Kamehla polyhedra: likarolo setshwani le sebakeng seo

Lipalo, geometry o motle hobane, ho fapana le aljibra, e hase kamehla ho hlakile hore na ke hobane'ng le seo u se nahana, fana ka ntho e itseng e bonwang. lefatšeng lena e babatsehang ea 'mele e sa tšoaneng khabisa polyhedra kamehla.

Kakaretso tlhahisoleseding e mabapi le polyhedra kamehla

Ho ea ka batho ba bangata, polyhedrons oa kamehla, kapa ha ba ntse ba ba bitsoa ba ise ba nyalane butle, rua thepa e ikhethang. Le dintho tsena tse amanang dikgakanyo 'maloa tsa saense. Ha u qala ho ithuta ya data thutatekanyo ea 'mele, u hlokomela hore hoo e ka bang ha ba tsebe letho ka khopolo e joalo e le polyhedra kamehla. The nehelano lintho tsena sekolong hase kamehla ho thahasellisang, kahoo batho ba bangata ha ba hopola seo ba neng ba bitsoang kateng. Ho ikhopotsa batho ba bangata ba bona ho leng feela sekotwaneng. Le ea mong ho le geometry 'mele ha e rua phetheho kang polyhedrons kamehla. mabitso kaofela lihloliloeng tsena tsa thutatekanyo simoloha ho tloha Greece ea boholo-holo. Li emela palo ea lifahleho tsa: ho tetraheterone - mane tsehelitseng, hexahedron - Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. Kaofela ha tsena 'mele thutatekanyo nka sebaka sa bohlokoa ka ho emoloa Plato o oa bokahohle. Ba bane ba bona ba akarelelitsoeng ho likarolo kapa mekhatlo: ho tetraheterone - mollo, le icosahedron - metsi sekotwaneng - lefatše, octahedron - moea. Dodecahedron akarelelitsoeng ho lintho tsohle. O ne a nkoa e le e khōlō, e le letšoao la bokahohleng.

The generalization tsa khopolo ea polyhedron ka

Polyhedron ke pokello e lekanyelitsoeng ea dikhutlontsi joalo hore:

• e mong le e ba mahlakoreng a leha e le efe ea dikhutlontsi ke ka nako e tšoanang lehlakoreng le 'ngoe feela ea khutlontsi mong ka lehlakoreng le e tšoanang;
• le e mong le ba dikhutlontsi o ka tsamaea ho ea ba bang ka ho fetisa haufi tlhokeng dikhutlontsi.

Dikhutlontsi molao, polyhedron emela lifahleho tsa lona le lehlakoreng la bona ba - likhopo. polyhedra divertise ke divertise tsa dikhutlontsi. Haeba poleloana e reng khutlontsi utloisisa a bataletseng polylines koetsoeng, ka nako eo tla ho tlhaloso e 'ngoe ea polyhedron ka. Tabeng ea moo ke lentsoe lena e ho ne ho bolela e le karolo ea sefofane hore e potapotoeng ke mela e robehile, o tla utloisisa hore holim'a metsi e entsoeng ka likotoana polygonal. Khonvekse polyhedron o bitsoa 'mele ho bua leshano ka lehlakoreng le leng la sefofane, haufi le lifahleho tsa lona.

E mong tlhaloso ya polyhedron le likarolo tsa lona

Polyhedron bitsoa holim entsoeng ka dikhutlontsi, e lekanyetsang mmele thutatekanyo. Ba:

• bao e seng khonvekse;
• khonvekse (nepahetseng le se fosahetseng).

Kamehla polyhedron - ke khonvekse polyhedron le ea pheletso setshwani. Metsoako ea polyhedra kamehla:

• Tetraheterone: 6 likhopo 4 lifahleho 5 divertise;
• hexahedron (sekotwaneng) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

Theorem Euler e

Ho theha le kamano e pakeng tsa palo ea metshetshe divertise 'me lifahleho tse topologically lekana le lefapheng lefe kapa lefe e le. Phaella ka ho re palo ea divertise 'me lifahleho (B + D) na fapaneng polyhedra kamehla le ho bapisa ba le palo ea likhopo, ho ka etsahala ho beha puso ea' ngoe: Balang ya nomoro sa lifahleho tsa lekana le palo ea divertise le metshetshe (P) eketseha ka 2. Ho ka etsahala hore ba e fumanang le moralo o balletsweng e bonolo:

• B + D = P + 2.

moralo ena ke utloahalang a hore polyhedra tsohle khonvekse.

litlhaloso mantlha

Khopolo ea ho polyhedron kamehla ke ha khoneha ho hlalosa kateng polelong e 'ngoe. Ho batla ho le bohlokoa le bophahamo ba modumo. 'mele e lokela ho nkoa e le joalo, ho ke ke ho hlokahala hore ho kopana' maloa a litlhaloso. Kahoo, e le 'mele tsa thutatekanyo tla ba polyhedron kamehla ha maemo ana a ile a kopana le:

• ke khonvekse;
• palo e tšoanang ea likhopo converges ka mong le e mong divertise lona;
• litšobotsi tse kaofela hae - dikhutlontsi kamehla, lekanang mong ho e mong;
• dikhutloteng All dihedral baa lekana.

Properties tsa polyhedra kamehla

Ho na le 5 mefuta e fapaneng ya polyhedra kamehla:

1. Sekotwaneng (hexahedron) - eo na le e sephara tlhorong angle e 90 °. E na le angle 3 feela. Palo sefahleho dikhutloteng ka tlhorong ea 270 °.
2. Tetraheterone - a bataletseng tlhorong angle ya - 60 °. E na le angle 3 feela. Palo sefahleho dikhutloteng ka tlhorong - 180 °.
3. Octahedron - a bataletseng tlhorong angle ya - 60 °. E na le hlaha ka lehlakoreng le a mane feela. Palo sefahleho dikhutloteng ka tlhorong - 240 °.
4. Dodecahedron - e sephara tlhorong angle ea 108 °. E na le angle 3 feela. Palo sefahleho dikhutloteng ka tlhorong - 324 °.
5. Icosahedron - eo na le e sephara tlhorong angle ya - 60 °. E na le hlaha ka lehlakoreng le tse hlano feela. Palo sefahleho dikhutloteng ka tlhorong ea ° 300.

Sebakeng polyhedra kamehla

Sebaka holim 'a' mele le tsa thutatekanyo (S) e balwa ka kamehla khutlontsi sebakeng eketseha ka palo ea lintho tse (G):

• S = (a: 2) x 2G ctg π / leq.

Moqolo oa polyhedron kamehla

boleng ena e balwa ka ho ata molumo oa phiramiti kamehla bao botlaaseng ke khutlontsi kamehla, palo ea lifahleho, 'me bophahamo ba eona e ke radius ngotsoe ya boemong (dagelever):

• V = 1: 3rS.

Meqolo ea polyhedra kamehla

Joaloka efe kapa efe tse ling tsa thutatekanyo tiileng, polyhedra kamehla na meqolo fapaneng. Ka tlaase mona ho mekhoa eo ka eona ba ka a bale:

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

Metsoako ea polyhedra kamehla

Hexahedron le octahedron ke kopanetsoe mele thutatekanyo. Ka mantsoe a mang, ba ka 'na fumana ho tsoa e mong le e tse ling tse ka ketsahalo eo ho centroid ea mong e nkiloeng e le tlhōrōng ea ba bang,' me sekgoeng. Hape ke kopanetsoe icosahedron le dodecahedron. Eo eena ka boeena feela tetraheterone e kopanetsoe. Ho ea ka mokhoa oa ho Euclid ka fumanoa tswa dodecahedron hexahedron ke ho haha "marulelo" ka lifahleho tsa sekotwaneng sa. The divertise tsa tetraheterone ba efe kapa efe divertise 4 of sekotwaneng, eseng lipara haufi hammoho bohale le. From hexahedron (sekotwaneng) ka fumanoa, 'me tse ling tse polyhedra kamehla. Ho sa tsotellehe 'nete ea hore dikhutlontsi kamehla ho na le tse ngata haholo, kamehla polyhedra, ho na le 5 feela.

The radii tsa dikhutlontsi kamehla

Le e mong le e ba lihloliloeng tsena tsa thutatekanyo tse hokahantsoeng ke concentric Makala 3:

• o ile a hlalosa fetang divertise ea;
• ngotsoe mabapi le e mong le e sa lifahleho tsa sona ka har'a eona;
• bohare malebana le metshetshe bohle ka mahareng.

The radius ea boemong hlalositsoeng ke moralo latelang e balwa:

• R = ka: 2: x tg π / g, x, tg θ: 2.

The radius ea boemong ngotsoe e balwa ka tsela e latelang:

• R = ka: 2: x ctg π / leq: x tg θ: 2,

moo θ - dihedral angle e pakeng tsa litšobotsi tse haufi.

The radius bohare ba boemong bo ka balwa sebedisa moralo tse latelang:

• ρ = e cos π / leq: 2 e leng sebe π / H,

moo H = ho boholo ba 4,6, 6,10, kapa 10. karo-karolelano ea ho radii ya ngotsoe e hlalositswe le symmetrically mabapi le leq q. E balwa ka tsela e latelang:

• R / dagelever = tg π / leq: x tg π / Q.

The setshwani tsa polyhedra

The setshwani tsa polyhedra kamehla thahasella mathomo ho lihloliloeng tsena tsa thutatekanyo. Ho e utloisisa ka motsamao oa 'mele ka sebaka, eo siea palo e tšoanang ea divertise, lifahleho le metshetshe. Ka mantsoe a mang, ka tlas'a tšusumetso ea setshwani lengths bohale, vertex, kapa sefahleho e lule e le boemong ba eona ba pele, kapa e susumelletsang ho home boemo ba khopo e 'ngoe, ho divertise ding kapa lifahleho.

Metsoako ea setshwani tsa polyhedra kamehla ke ho tloaelehile hore ho tsohle mefuta ya-butle thutatekanyo. Mona e tshwarwa ka phetoho boitsebiso, e siea e 'ngoe ea lintlha tse ka boemo ba pele. Ho joalo, ha u reteletse porisime polygonal ka fumana ba bang ba symmetries. Leha e le efe ba bona ba ka ho hlahisoa e le sehlahisoa sa bonahatsa. Setshwani, e leng sehlahisoa sa esita le palo ea ke lipontso, tse bitsoang tobileng. Haeba ho ke ke sehlahisoa sa makatsa nomoro ya ke lipontso, ka nako eo e bitsoa poeletsotlaleho. Kahoo, ba chenchana ho pota-pota mohala emela otlolohileng setshwani. Leha e le ponahatso polyhedron - ke pitikololo setshwani.

Ho utloisisa hantle hore na boemo ba leholimo setshwani tsa polyhedra kamehla, o ka nka mohlala oa tetraheterone ena. mola ofe kapa hore o tla feta ka e nngwe ya divertise le setsi sa ea sebopeho thutatekanyo, o tla nka sebaka, 'me ka ho e le setsi sa bohale bo fapaneng ho eona. E mong le e ba chenchana 120 le 240 ° ka bophara mola ke ea ka bongateng tetrahedral setshwani. Kaha e 4 divertise 'me lifahleho, re fumana kakaretso ea robeli symmetries tobileng. E 'ngoe ea mela e fetang bohareng ba metshetshe le setsi sa ea' mele, o feta bohareng ba bohale bo fapaneng. chenchana efe kapa efe ya 180 °, e bitsoang halofo-ba sokolohe ka setshwani otlolohile. Ho tloha tetraheterone na lipara tse tharo tsa likhopo, o fumana mela e meraro ea setshwani. E Thehiloe ho tse ka holimo, re ka fihlela qeto ea hore palo yohle ya setshwani ho toba, 'me ho akarelletsa le oa ho fetola boitsebiso, e tla ba ho fihlela ho leshome le metso e. Tse ling tse ka ho toba setshwani tetraheterone ha a eo, empa e na le 12 pitikololo setshwani. Ka lebaka leo, 24 feela tšoauoa tetraheterone symmetries. Ka ho hlaka, re ka haha le mohlala tsa tetraheterone kamehla entsoe ka lebokoso le etsa bonnete ba hore ho ke ke 'mele thutatekanyo hlile e na le 24 feela setshwani.

Dodecahedron le icosahedron - haufi-ufi ho ea sebakeng sa 'mele. Icosahedron o na le palo e khōlō ka ho fetisisa sa lifahleho, e hlaha ka lehlakoreng le dihedral le ho fetisisa ho tsohle ka ka thata-thata le khomarele ho boemong ngotsoe. Dodecahedron na le tlaase-tlaase angular sekoli kholo ka ho fetisisa tiileng angle ka vertex ena. E ka e fumantsa ho tlatsa ka boemong maemo ho.

anthivaerase polyhedra

Kamehla polyhedra scan, eo kaofela ha rōna re ile a khomarela hammoho bongoaneng, ba le e ngata likhopolo tsa eona. Haeba ho na le sete ya dikhutlontsi, e mong le e-morao ea eona e amahanngoa le lehlakoreng le leng feela la polyhedron, e leng boitsebahatso ba mekga e lokela ho iphaphatha le maemo a mabeli:

• ea e mong le e khutlontsi, o ka tsamaea ho khutlontsi ho ba le boitsebiso ba lehlakoreng;
• identifiable lehlakoreng lokela ho na le nako e lekanang.

Ho sete ya dikhutlontsi hore kopana le maemo a bana, 'me e bitsoa polyhedron scan. E mong le e 'mele ea tsena e na le e' maloa ea bona. Ka mohlala, sekotwana tsa tsona ho na le likotoana tse 11.