Ezimacala rhoqo: izinto wolingano-ndawo

Geometry sihle ngenxa yokuba, ngokungafaniyo aljebra, nto leyo akusoloko kucacile ukuba kutheni nento ocinga, inika into olubonakalayo. Le ihlabathi elimangalisayo le kwimibutho eyahlukeneyo ahombise ezimacala rhoqo.

ulwazi ngokubanzi malunga ezimacala rhoqo

Ngokutsho abaninzi, polyhedrons rhoqo, okanye njengoko ibizwa eziqinileyo zikaPlato, ilifa iimpawu ezizodwa. Ngezi zinto enxulumene yeengcinga eziliqela zenzululwazi. Xa uqala ukufunda idata zejiyometri lomzimba, ufumanisa ukuba phantse abazi nto malunga ingqiqo njengoko ezimacala ehlelelekile. Le ntetho kwezi zinto esikolweni akusoloko umdla, ngoko abaninzi nkqu ukukhumbula musa ukuba nabizelwa ke. In memory of abantu abaninzi ukuba nje ityhubhu. Akukho namnye wew umzimba angenayo imfezeko ezifana polyhedrons rhoqo. Onke amagama ezi imizimba zejiyometri nesuka Greece yamandulo. Zimele inani ubuso: i tetrahedron --ezine shape-, hexahedron - Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. Zonke ezi umzimba zejometri libambe indawo ebalulekileyo uPlato lwaluzibona iphela. Abane kubo ehlanganiswe elements okanye amaqumrhu: i tetrahedron - umlilo, lo icosahedron - cube amanzi - umhlaba, octahedron - moya. Dodecahedron kuthwelwe zonke izinto. Waye ingqalelo ephambili, njengoko njengomfuziselo iphela.

Le gabalala ngayo ingqiqo emacala

Emacala yingqokelela anesiphelo yeepolygons ezinjalo:

• ngalinye emacaleni nayiphi na iimilo ke ngelo xesha elinye icala kuphela elinye yepholigoni kwicala efanayo;
• ngalinye iimilo unako ukuhamba ukuya kwezinye ngokugqithisa kufutshane noko maninzi.

Eneepoligoni okumisele Ipolihedron zimela ubuso bayo kwicala labo - ezimbanjeni. eziphezulu ezimacala ezi eziphezulu zepolygons. Ukuba polygon elithi ukuqonda polylines tyaba uvaliwe, ngoko uze kwenye inkcazo emacala. Kwimeko apho ngalo gama kwakuthetha yinxalenye moya lowo lingqongwe imigca baphuke, kuya kuqondwa umphezulu eziquka ngamaqhekeza polygonal. Ipolihedron khaxa kuthiwa umzimba elele kwelinye icala moya, kufutshane ubuso bayo.

Enye inkcazo emacala kunye nezinto zalo

Emacala wabiza indawo equka weepholigoni, enqamla umzimba yemigca. Zezi:

• non-khaxa;
• khaxa (elungileyo nengalunganga).

Ipolihedron rhoqo - yinto Ipolihedron khaxa kunye macala nyanga ezinintsi. Elements of ezimacala rhoqo:

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

theorem Euler kaThixo

Useka ubudlelwane phakathi kwenani leziphelo, eziphezulu kunye Ubuso topologically elilingana sphere. Ukongeza inani eziphezulu nobuso (B + D) ube ezimacala ezahlukeneyo rhoqo baze bathelekise nabo inani iimbambo, unako ukubeka ulawulo omnye: isixa inani ubuso lilingane nenani eziphezulu kunye nokuphela kwegqabi (P) inyuke nge 2. Kungenzeka ukuba kuveliswe indlela yokubala elula:

• B + D = P + 2.

Le fomyula sisebenza kubo bonke ezimacala khaxa.

iinkcazelo esisiseko

Ingcamango emacala rhoqo akunakwenzeka ukuchaza kwisivakalisi mnye. Kuyinto iyathandwa ngakumbi kunye nomthamo. A umzimba ukuba owaziwa ngolo hlobo, azinakuba zingezi ikhawulelana iqela iinkcazelo. Ngenxa yoko, umzimba zejiyometri iya kuba Ipolihedron rhoqo xa ezi meko ithotyelwe:

• kuba khaxa;
• inani elifanayo iimbambo converges kwi nganye eziphezulu yayo;
• zonke iinkalo yakhe - yeepoligoni rhoqo, lingana omnye komnye;
• Zonke angles dihedral bayalingana.

Iimpawu ezimacala rhoqo

Kukho 5 iintlobo ezahlukeneyo ezimacala rhoqo:

1. Cube (hexahedron) - it has a angle tyaba enkcochoyini yi ° 90. It has a engile-3 kabini. Isixa-mali ubuso angles njengezona ° 270.
2. Tetrahedron - flat engile njengezona - ° 60. It has a engile-3 kabini. Isixa-mali ubuso angles njengezona - 180 °.
3. Octahedron - flat engile njengezona - ° 60. It has a engile-ezine macala onke. Isixa-mali ubuso angles njengezona - 240 °.
4. Dodecahedron - a engile ezimcaba njengezona 108 °. It has a engile-3 kabini. Isixa-mali ubuso angles njengezona - 324 °.
5. Icosahedron - isiphathele engile ezimcaba njengezona - ° 60. It has a engile-kabini ezintlanu. Isixa-mali ubuso angles njengezona ° 300.

Ummandla ezimacala rhoqo

Ummandla ubuso imizimba zejometri (S) zibalwe indawo yipolygon liphindaphindwe ngenani facets (G):

• S = (a: 2) x 2G ctg π / p.

Umthamo we emacala rhoqo

Eli xabiso ibalwa phinda umthamo yephiramidi rhoqo ogama isiseko yi yipolygon, inani ubuso, ukuphakama kwayo radius ebhalwe kwisigaba (r):

• V = 1: 3rS.

Komthamo ezimacala rhoqo

Njengazo naziphi na ezinye zejiyometri eqinileyo, ezimacala rhoqo kufuneka imiqulu ezahlukeneyo. Ngezantsi kukho ifomula 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.

Elements of ezimacala rhoqo

Hexahedron kunye octahedron imizimba dual yemigca. Ngamanye amazwi, ukuze siphume kwenye nenye xa kuthe ukuba centroid ka omnye amkelwe njengoko phezulu komnye, vice versa. Kwakhona ezi icosahedron ezimbini kunye dodecahedron. Yena tetrahedron kuphela ezimbini. Ngokutsho indlela Euclid zingafumaneka ukusuka hexahedron dodecahedron ngokuthi ukwakha 'uphahla' kwi ubuso ityhubhu. Le eziphezulu ze-tetrahedron na eziphezulu 4 ityhubhu, hayi ngababini ezikufutshane ecaleni buthelezi. Ukusuka hexahedron (cube) zingafumaneka, kunye nezinye ezimacala rhoqo. Ngaphandle kokuba yeepoligoni kukho zingenakubalwa, ezimacala rhoqo, kukho 5 kuphela.

Le radii yeepoligoni

Ngasinye kwezi imizimba zejiyometri iintlobo ezidityanisiweyo concentric 3:

• wachaza edlula eziphezulu;
• sibhalwe malunga ngalinye ubuso bayo phakathi kwayo;
• udibaniso zonke kungqameko phakathi.

Embindini kwisigaba ochazwe yi le fomyula ilandelayo ibalwa:

• R = a: 2 x kubatshaba- π / g x kubatshaba- θ: 2.

Embindini kwisigaba sibhalwe ibalwa ngolu hlobo lulandelayo:

• R = a: 2 x ctg π / p x kubatshaba- θ: 2,

apho θ - angle dihedral eliphakathi facets ezikufutshane.

Embindini median kwisigaba kungabalelwa usebenzisa ifomula ilandelayo:

• ρ = a cos π / p: π 2 isono / h;

apho h = ubukhulu 4.6, 6.10, okanye 10. Umlinganiselo radii le kubhalwe echazwe kwaye symmetrically ngokuphathelele p kunye q. Oku kubalwa ngale ndlela ilandelayo:

• R / r = kubatshaba- π / p x kubatshaba- π / q.

Le lohlukanisa ezinamacala amaninzi

Le lohlukanisa le ezimacala rhoqo umdla iphambili kwezi imizimba yemigca. It kuqondwa intshukumo yomzimba kwisithuba, nto leyo eshiya inani elifanayo eziphezulu, ubuso kunye nokuphela kwegqabi. Ngamanye amazwi, phantsi kwempembelelo wolingano utshintshwano buthelezi, enekona, okanye ubuso igcina kwindawo yaso yesiqhelo, okanye ihamba kwisikhundla kwikhaya lomnye ubambo, ezinye eziphezulu okanye ubuso.

Elements of yolingano le ezimacala rhoqo ziqhelekile zonke iintlobo eziqinileyo yemigca. Apha ke lwenziwa kwi inguqu ungubani, nto leyo ishiya naziphi na iingongoma kwindawo yokuqala. Ngoko ke, xa uvula le bemilo polygonal Ungafumana symmetries. Nayiphi na kubo zimelwe njengoko imveliso ye reflection. Symmetry, nto leyo ke imveliso ngengozi inani ukukhanya, ebizwa ngqo. Ukuba isisiphumo inani elingumnqakathi zezinto, ngoko ibizwa ngokuba ingxelo. Ngenxa yoko, onke amathuba ngeenxa umgca zimele wolingano- ngqo. Nayiphi na imilo emacala umfanekiso - ke isimetri nguqu.

Ukuze siqonde ngcono iziqalelo wolingano le ezimacala rhoqo, ungathatha mzekelo tetrahedron. Nayiphi line ukuba uya kucanda omnye eziphezulu kunye embindini imilo zejiyometri, kuya kwenzeka, yaye embindini weli cala malunga kuyo. Ngalinye Akubuya 120 no 240 ° ehlabathini umgca bobabo ukuya isimetri isininzi tetrahedral. Ekubeni eziphezulu 4 ubuso, sifumana isimbuku symmetries ngqo ezisibhozo. Naluphi na imigca badlula phakathi kungqameko kunye nombindi umzimba, udlula i kumbindi weli cala esahlukileyo. Nayiphi ukujikeleza 180 °, ebizwa enesiqingatha-turn ezijikeleze macala ngqo. Ekubeni tetrahedron iye ngababini ezintathu iimbambo, ufumana imigca ezintathu macala. Ngenxa yoku kungasentla, sinokugqiba ukuba inani elipheleleyo macala ngqo, kwaye kuquka inguqu yesazisi, uya kuba phezulu ukuya kwezilishumi elinambini. Okunye lohlukanisa ngqo tetrahedron ayikho, kodwa 12 ulingano nguqu. Ngenxa yoko, 24 kuphela ebonakala symmetries tetrahedron. Ecacileyo, singakwazi ukwakha umfuziselo tetrahedron rhoqo eyenziwe ngekhadibhodi kwaye uqinisekise ukuba umzimba zejometri nempela ulingano-24 kuphela.

Dodecahedron kunye icosahedron - ikufutshane kwindawo emzimbeni. Icosahedron na nani likhulu ubuso, i-engile dihedral kwaye uninzi bonke ngokuqinileyo unamathele kwinqanaba sibhalwe. Dodecahedron na emboxo isiphako esikhulu engile asezantsi eqinileyo kule enekona. Ku nga nkulu ugcwalise kwinqanaba nezisekelwe.

ukuskena ezimacala

Rhoqo scan ezimacala, apho thina sonke wanamathela ndawonye ebuntwaneni, ezininzi iingqiqo. Ukuba kukho iseti yeepolygons, kwicala ngalinye apho bunxulunyaniswa elinye icala kuphela Ipolihedron, ukuphawulwa amaqela kufuneka athobele imiqathango ezimbini:

• of polygon nganye, yiya ku yepholigoni ukuba ukuqatshelwa icala;
• icala olubonakalayo kufuneka ubude efanayo.

It is a iseti yeepolygons ezidibana ezi meko, yaye kuthiwa scan emacala. Nganye kwezi uye eziliqela kubo. Umzekelo, ukuba amagaqa apho kukho ngamaceba-11.