Polyhedra ejwayelekile: izakhi wesimethri futhi endaweni

Jiyomethri kuyinto enhle ngoba, ngokungafani nokubili algebra, engesiyo kucaca ukuthi kungani nokuthi yini ocabanga ukuthi, kunikeza into ezibukwayo. Leli zwe elihle zemizimba ahlukahlukene kuhlobisa polyhedra njalo.

Ukwaziswa Okumayelana polyhedra njalo

Ngokusho eziningi, polyhedrons njalo, noma njengoba bebizwa ukudla okuqinile kaPlato, ifa izakhiwo esiyingqayizivele. Ngalezi ezintweni ezihlobene imicabango eziningana ngokwesayensi. Lapho uqala ukufunda idatha weJiyomethri yomzimba, uyaqaphela ukuthi cishe angazi lutho mayelana umqondo ezifana polyhedra njalo. Ukuziletha letintfo esikoleni ngaso sonke isikhathi ezithakazelisayo, ngakho musa eziningi ngisho ukukhumbula lokho bebizelwa. Kukhunjulwa abantu abaningi kumane cube. Akekho geometry umzimba akusho ifa ukuphelela ezifana polyhedrons njalo. Zonke amagama lezi zindikimba weJiyomethri uvela eGrisi lasendulo. Zimelela inani ubuso: i Tetrahedron - ezine emaceleni, hexahedron - U-Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. Konke lokhu umzimba weJiyomethri zisendaweni ebalulekile ekuhlaleni kwesisu kaPlato universe. Abane kubona ehlanganiswe izakhi noma izinhlangano: i Tetrahedron - komlilo, icosahedron - amanzi cube - emhlabeni, octahedron - emoyeni. Dodecahedron komculo zonke izinto. Wabalwa main, njengoba iwuphawu yonke.

I nje okungaqondile ka nomqondo polyhedron

Polyhedron iqoqo ezilinganiselwe ka ngemapholigoni yokuthi:

• ngamunye ezinhlangothini yimiphi ngemapholigoni kuyinto ngesikhathi esifanayo uhlangothi olulodwa kuphela omunye ipholigoni ohlangothini olufanayo;
• kusukela ngamunye ngemapholigoni ungahamba ukuba enye odlulayo eduze nalokho emapholigoni.

Polygons akha lesi polyhedron ukumela ubuso bayo kanye side yabo - izimbambo. polyhedra vertices kukhona vertices of emapholigoni. Uma ipholigoni eside ukuqonda polylines flat avaliwe, andukuba uze komunye incazelo polyhedron. Esimweni lapho yilo gama lalibhekisela ingxenye indiza ukuthi iphahlwe imigqa siphukile, kuyothiwa ebusweni waqonda ehlanganisa izicucu zikanhlangothiningi. polyhedron convex ibizwa ngokuthi umzimba elele Kolunye uhlangothi indiza eyayiseduze ubuso bayo.

Enye incazelo polyhedron nesimo sezulu

Polyhedron ngokuthi ebusweni ehlanganisa ngemapholigoni, okwenza zilinganiselwe izinhlobo umzimba weJiyomethri. Ziyakwazi:

• non-convex;
• ayizindunduma (okulungile nokungalungile).

polyhedron ejwayelekile - kuyinto polyhedron ayizindunduma nge wesimethri maximal. Tincenye polyhedra njalo:

• Tetrahedron: 6 izimbambo 4 ubuso 5 vertices;
• hexahedron (cube) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

ifayela echaza ifomu Euler

It usungula ubudlelwano phakathi inani imiphetho, vertices futhi ubuso topologically okulingana uyindilinga. Ukungeza inombolo vertices nobuso (B + D) ehlukile polyhedra njalo futhi nalokho inani izimbambo, kungenzeka ukusetha koMbuso owodwa: isamba inani ubuso ilingana nenani vertices futhi imiphetho (P) kukhuphuke ngo 2. Kungenzeka ukwenzela ukufinyelela eempendulweni ifomula elula:

• B + D = P + 2.

Lokhu ifomula ngaso sonke polyhedra convex.

izincazelo eyisisekelo

Umqondo polyhedron ejwayelekile akunakwenzeka ukuchaza umusho owodwa. Kuyinto zenani ngaphezulu nevolumu. Umgwamanda ohlonishwa kanjalo, kubalulekile ukuthi lihlangabezana eziningi izincazelo. Ngakho, umzimba weJiyomethri kuyoba polyhedron ejwayelekile lapho nale mibandela:

• kuba ayizindunduma;
• inombolo efanayo lwezimbambo converges ngasinye vertices zayo;
• zonke izici zakhe - ngemapholigoni njalo, ilingana nomunye;
• Zonke engeli dihedral bayalingana.

Properties of polyhedra njalo

Kukhona 5 izinhlobo ezahlukene polyhedra njalo:

1. Cube (hexahedron) - it has a engela flat isihloko kuyinto 90 °. It has a engela 3 emaceleni. Inani ubuso angles ngesikhathi sokudlondlobala 270 °.
2. Tetrahedron - engela flat sokudlondlobala - 60 °. It has a engela 3 emaceleni. Inani ubuso angles ngesikhathi sokudlondlobala - 180 °.
3. Octahedron - engela flat sokudlondlobala - 60 °. It has a engela ezine emaceleni. Inani ubuso angles ngesikhathi sokudlondlobala - 240 °.
4. Dodecahedron - engela flat sokudlondlobala 108 °. It has a engela 3 emaceleni. Inani ubuso angles ngesikhathi sokudlondlobala - 324 °.
5. Icosahedron - it has a engela flat sokudlondlobala - 60 °. It has a engela amahlanu emaceleni. Inani ubuso angles ngesikhathi sokudlondlobala 300 °.

Endaweni polyhedra njalo

Le ndawo ebusweni izidumbu yejeyomethri (S) ibalwa njengoba njalo ipholigoni endaweni siphindwe ngesibalo sinezici (G):

• S = (a: 2) x 2G ctg π / p.

Umthamo we polyhedron ejwayelekile

Lokhu value ibalwa ngokuvele azalane ngevolumu iphiramidi njalo kabani base kuyinto ipholigoni njalo, isibalo ubuso, futhi ukuphakama kwalo iyona engaba elalibhalwe we sphere (r):

• V = 1: 3rS.

Umqulu ka polyhedra njalo

Njenganoma iyiphi enye weJiyomethri okuqinile, polyhedra njalo babe amavolumu ezahlukene. Ngezansi amafomula ngawo bakwazi ukubala:

• Tetrahedron: α x 3√2: 12;
• octahedron: α x 3√2: 3;
• icosahedron; α x 3;
• hexahedron (cube): α x 5 x 3 x (3 + √5): 12;
• dodecahedron: α x 3 (15 + 7√5): 4.

Tincenye polyhedra njalo

Hexahedron futhi octahedron kukhona ezimbili izidumbu weJiyomethri. Ngamanye amazwi, kungenzeka ukuphuma nomunye uma kwenzeka ukuthi centroid elilodwa sithathwa phezulu kwezinye, futhi okuphambene nalokho. Futhi kukhona icosahedron elimbaxambili futhi dodecahedron. Yena kuphela Tetrahedron kuyinto ezimbili. Ngokwe-indlela Euclid ingatholakala kusuka hexahedron dodecahedron ngokwakha "uphahla" ebusweni cube. Vertices the Tetrahedron kukhona noma iyiphi 4 vertices cube, hhayi ngazimbili eduze kanye onqenqemeni. Kusukela hexahedron (cube) ingatholakala, nezinye polyhedra njalo. Naphezu kweqiniso lokuthi ngemapholigoni njalo kukhona azinakubalwa, polyhedra njalo, kukhona 5 kuphela.

I radii ka ngemapholigoni njalo

Nge ngamunye lezi zindikimba weJiyomethri kukhona exhunyiwe sphere eyindilinga 3:

• echazwe edabula vertices;
• elalibhalwe ngokuphathelene nendaba ngayinye engamanga ubuso bayo phakathi kwalo;
• lesemkhatsini ngokuphathelene nabo bonke imiphetho phakathi.

Lo engaba sphere esachazwa fomula elandelayo ibalwa:

• R = a: 2 x TG π / g x TG θ: 2.

Lo engaba sphere elalibhalwe ibalwa kanje:

• R = a: 2 x ctg π / p x TG θ: 2,

lapho θ - engela dihedral esiphakathi facets eduze.

Irediyasi imidiyeni sphere ingabalwa usebenzisa le ndlela elandelayo:

• ρ = a cos π / p: 2 isono π / h,

lapho h = ubukhulu 4.6, 6.10, noma 10. I isilinganiso radii we elalibhalwe echazwe futhi afanako mayelana p futhi q. It ibalwa kanje:

• R / r = TG π / p x TG π / q.

I wesimethri ka polyhedra

I wesimethri we polyhedra njalo iyathakazelisa eyinhloko lezi zindikimba weJiyomethri. Kuyaqondakala njengoba ukunyakaza komzimba emkhathini, uyaphuma inani elifanayo vertices, ubuso, imiphetho. Ngamanye amazwi, ngaphansi kwethonya wesimethri kugucuka onqenqemeni, vertex, noma ubuso azigcina endaweni yalo, noma uhamba endaweni ekhaya lomunye ubambo, omunye vertices noma ubuso.

Elements wesimethri we polyhedra njalo evame ukubuzwa yibo bonke izinhlobo ukudla okuqinile weJiyomethri. Lapha iqhutshwa ukuthi ungubani ukuguqulwa, uyaphuma yimiphi amaphuzu e-esikhundleni sazo sokuqala. Ngakho, lapho uvula iphrizimu zikanhlangothiningi ungathola ezinye symmetries. Noma yimuphi wabo kungenziwa imelelwa umkhiqizo reflection. Kufanane, okuyinto umkhiqizo i ngisho inani zibona, ngokuthi oqondile. Uma umkhiqizo inombolo obungavamile zibona, khona-ke kubizwa ngokuthi impendulo. Ngakho, bonke ematfuba emhlabeni umugqa amelela wesimethri ngqo. Noma yimuphi polyhedron kuveza - iyona wesimethri ephambene.

Ukuze uqonde kangcono izici wesimethri we polyhedra njalo, ungathatha isibonelo Tetrahedron. Noma yimuphi umugqa siyodlula omunye vertices kanye maphakathi ukuma weJiyomethri, kuyokwenzeka, futhi, unqamula enkabeni okuphambene onqenqemeni kukho. Ngamunye ematfuba 120 kanye 240 ° emhlabeni umugqa kungokwalabo plural wesimethri tetrahedral. Njengoba 4 vertices nobuso, sithola ingqikithi symmetries oqondile eziyisishiyagalombili. Noma imiphi yale imigqa edabula maphakathi emaphethelweni kanye maphakathi umzimba, udabula phakathi onqenqemeni okuphambene. Noma yimuphi ukujikeleza 180 °, ngokuthi nengxenye-kuguqulwe a wesimethri ngqo. Kusukela Tetrahedron usenemigqa ezintathu izimbambo, uthola obuthathu wesimethri. Ngokusekelwe kulokhu okungenhla, singaphetha ngokuthi inani eliphelele wesimethri ngqo, futhi kubandakanya ubunikazi ukuguqulwa, kuyoba kuze nambili. Okunye wesimethri oqondile Tetrahedron alikho, kodwa 12 wesimethri ephambene. Ngenxa yalokho, 24 kuphela libhekene Tetrahedron symmetries. Ngokucaca, singakha ngazo indlu imodeli we Tetrahedron njalo eyenziwe amakhadibhodi uqiniseke ukuthi umzimba weJiyomethri ayilahlekelwa wesimethri 24 kuphela.

Dodecahedron futhi icosahedron - abaseduze endaweni umzimba. Icosahedron inesibalo esikhulu sobuso, i-engeli dihedral futhi kunayo yonke kungaba uvalwe basingatha sphere olotshiwe. Dodecahedron has ephansi emincane isici engela ngobukhulu okuqinile ngesikhathi vertex. Kungaba ukukhulisa ukugcwalisa emkhakheni circumscribed.

ukuskena polyhedra

Avamile polyhedra scan, okuyinto sonke alahlane ebuntwaneni, Ziningi imiqondo. Uma kukhona iqoqo ngemapholigoni, ohlangothini lapho ngamunye kufaniswa eyodwa kuphela ohlangothini polyhedron, ukuhlonza amaqembu kumele athobele izimo ezimbili:

• ka ipholigoni ngamunye, ungakwazi ukuya ipholigoni kokuba ukuhlonza ecaleni;
• uhlangothi okukwazi ukubona kufanele abe nobude obufanayo.

Kuyinto iqoqo ngemapholigoni ezihlangabezana mibandela, futhi ibizwa ngokuthi scan polyhedron. Ngayinye yalezi emizimbeni has abaningana kubo. Ngokwesibonelo, iqhuzu lapho kukhona 11 izingcezu.