Regular polyhedra: abubuwa fasali da kuma yankin

Lissafi da kyau domin, sabanin aljabara, wanda shi ne ba ko da yaushe bayyana dalilin da abin da kuke tunani, ba a gani abu. Wannan ban mamaki duniyar daban-daban da jikin ƙawata yau da kullum polyhedra.

Janar bayani a kan yau da kullum polyhedra

Kamar yadda mutane da yawa, na yau da kullum polyhedrons, ko kamar yadda aka kira su Platonic daskararru, mallaka musamman Properties. Tare da wadannan abubuwa da alaka da dama kimiyya shiriritar. A lokacin da ka fara zuwa karatu da lissafi data na jiki, ka gane cewa kusan ba su sani ba wani abu game da irin wannan ra'ayi kamar yadda na yau da kullum polyhedra. The gabatar da wadannan abubuwa a cikin makaranta ba ko da yaushe ban sha'awa, don haka da yawa ba su ma tuna abin da suka kasance sunã kira. A ƙwaƙwalwar ajiyar da mafi yawan mutane shi ne kawai wani shigen sukari. Babu wani daga cikin jiki lissafi bai mallaki irin kamala kamar yadda na yau da kullum polyhedrons. All sunayen wadannan lissafi gawarwakin samo asali daga zamanin tsohuwar Girka. Su wakilci yawan fuskoki: da tetrahedron - hudu-gefe, hexahedron - Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. Duk wadannan na lissafi jiki bautar wani muhimmin wuri a cikin Plato ta ganewa na sararin samaniya. Hudu daga cikinsu suna aiwatuwa abubuwa ko abokai: da tetrahedron - da wuta, da icosahedron - ruwa shigen sukari - duniya, octahedron - iska. Dodecahedron aiwatuwa dukan kõme. Ya aka dauke da main, a matsayin wata alama na sararin samaniya.

The hakan na da manufar a polyhedron

Polyhedron ne mai iyaka tarin polygons irin wannan cewa:

• kowane daga cikin bangarorin da wani daga cikin polygons ne a lokaci guda kawai a gefe daya na wani polygon a kan wannan gefe.
• daga kowane daga cikin polygons za ka iya tafiya da sauran ta wucewa m cikinta polygons.

Polygons da ke cikin polyhedron wakiltar ta fuskõkinsu da gefen - hakarkarinsa. polyhedra vertices ne vertices na polygons. Idan ajali polygon fahimta lebur rufaffiyar polylines, to, ku zo wa daya definition of a polyhedron. A cikin akwati inda ta wannan lokaci da ake nufi wani ɓangare daga cikin jirgin saman da aka daure ta karya Lines, shi za a gane surface kunshi polygonal guda. Convex polyhedron ake kira jiki kwance a gefe daya daga cikin jirgin sama, dab da ta fuskoki.

Wani definition of a polyhedron da abubuwa

Polyhedron kira surface kunshi polygons, wanda ya takaita lissafi jiki. Su ne:

• ba-convex.
• convex (daidai da daidai ba).

Regular polyhedron - shi ne mai convex polyhedron tare da iyaka fasali. Abubuwa na yau da kullum da polyhedra:

• Tetrahedron: 6 hakarkarinsa 4 fuskoki 5 vertices.
• hexahedron (shigen sukari) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

Euler ta Theorem

Yana tabbatar da dangantaka tsakanin yawan gefuna, vertices da fuskoki ne topologically daidai da wata Sphere. Ƙara yawan vertices da fuskoki (B + D) da daban-daban na yau da kullum polyhedra da kuma gwada su, da yawan hakarkarinsa, yana yiwuwa su kafa wata doka: jimlar yawan fuskoki daidai da adadin vertices da gefuna (P) ya karu da 2. Yana yiwuwa a samu wani sauki dabara:

• B + D = P + 2.

Wannan dabara yana aiki ga duk convex polyhedra.

asali ma'anar

A ra'ayi na yau da kullum polyhedron shi yiwuwa a bayyana a cikin daya jumla. Shi ne mafi daraja da girma. A jikin za a gane kamar yadda irin wannan, shi ne cewa wajibi ne ta hadu da wani yawan fassarorin. Saboda haka, a lissafi jiki zai zama yau da kullum polyhedron lokacin da wadannan yanayi ne ya sadu da:

• shi ne convex.
• wannan adadin hakarkarinsa converges a kowane na da vertices.
• duk fuskoki dabam-dabam na - na yau da kullum polygons, daidai da juna.
• All dihedral kusassari ne daidai.

Properties na yau da kullum da polyhedra

Akwai 5 daban-daban na yau da kullum da polyhedra:

1. Cube (hexahedron) - yana da wani lebur koli kwana ne 90 °. Yana yana da wani 3-gefe kwana. Adadin fuska kusassari a koli na 270 °.
2. Tetrahedron - lebur koli kwana na - 60 °. Yana yana da wani 3-gefe kwana. Adadin fuska kusassari a koli - 180 °.
3. Octahedron - lebur koli kwana na - 60 °. Shi yana da hudu-gefe kwana. Adadin fuska kusassari a koli - 240 °.
4. Dodecahedron - wani lebur koli kwana na 108 °. Yana yana da wani 3-gefe kwana. Adadin fuska kusassari a koli - 324 °.
5. Icosahedron - shi yana mai lebur koli kwana na - 60 °. Yana yana da wani biyar-gefe kwana. Adadin fuska kusassari a koli na 300 °.

A yankin na yau da kullum polyhedra

A surface yankin na geometrical jikuna (S) da aka lasafta a matsayin na yau da kullum polygon yanki yawaita da yawan fuskoki dabam-dabam (G):

• S = (a: 2) x 2G ctg π / p.

The girma na yau da kullum polyhedron

Wannan darajar da aka lasafta ta halitta da girma na yau da kullum da dala wanda tushe ne na yau da kullum polygon, yawan fuskoki, da kuma rufinta ne rubũtacce radius daga cikin Sphere (r):

• V = 1: 3rS.

Kundin na yau da kullum da polyhedra

Kamar wani lissafi m, na yau da kullum polyhedra da daban-daban kundin. Kasa ne dabarbari da abin da suka iya lissafta:

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

Abubuwa na yau da kullum da polyhedra

Hexahedron da octahedron ne dual lissafi jikinsu. A wasu kalmomin, da suka iya samun fita daga juna a cikin taron cewa centroid na daya da aka dauka a matsayin saman da sauran, kuma mataimakin versa. Har ila yau ne dual icosahedron da dodecahedron. Kansa kawai tetrahedron ne dual. Bisa ga Hanyar Euclid za a iya samu daga wani dodecahedron hexahedron ta gina "rufi" a kan fuskokin da shigen sukari. A vertices na tetrahedron ne wani 4 vertices na shigen sukari, ba m da nau'i-nau'i tare gefen. Daga hexahedron (shigen sukari) za a iya samu, da kuma sauran yau da kullum polyhedra. Duk da cewa yau da kullum polygons akwai m, na yau da kullum polyhedra, akwai kawai 5.

A radii na yau da kullum da polygons

Tare da kowanne daga cikin wadannan na lissafi jikin su ne alaka concentric duniyoyin 3:

• bayyana wucewa ta cikin vertices.
• rubũtacce game da kowane daga ta fuskoki a tsakiyar shi.
• tsakãtsaki ce a game da duk da gefuna a tsakiyar.

A radius daga cikin Sphere aka bayyana da wadannan dabara da aka lasafta:

• R = wani: 2 x KU π / g x KU θ: 2.

A radius daga cikin rubũtacce Sphere da aka lasafta kamar haka:

• R = wani: 2 x ctg π / p x KU θ: 2,

inda θ - dihedral kwana wanda yake tsakanin m fuskoki dabam-dabam.

A tsakãtsaki radius daga cikin Sphere za a iya lasafta ta yin amfani da wadannan dabara:

• ρ = wani cos π / p: 2 zunubi π / h,

inda h = girma da 4.6, 6.10, ko 10. A rabo daga cikin radii na rubũtacce bayyana da kuma symmetrically game da p da q. Yana da aka lasafta kamar haka:

• R / r = KU π / p x KU π / q.

The fasali na polyhedra

The fasali na yau da kullum da polyhedra ne na farko da sha'awa ga wadannan lissafi jikinsu. An fahimci matsayin wani motsi na jiki a sarari, wanda ya fita da wannan adadin vertices, fuskokin da kuma gefuna. A wasu kalmomin, a ƙarƙashin rinjayar fasali rikirkida baki, kokuwa, ko fuska rike ta asali matsayi, ko motsa zuwa gida matsayi na wani haƙarƙari, da sauran vertices ko fuskoki.

Abubuwa na fasali na yau da kullum da polyhedra ne ga kowa da kowa iri na lissafi daskararru. Ga yadda aka gudanar a kan ainihi canji, wanda ya bar wani abu daga cikin maki a asali matsayi. Saboda haka, a lokacin da ka kunna polygonal Prism iya samun wasu symmetries. Duk wani daga cikinsu za a iya wakilta a matsayin samfurin da tunani. Fasali, wanda yake shi ne samfurin na da wani ko da yawan tunani, da ake kira kai tsaye. Idan shi ne samfurin na wani m yawan tunani, to, shi ne ake kira feedback. Saboda haka, duk da jũya a kusa da layin wakiltar mike fasali. Wani tunani polyhedron - ne kishiya fasali.

Don fahimtar fasali abubuwa na yau da kullum da polyhedra, za ka iya yi da misali na tetrahedron. Wani layi cewa zai ratsa ta daya daga cikin vertices da cibiyar da lissafi siffar, zai faru, da kuma ta hanyar tsakiyar gefen daura da shi. Kowace daga cikin jũya 120 da kuma 240 ° a kusa da layin nasa ne da jam'i tetrahedral fasali. Tun da shi 4 vertices da fuskoki, mun samu a total na takwas kai tsaye symmetries. Duk wani daga cikin Lines wucewa ta cikin tsakiyar gefuna da cibiyar na jiki, shi ya wuce ta tsakiyar m baki. Duk wani juyawa na 180 °, da ake kira mai rabin-juya a mike fasali. Tun da tetrahedron yana da uku nau'i-nau'i daga hakarkarinsa, ka samu uku Lines na fasali. Bisa ga sama, za mu iya kammala da cewa, jimlar yawan kai tsaye fasali, da kuma ciki har da ainihi canji, zai kasance har zuwa goma sha biyu. Sauran kai tsaye fasali tetrahedron ba ya wanzu, amma shi yana da 12 kishiya fasali. Saboda haka, kawai 24 halin tetrahedron symmetries. Domin tsabta, ba za mu iya gina wata model na yau da kullum tetrahedron sanya kwali da kuma tabbatar da shi ne lissafi jiki sosai yana da kawai 24 fasali.

Dodecahedron da icosahedron - mafi kusa ga jiki yankin. Icosahedron na da most yawan fuskoki, da dihedral kwana da kuma mafi yawansu duk iya tam jingina zuwa ga rubũtacce Sphere. Dodecahedron yana da mafi ƙasƙanci kusurwa aibi most m kwana a kokuwa. Yana iya kara cika a cikin circumscribed Sphere.

scanning polyhedra

Regular polyhedra scan, wanda dukan mu muna makale tare a cikin shimfiɗar jariri, da yawa Concepts. Idan akwai wani sa na polygons, kowane gefe na wanda aka gano tare da kawai a gefe daya daga cikin polyhedron, da ganewa daga cikin jam'iyyun dole ne bi da yanayi biyu:

• na kowane polygon, za ka iya zuwa wani polygon da ciwon da katin shaida na gefe.
• tabbatarwa gefen kamata da wannan tsawon.

Shi ne mai sa na polygons cewa hadu wadannan yanayi, da ake kira mai polyhedron scan. Kowace daga cikin wadannan gawarwaki da dama daga gare su. Alal misali, wani shigen sukari wanda akwai 11 guda.