiri. Imati: "Kupangidwa kwa Masamu: The Pythagoreans".

Nthawi zambiri timakambilana ngati masamu akupezeka (monga maiko osadziwika) kapena kupangidwa (monga makina omwe kunalibe kale). Akatswiri ena a masamu amadziona ngati ochita kafukufuku, ena monga oyambitsa kapena okonza, omwe nthawi zambiri amawerengera.

Koma mlembi wa bukhuli akulemba za kupangidwa kwa masamu ambiri.

Kuchokera kukokomeza kufika pachinyengo

Pambuyo pa gawo lalitali loyambilirali, ndipitilira mpaka poyambira. geometrykufotokoza momwe kudalira kwambiri geometry kungasokeretse wasayansi. Johannes Kepler amadziwika mu physics ndi zakuthambo monga wotulukira malamulo atatu a kayendedwe ka zinthu zakuthambo. Choyamba, planeti lililonse la mapulaneti ozungulira dzuŵa limayenda mozungulira dzuŵa m’njira yozungulira yozungulira, mbali ina ya mbali yake imene ili dzuŵa. Kachiwiri, nthawi ndi nthawi kuwala kotsogolera padziko lapansi, kotengedwa kuchokera ku Dzuwa, kumakoka minda yofanana. Chachitatu, chiŵerengero cha masikweya a nthawi ya kusinthika kwa pulaneti mozungulira Dzuwa kupita ku kiyibodi ya theka la axis yayikulu kwambiri ya kanjira kake (ie, mtunda wapakati kuchokera ku Dzuwa) ndi wokhazikika kwa mapulaneti onse ozungulira dzuŵa.

Mwinamwake ili linali lamulo lachitatu - linafunikira deta yambiri ndi mawerengedwe kuti akhazikitse, zomwe zinapangitsa Kepler kuti apitirize kufufuza machitidwe ndi malo a mapulaneti. Mbiri ya "kutulukira" kwake kwatsopano ndi yophunzitsa kwambiri. Kuyambira kalekale, takhala tikusilira osati polyhedra wokhazikika, komanso mikangano yosonyeza kuti pali asanu okha m'mlengalenga. Polyhedron yamitundu itatu imatchedwa yokhazikika ngati nkhope zake zili zofanana ndi ma polygons ndipo vertex iliyonse ili ndi nambala yofanana ya m'mphepete. Mwachifanizo, ngodya iliyonse ya polyhedron yokhazikika iyenera "kuwoneka chimodzimodzi". Polyhedron wotchuka kwambiri ndi kyubu. Aliyense wawona bondo wamba.

Tetrahedron wokhazikika sadziwika bwino, ndipo kusukulu amatchedwa piramidi ya triangular yokhazikika. Zikuwoneka ngati piramidi. Otsala atatu okhazikika polihedra sadziwika bwino. Octahedron imapangidwa tikamagwirizanitsa pakati pamphepete mwa cube. Dodecahedron ndi icosahedron zimawoneka ngati mipira. Zopangidwa kuchokera ku zikopa zofewa, zimakhala zosavuta kukumba. Lingaliro lakuti palibe polihedra wokhazikika kupatulapo zolimba zisanu za Plato ndi zabwino kwambiri. Choyamba, timazindikira kuti ngati thupi limakhala lokhazikika, ndiye kuti nambala yomweyo (siyani q) ya ma polygons ofanana ayenera kusinthika pa vertex iliyonse, izi zikhale ma-angles. Tsopano tiyenera kukumbukira chomwe ngodya mu polygon wokhazikika. Ngati wina sakumbukira kusukulu, tikukumbutsani momwe mungapezere chitsanzo choyenera. Tinayenda ulendo kuzungulira ngodya. Pa vertex iliyonse timatembenukira ku ngodya yofanana a. Tikamazungulira poligoni ndikubwerera koyambira, tapanga p kutembenuka koteroko, ndipo tonse tatembenuza madigiri 360.

Koma α ndi 180 madigiri 'wothandizira pa ngodya yomwe tikufuna kuwerengera, ndipo motero

Tapeza ndondomeko ya ngodya (katswiri wa masamu anganene kuti: miyeso ya ngodya) ya polygon wokhazikika. Tiyeni tiwone: mu makona atatu p = 3, palibe a

Ngati chonchi. Pamene p = 4 (square), ndiye

madigiri nawonso ali bwino.

Kodi timapeza chiyani pa pentagon? Ndiye chimachitika ndi chiyani pakakhala q ma polygons, p iliyonse imakhala ndi ngodya zofanana

 madigiri kutsika pa vertex imodzi? Ngati inali pa ndege, ndiye kuti ngodya ikanapanga

madigiri ndipo sangakhale oposa madigiri 360 - chifukwa ma polygons amalumikizana.

Komabe, popeza ma polygons amakumana mumlengalenga, ngodyayo iyenera kukhala yochepa kuposa ngodya yonse.

Ndipo apa pali kusiyana komwe kumatsatira zonse:

Gawani ndi 180, chulukitsani magawo onse awiri ndi p, dongosolo (p-2) (q-2) < 4. Chotsatira ndi chiyani? Tidziwe kuti p ndi q ayenera kukhala manambala achilengedwe ndipo p > 2 (chifukwa chiyani Ndiwalemba onse mu tebulo 2.

Sindimayika zojambula, aliyense akhoza kuwona ziwerengerozi pa intaneti ... Pa intaneti ... Sindingakane kudodometsa kwanyimbo - mwinamwake ndizosangalatsa kwa owerenga achinyamata. Mu 1970 ndinalankhula pa msonkhano. Nkhaniyi inali yovuta. Ndinali ndi nthawi yochepa yokonzekera, ndinakhala madzulo. Nkhani yayikulu idawerengedwa pokha. Malowa anali abwino, okhala ndi malo ogwirira ntchito, chabwino, adatseka seveni. Kenako mkwatibwi (tsopano mkazi wanga) adadzipereka kuti andilemberanso nkhani yonse: pafupifupi masamba khumi ndi awiri osindikizidwa. Ndinakopera (ayi, osati ndi cholembera, tinali ndi zolembera), nkhaniyo inali yopambana. Lero ndinayesa kupeza bukuli, lomwe ndi lachikale. Ndikukumbukira dzina lokha la wolemba ... Kusaka pa intaneti kunatenga nthawi yayitali ... mphindi khumi ndi zisanu zonse. Ndikuganiza za izo ndi kuseka ndi chisoni pang'ono popanda chifukwa.

Timabwerera ku Keplera ndi geometry. Mwachiwonekere, Plato ananeneratu za kukhalako kwa mawonekedwe achisanu okhazikika chifukwa analibe kanthu kena kogwirizanitsa, kuphimba dziko lonse. Mwina n’chifukwa chake analangiza wophunzira (Theajtet) kuti amufufuze. Monga izo zinali, kotero izo zinali, pa maziko a dodecahedron anapezeka. Izi timazitcha kuti Plato pantheism. Asayansi onse, mpaka ku Newton, anagonja kwa izo kumlingo waukulu kapena wocheperapo. Kuyambira zaka za zana la khumi ndi zisanu ndi zitatu zomveka bwino, chikoka chake chachepa kwambiri, ngakhale kuti sitiyenera kuchita manyazi kuti tonsefe timagonja kwa izo mwanjira ina.

Mu lingaliro la Kepler la kumanga dongosolo la dzuwa, chirichonse chinali cholondola, deta yoyesera inagwirizana ndi chiphunzitsocho, chiphunzitsocho chinali chogwirizana, chokongola kwambiri ... koma chonyenga kwathunthu. M'nthawi yake, mapulaneti asanu ndi limodzi okha ankadziwika: Mercury, Venus, Earth, Mars, Jupiter ndi Saturn. N’chifukwa chiyani pali mapulaneti XNUMX okha? Kepler anafunsa. Ndipo ndi nthawi yotani yomwe imatsimikizira mtunda wawo kuchokera ku Dzuwa? Iye ankaganiza kuti chirichonse chinali kugwirizana, kuti geometry ndi cosmogony ndi ogwirizana wina ndi mzake. Kuchokera m’zolemba za Agiriki akale, iye anadziŵa kuti panali ma polyhedra asanu okha okhazikika. Anaona kuti pakati pa mayendedwe asanu ndi limodziwo panali mikwingwirima isanu. Ndiye mwina malo aliwonse aulerewa amafanana ndi polyhedron wamba?

Patapita zaka zingapo kuonerera ndi ntchito yongopeka, iye analenga chiphunzitso zotsatirazi, ndi thandizo limene iye masamu molondola miyeso ya kapitawo, amene anapereka m'buku lakuti "Mysterium Cosmographicum", lofalitsidwa mu 1596: Tangoganizani chimphona gawo. m’mimba mwake ndi m’mimba mwake wa kanjira ka Mercury m’mayendedwe ake apachaka mozungulira dzuwa. Ndiye taganizirani kuti pa chigawo ichi pali octahedron wokhazikika, pa izo bwalo, pa izo icosahedron, pa izo kachiwiri bwalo, pa izo dodecahedron, pa gawo lina, pa izo tetrahedron, ndiye kachiwiri bwalo, cube. ndipo, potsiriza, pa kyubu iyi mpira akufotokozedwa.

Kepler ananena kuti ma diameter a mizere yotsatizanawa anali ma diameter a mizere ya mapulaneti ena: Mercury, Venus, Earth, Mars, Jupiter, ndi Saturn. Chiphunzitsocho chinawoneka kukhala cholondola kwambiri. Tsoka ilo, izi zidagwirizana ndi data yoyeserera. Ndipo ndi umboni wotani wa kulondola kwa chiphunzitso cha masamu kuposa kulemberana makalata ndi deta yoyesera kapena deta yowonera, makamaka "yotengedwa kuchokera kumwamba"? Ndikunena mwachidule mawerengedwewa mu Gulu 2. Ndiye Kepler anachita chiyani? Ndinayesa ndikuyesera mpaka zinatheka, ndiko kuti, pamene kasinthidwe (dongosolo la ma spheres) ndi zotsatira zake zimagwirizana ndi deta yowonera. Nazi ziwerengero zamakono ndi mawerengedwe a Kepler:

Munthu akhoza kugonjera ku chidwi cha chiphunzitsocho ndikukhulupirira kuti miyeso yakumwamba ndi yolakwika, osati mawerengedwe opangidwa mwakachetechete wa msonkhano. Tsoka ilo, lero tikudziwa kuti pali mapulaneti osachepera asanu ndi anayi ndikuti zonse zomwe zimachitika mwangozi zimangochitika mwangozi. Zachisoni. Zinali zokongola kwambiri...