Nā ala geometric a me nā mānoanoa

I koʻu kākau ʻana i kēia ʻatikala, hoʻomanaʻo wau i kahi mele kahiko loa a Jan Pietrzak, āna i mele ai ma mua o kāna hana hoʻomākeʻaka ma ka cabaret Pod Egidą, i ʻike ʻia ma ka Polani Poʻe Repubalika ma ke ʻano he valve palekana; hiki i kekahi ke ʻakaʻaka maoli i nā paradoxes o ka ʻōnaehana. Ma kēia mele, ua paipai ka mea kākau i ke komo ʻana i ka politika socialist, e hoʻohenehene ana i ka poʻe makemake e lilo i ka politika a me ka hoʻopau ʻana i ka lekiō ma ka nūpepa. "ʻOi aku ka maikaʻi o ka hoʻi ʻana i ke kula heluhelu," i mele ʻo Petshak i ka makahiki XNUMX makahiki.

E hoʻi ana au i ke kula heluhelu. Ke heluhelu hou nei au (ʻaʻole no ka manawa mua) i ka puke na Shchepan Yelensky (1881-1949) "Lylavati". No nā mea heluhelu liʻiliʻi, ʻōlelo ka huaʻōlelo i kekahi mea. ʻO kēia ka inoa o ke kaikamahine a ka mea makemakika Hindu kaulana i kapa ʻia ʻo Bhaskara (1114-1185), ʻo Akaria ka inoa, a i ʻole ke kanaka naʻauao nāna i kapa i kāna puke ma ka algebra me kēlā inoa. Ma hope mai ua lilo ʻo Lilavati i mea makemakika kaulana a me ke akeakamai. Wahi a nā kumu ʻē aʻe, ʻo ia ka mea nāna i kākau i ka puke.

Ua hāʻawi ʻo Szczepan Yelensky i ka inoa like i kāna puke ma ka makemakika (paʻi mua, 1926). He mea paʻakikī paha ke kapa ʻana i kēia puke he hana makemakika - ʻoi aku ka nui o nā puʻupuʻu, a ua kākau hou ʻia mai nā kumu Farani (ʻaʻole i loaʻa nā kuleana kope i kēia manawa). I kekahi hihia, no nā makahiki he nui, ʻo ia wale nō ka puke Polani kaulana ma ka makemakika - ma hope ua hoʻohui ʻia ka puke lua a Jelensky, ʻo Pythagoras's Sweets. No laila, ʻaʻohe mea e koho ai ka poʻe ʻōpio makemake i ka makemakika (ʻo ia kaʻu mea ma mua) ...

ma ka ʻaoʻao ʻē aʻe, ʻike ʻia ʻo "Lilavati" kokoke ma ka puʻuwai ... ʻAe, aia kekahi manawa ... ʻO kā lākou pōmaikaʻi nui loa ʻo wau ... he ʻōpio ia manawa. I kēia lā, mai ka manaʻo o kahi mea makemakika i hoʻonaʻauao maikaʻi ʻia, ke nānā aku nei au iā Lilavati ma kahi ʻano ʻokoʻa loa - e like paha me ka mea piʻi ma nā piko o ke ala i Shpiglasova Pshelench. ʻAʻole nalowale kekahi a ʻaʻole hoʻi i kekahi i kona nani ... Ma kāna ʻano hiʻohiʻona, ʻo Shchepan Yelensky, nāna i haʻi i nā manaʻo i kapa ʻia ʻo ka lāhui i kona ola pilikino, kākau ʻo ia ma ka ʻōlelo mua:

Me ka ʻole o ka hoʻopā ʻana i ka wehewehe ʻana i nā ʻano ʻāina, e ʻōlelo wau ma hope o kanaiwa mau makahiki, ʻaʻole i nalowale nā ​​​​ʻōlelo a Yelensky e pili ana i ka makemakika. Aʻo ka makemakika iā ʻoe e noʻonoʻo. He mea oiaio. Hiki iā mākou ke aʻo iā ʻoe e noʻonoʻo ʻokoʻa, ʻoi aku ka maʻalahi a me ka nani? Malia paha. ʻAʻole hiki iā mākou. Ke wehewehe nei au i kaʻu mau haumāna makemake ʻole i ka makemakika he hōʻike hoʻi kēia o ko lākou naʻauao. Inā ʻaʻole hiki iā ʻoe ke aʻo i ka manaʻo makemakika maʻalahi, a laila... ʻoi aku paha ka ʻino o kou noʻonoʻo ma mua o kā māua makemake ʻelua...?

Nā hōʻailona ma ke one

A eia ka moʻolelo mua ma "Lylavati" - he moʻolelo i hōʻike ʻia e ke akeakamai Farani ʻo Joseph de Maistre (1753-1821).

Ua kiola ʻia kekahi luina o ka moku ʻāhaʻi e ka ʻale i kahakai ʻole, a ua manaʻo ʻo ia he kanaka ʻole. ʻEā, ma ke one kahakai, ʻike ʻo ia i kahi ʻano o kahi kiʻi geometric i kaha ʻia i mua o kekahi. Ia manawa i hoomaopopo iho ai oia aole mehameha ka moku!

Ma ka haʻi ʻana iā de Mestri, kākau ʻo Yelensky: kiʻi geometricHe ʻōlelo leo ʻole ia no ka poʻe pōʻino, moku, ʻaʻole naʻe, akā ua hōʻike ʻo ia iā ia i ka nānā ʻana i ka nui a me ka helu, a ua hoʻolaha kēia i kahi kanaka naʻauao. Nui no ka mōʻaukala.

E hoʻomanaʻo e hana ka luina i ka hopena like, no ka laʻana, ma ke kaha ʻana i ka huapalapala K, ... a me nā ʻāpana ʻē aʻe o ke alo o ke kanaka. Maʻaneʻi ua kūpono ke geometry.

Eia naʻe, ua manaʻo ka mea kilo kilo ʻo Camille Flammarion (1847-1925) e aloha aku nā lāhui kanaka i kekahi i kekahi mai kahi mamao aku me ka hoʻohana ʻana i ka geometry. Ua ʻike ʻo ia i kēia wale nō ka hoʻāʻo pololei a hiki ke kamaʻilio. E hōʻike kāua i kēlā poʻe Martian i nā triangle Pythagorean ... e pane lākou iā mākou me Thales, e pane mākou iā lākou me nā hiʻohiʻona Vieta, e komo ko lākou pōʻai i kahi triangle, no laila ua hoʻomaka kahi hoaaloha ...

Ua hoʻi nā mea kākau e like me Jules Verne lāua ʻo Stanislav Lem i kēia manaʻo. A i ka makahiki 1972, ua kau ʻia nā tile me nā hiʻohiʻona geometric (a ʻaʻole wale) ma luna o ka Pioneer probe, e hele mau ana i nā ākea o ka lewa, i kēia manawa kokoke i 140 ʻāpana astronomical mai mākou (1 ʻo wau ka awelika mamao o ka Honua mai ka Honua) . Lā, ʻo ia hoʻi, ma kahi o 149 miliona km). Ua hoʻolālā ʻia ka tile, ma kahi hapa, e ka astronomer Frank Drake, ka mea nāna i hoʻokumu i ke kānāwai hoʻopaʻapaʻa e pili ana i ka helu o nā civilization extraterrestrial.

He mea kupanaha ke Geometry. Ua ʻike mākou a pau i ka manaʻo nui i ke kumu o kēia ʻepekema. Ua hoʻomaka mākou (ʻo mākou nā kānaka) e ana i ka ʻāina (a ma hope o ka ʻāina) no nā kumu utilitarian. ʻO ka hoʻoholo ʻana i nā mamao, ke kahakaha ʻana i nā laina pololei, ka māka ʻana i nā kihi ʻākau a me ka helu ʻana i nā leo i lilo i mea pono. No laila ka mea a pau ʻomaka ("Ana o ka honua"), no laila ka makemakika a pau ...

Eia nō naʻe, no kekahi manawa ua uhi mai kēia kiʻi maopopo o ka mōʻaukala ʻepekema iā mākou. No ka mea, inā makemake wale ʻia ka makemakika no nā hana hana, ʻaʻole mākou e komo i ka hōʻoia ʻana i nā theorems maʻalahi. "Ke ike nei oe he oiaio keia," wahi a kekahi ma hope o ka nana ana i loko o kekahi mau huinakolu kupono, ua like ka huina o na huinahalike o na hypotenuse me ka huinahalike o ka hypotenuse. No ke aha ia ʻano formalism?

Pono ka pai Plum e ʻono, pono ka polokalamu kamepiula, pono ka mīkini. Inā helu au i ka nui o ka pahu he kanakolu manawa a ua hoʻonohonoho pono nā mea a pau, no ke aha lā?

I kēia manawa, ua manaʻo ka poʻe Helene kahiko e loaʻa kekahi mau hōʻike kūhelu.

No laila, hoʻomaka ka makemakika me Thales (625-547 BC). Ua manaʻo ʻia ʻo Mileto ka mea i hoʻomaka e kahaha no ke aha. ʻAʻole lawa ka poʻe akamai i ka ʻike ʻana i kekahi mea, ua manaʻoʻiʻo lākou i kekahi mea. Ua ʻike lākou i ka pono o ka hōʻoiaʻiʻo, kahi ʻano kūpono o nā manaʻo mai ka manaʻo a hiki i ka thesis.

Ua makemake hou lakou. ʻO Thales paha ka mea nāna i hoʻāʻo mua e wehewehe i ke ʻano o ke kino ma ke ʻano kūlohelohe, me ka ʻole o ke kōkua akua. Ua hoʻomaka ka philosophy ʻEulopa me ka philosophy of nature - me ka mea i hala ma hope o ka physics (no laila ka inoa: metaphysics). Akā na nā Pythagoreans (Pythagoras, c. 580-c. 500 BC).

Ua hoʻokumu ʻo ia i kāna kula ponoʻī ma Crotone ma ka hema o ka Apennine Peninsula - i kēia lā e kapa mākou iā ia he sect. ʻEpekema (ma ke ʻano o ka huaʻōlelo o kēia manawa), pili pono ka mysticism, ka hoʻomana a me ka noʻonoʻo. Ua hōʻike maikaʻi loa ʻo Thomas Mann i nā haʻawina o ka makemakika ma kahi hale haʻuki Kelemania ma ka puke moʻolelo ʻo Kauka Faustus. Unuhi ʻia e Maria Kuretskaya lāua ʻo Witold Virpsha, heluhelu ʻia kēia ʻāpana:

Ma ka puke hoihoi a Charles van Doren, The History of Knowledge from the Dawn of History to the Present Day, ua loaʻa iaʻu kahi manaʻo hoihoi loa. Ma kekahi o nā mokuna, wehewehe ka mea kākau i ke koʻikoʻi o ke kula Pythagorean. ʻO ke poʻo inoa o ka mokuna i hoʻokūpaʻa iaʻu. Heluhelu ia: "The Invention of Mathematics: The Pythagoreans".

Kūkākūkā pinepine mākou inā ʻike ʻia nā manaʻo makemakika (e laʻa me nā ʻāina ʻike ʻole ʻia) a i ʻole i haku ʻia (e laʻa me nā mīkini i loaʻa ʻole ma mua). ʻIke kekahi poʻe makemakika makakū iā lākou iho he poʻe noiʻi, ʻo nā mea ʻē aʻe ma ke ʻano he mea hana a mea hoʻolālā paha, ʻaʻole i helu pinepine ʻia.

Akā, kākau ka mea kākau o kēia puke e pili ana i ka mea i haku ʻia o ka makemakika ma ka laulā.

Mai ka hoʻonui a hiki i ka hoʻopunipuni

Ma hope o kēia māhele hoʻolauna lōʻihi, e neʻe au i ka hoʻomaka. ʻomakae wehewehe i ke ʻano o ka hilinaʻi nui ʻana i ka geometry hiki ke alakaʻi hewa i kahi ʻepekema. Ua ʻike ʻia ʻo Johannes Kepler ma ka physics a me ka astronomy ma ke ʻano he mea ʻike i nā kānāwai ʻekolu o ka neʻe ʻana o nā kino lani. ʻO ka mua, e neʻe ana kēlā me kēia honua ma ka ʻōnaehana solar a puni ka lā ma kahi orbit elliptic, ma kekahi o ka piko o ka lā. ʻO ka lua, i nā manawa maʻamau, e huki ana ke kukuna alakaʻi o ka honua, i huki ʻia mai ka Lā mai, i nā kaha like. ʻO ke kolu, ʻo ka lakio o ka huinahā o ka manawa o ka hoʻololi ʻana o ka honua a puni ka Lā i ka pahu kupapaʻu o ka axis semi-nui o kona orbit (ʻo ia hoʻi, ka mamao awelika mai ka Lā) he mau ia no nā hīkū a pau o ka ʻōnaehana solar.

ʻO ke kolu paha kēia o ke kānāwai - he nui ka ʻikepili a me ka helu ʻana no ka hoʻokumu ʻana, ʻo ia ka mea i koi ai iā Kepler e hoʻomau i ka ʻimi ʻana i nā mamana o ka neʻe ʻana a me ke kūlana o nā hōkū. ʻO ka moʻolelo o kāna "ʻike" hou he mea aʻo nui. Mai ka wā kahiko, mahalo mākou ʻaʻole wale i ka polyhedra maʻamau, akā ʻo nā hoʻopaʻapaʻa pū kekahi e hōʻike ana he ʻelima wale nō o lākou i ka lewa. Kapa ʻia ka polyhedron ʻekolu manaʻo maʻamau inā he mau polygon maʻamau kona mau helehelena a ua like ka helu o nā ʻaoʻao o kēlā me kēia vertex. Illustratively, kēlā me kēia kihi o ka polyhedron maʻamau e "nānā like". ʻO ka polyhedron kaulana loa ʻo ia ka cube. Ua ʻike nā kānaka a pau i kahi kuʻekuʻe wāwae maʻamau.

ʻAʻole ʻike nui ʻia ka tetrahedron maʻamau, a ma ke kula ua kapa ʻia ʻo ia ka pyramid triangular maʻamau. Ua like ia me ka pyramid. ʻAʻole i ʻike maikaʻi ʻia nā polyhedra maʻamau ʻekolu i koe. Hoʻokumu ʻia kahi octahedron ke hoʻohui mākou i nā kikowaena o nā ʻaoʻao o kahi pahu. Ua like ke ʻano o ka dodecahedron a me ka icosahedron me he mau pōpō. Hana ʻia mai ka ʻili palupalu, ʻoluʻolu lākou e ʻeli. ʻO ka manaʻo ʻaʻohe polyhedra maʻamau ma mua o nā paʻa Platonic ʻelima maikaʻi loa. ʻO ka mua, ʻike mākou inā he maʻamau ke kino, a laila e hui pū ka helu like (e q) o nā polygon maʻamau like i kēlā me kēia wēlau, e lilo kēia i mau kihi p. I kēia manawa pono mākou e hoʻomanaʻo i ke ʻano o ka huina i loko o kahi polygon maʻamau. Inā hoʻomanaʻo ʻole kekahi mai ke kula, hoʻomanaʻo mākou iā ʻoe pehea e ʻike ai i ke kumu kūpono. Hele mākou i kahi huakaʻi a puni ke kihi. Ma kēlā me kēia piko, huli mākou ma ka ʻaoʻao like a. Ke hele mākou a puni ka polygon a hoʻi i kahi i hoʻomaka ai, ua hana mākou i p ia mau huli ʻana, a ʻo ka huina ua huli mākou i 360 degere.

Akā ʻo ka α he 180 degere ka hoʻokō ʻana i ka huina a mākou e makemake ai e helu, a no laila

Ua loaʻa iā mākou ke kumu no ka huina (e ʻōlelo ka mea makemakika: nā ana o kahi kihi) o kahi polygon maʻamau. E nānā kākou: ma ka huinakolu p = 3, ʻaʻohe a

E like me kēia. Ke p = 4 (pālike), a laila

maikai no hoi na degere.

He aha kā mākou e loaʻa ai no ka pentagon? No laila he aha ka hopena inā loaʻa nā q polygons, ua like nā kihi o kēlā me kēia p

 degere e iho ana ma ka piko hoʻokahi? Inā ma luna o ka mokulele, a laila e hana ʻia kahi kihi

degere a ʻaʻole hiki ke ʻoi aku ma mua o 360 degere - no ka mea, ʻāwili nā polygons.

Eia naʻe, no ka hui ʻana o kēia mau polygons ma ke ākea, pono e emi ka huina ma mua o ka huina piha.

A eia ka like ole o keia mau mea a pau.

E puunaue me 180, e hoonui i na apana elua me ka p, ka hooponopono (p-2) (q-2) < 4. He aha ka mea mahope? E hoʻomaopopo kākou he mau helu maoli ka p a me q a ʻo ka p > 2 (no ke aha? A he aha ka p?) a me q > 2. ʻAʻole nui nā ala e hoʻemi ai i ka huahelu o nā helu kūlohelohe ʻelua ma mua o 4. E papa inoa iā lākou a pau ma ka papa 1.

ʻAʻole au e kau i nā kiʻi kiʻi, hiki i nā mea a pau ke ʻike i kēia mau kiʻi ma ka Pūnaewele. I ka makahiki 1970 ua kamaʻilio au ma kahi seminar. Ua paʻakikī ke kumuhana. He liʻiliʻi koʻu manawa e hoʻomākaukau ai, noho au i ke ahiahi. Heluhelu wale 'ia ka 'atikala nui ma kahi. Ua ʻoluʻolu ka wahi, me ka lewa hana, pono, ua pani ʻia i ka hola ʻehiku. A laila ua hāʻawi mai ka wahine mare (ʻo kaʻu wahine i kēia manawa) e kākau hou i ka ʻatikala holoʻokoʻa noʻu: ma kahi o ʻelua mau ʻaoʻao i paʻi ʻia. Ua kope au (ʻaʻole, ʻaʻole me ka peni quill, loaʻa iā mākou nā peni), ua kūleʻa ka haʻiʻōlelo. I kēia lā ua hoʻāʻo wau e ʻimi i kēia puke, kahi mea kahiko. Hoʻomanaʻo wau i ka inoa wale nō o ka mea kākau ... Ua lōʻihi ka huli ʻana ma ka Pūnaewele ... he ʻumikūmālima minuke piha. Ke noʻonoʻo nei au me ka ʻakaʻaka a me ka mihi iki ʻole.

Hoʻi mākou i Keplera a me ka geometry. ʻIke ʻia, wānana ʻo Plato i ke ola ʻana o ke ʻano maʻamau ʻelima no ka mea ua nele ʻo ia i kahi mea hoʻohui, e uhi ana i ka honua holoʻokoʻa. ʻO ia paha ke kumu i kauoha ai ʻo ia i kekahi haumāna (Theajtet) e ʻimi iā ia. E like me ia, pēlā nō, ma ke kumu i ʻike ʻia ai ka dodecahedron. Kapa mākou i kēia ʻano o Plato pantheism. ʻO nā kānaka ʻepekema a pau, a hiki i Newton, ua hāʻule i ka nui a i ʻole ka liʻiliʻi. Mai ke kenekulia ʻumikūmāwalu noʻonoʻo nui, ua emi loa kona mana, ʻoiai ʻaʻole pono mākou e hilahila i ka ʻoiaʻiʻo ua hāʻule mākou a pau iā ia ma kekahi ʻano a i ʻole.

I ka manaʻo o Kepler no ke kūkulu ʻana i ka ʻōnaehana solar, ua pololei nā mea a pau, ua hui pū ka ʻikepili hoʻokolohua me ka manaʻo, ua kūlike ke kumumanaʻo, nani loa ... akā hoʻopunipuni loa. I kona manawa, ʻeono wale nō mau hokele i ʻike ʻia: Mercury, Venus, Earth, Mars, Jupiter a me Saturn. No ke aha he ʻeono wale nō nā hōkūhele? i ninau aku ai o Kepler. A he aha ka mea maʻamau e hoʻoholo ai i ko lākou mamao mai ka Lā? Ua manaʻo ʻo ia ua pili nā mea a pau, kēlā geometry a me ka cosmogony pili loa kekahi i kekahi. Mai nā palapala a ka poʻe Helene kahiko, ua ʻike ʻo ia he ʻelima wale nō polyhedra maʻamau. Ua ʻike ʻo ia he ʻelima puka ma waena o nā ʻōpuni ʻeono. No laila, pili paha kēlā me kēia o kēia mau wahi manuahi i kekahi polyhedron maʻamau?

Ma hope o nā makahiki he nui o ka nānā ʻana a me ka hana theoretical, ua hana ʻo ia i kēia kumumanaʻo, me ke kōkua o kāna helu pololei ʻana i nā ana o nā orbits, āna i hōʻike ai ma ka puke "Mysterium Cosmographicum", i paʻi ʻia ma 1596: E noʻonoʻo i kahi pōʻai nunui. ʻo ke anawaena ʻo ia ke anawaena o ka orbit o Mercury i kona neʻe ʻana a puni ka lā. A laila e noʻonoʻo ʻoe aia ma luna o kēia pōʻai he octahedron maʻamau, ma luna ona he pōʻai, ma luna ona he icosahedron, ma luna o laila he pōʻai, ma luna ona he dodecahedron, ma luna o kekahi pōʻai, ma luna ona he tetrahedron, a laila he pōʻai, he cube. a ʻo ka hope loa, ua wehewehe ʻia ke kinipōpō ma luna o kēia pahu.

Ua hoʻoholo ʻo Kepler, ʻo nā anawaena o kēia mau pōʻai hope, ʻo ia nā anawaena o nā ʻōpuni o nā hōkūhele ʻē aʻe: Mercury, Venus, Earth, Mars, Jupiter, a me Saturn. Me he mea la ua pololei loa ka manao. ʻO ka mea pōʻino, ua hui pū kēia me ka ʻikepili hoʻokolohua. A he aha nā hōʻike maikaʻi aʻe o ka pololei o ka manaʻo makemakika ma mua o ka pili ʻana me ka ʻikepili hoʻokolohua a i ʻole ka ʻikepili nānā, ʻoi aku ka "lawe ʻia mai ka lani"? Hōʻuluʻulu au i kēia mau helu ma ka Papa 2. No laila he aha ka hana a Kepler? Ua ho'āʻo wau a ho'āʻo a hiki i ka hanaʻana,ʻo ia hoʻi, i ka manawa i hoʻonohonohoʻia ai ka hoʻonohonoho (ka hoʻonohonoho o nā spheres) a me nā helu i loaʻa me kaʻikepili nānā. Eia nā helu Kepler hou a me nā helu.

Hiki i kekahi ke hāʻule i ka hoihoi o ke kumumanaʻo a manaʻoʻiʻo ʻaʻole pololei nā ana i ka lewa, ʻaʻole nā ​​helu i hana ʻia i ka noho mālie o ka hale hana. ʻO ka mea pōʻino, i kēia lā, ʻike mākou aia ma kahi o ʻeiwa mau hōkūhele a ʻo nā hopena a pau he coincidence wale nō. He minamina. He nani loa ia...