Kugara polyhedra: zvinhu symmetry uye nharaunda

Geometry akanaka nokuti kusiyana nemasvomhu, zvisina pachena nguva dzose sei uye chii unofunga, anopa kuona chinhu. Ino miviri siyana zvinoshamisa vashongedze dzose polyhedra.

General mashoko nokugara polyhedra

Maererano zhinji, dzose polyhedrons, kana sezvo dzinonzi Platonic nezvishoma, nhaka nemamwe ehupfumi. With izvi zvinhu chokuita hypotheses akawanda yesayenzi. Paunotanga kudzidza geometric Data muviri, iwe unoziva kuti zvinenge havazivi chinhu kwakadaro pfungwa sezvo dzose polyhedra pamusoro. The mharidzo izvi zvinhu muzvikoro nguva inonakidza, saka vakawanda havatombobvumi kurangarira zvavakanga makadanirwa. Pakuyeuka vanhu vakawanda rinongova kuutio. Hapana muviri geometry asingadi nhaka zvakadai zvakakwana sezvo dzose polyhedrons. All mazita geometric mitumbi izvi rakabva Greece yekare. Dzinomirira nhamba zviso: ari tetrahedron - mativi mana, hexahedron - Allen, octahedron - Zimbabwe Educational, dodecahedron - dodecahedral, icosahedron - icosahedral. Vose muviri izvi geometric kune nzvimbo inokosha hwaPlato pamuviri zvakasikwa. Four vavo mudzidziso zvinhu kana masangano: kuti tetrahedron - kumoto icosahedron - water kuutio - pasi, octahedron - mhepo. Dodecahedron mudzidziso zvose zvinhu. Aionekwa huru, sechiratidzo zvakasikwa.

The zvavo pamusoro pfungwa imwe polyhedron

Polyhedron ari finite wemabhuku polygons zvokuti:

• mumwe nomumwe kumativi chero polygons riri panguva imwe chete rumwe rutivi nomumwe polygon kudivi rimwe;
• kubva nomumwe polygons unogona kufamba kune dzimwe nokuisa pedyo pawo polygons.

Polygons vanoumba ari polyhedron zvinomiririra zviso ayo uye zvavo divi - mbabvu. polyhedra vertices ndiwo vertices ose polygons. Kana rokuti polygon kunzwisisa sandara namahwindo polylines, chiuyai mumwe tsanangudzo polyhedron. Kana apo iri izwi iri raireva chikamu ndege iri rakatenderedzwa kuputswa mitsetse, zvichava kunzwisiswa pamusoro zvinosanganisira polygonal zvidimbu. Convex polyhedron anonzi muviri avete rumwe ndege, yakabatana zviso ayo.

Imwe tsanangudzo polyhedron nezvinhu zvaro

Polyhedron akadana pamusoro zvinosanganisira polygons, iyo ichikutadzisa ari geometric muviri. Ivo:

• asiri convex;
• convex (nezvakaipa).

Kugara polyhedron - munhu convex polyhedron pamwe maximal symmetry. Zvinhu dzose polyhedra:

• Tetrahedron: 6 mbabvu 4 zviso 5 vertices;
• hexahedron (kuutio) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

Euler kuti theorem no

It anovamba ukama pakati nhamba micheto, vertices uye zviso vari topologically yakaenzana rimwe bhora. VaGumbo uwandu vertices uye zviso (B + D) vane zvakasiyana dzose polyhedra uye kuenzanisa navo nhamba yembabvu, zvinokwanisika kuti kuisa chimwe ushe: Verengai nhamba zviso zvakaenzana nokuwanda vertices uye micheto (P) yakawedzera 2. Zvinobvira zvinowana iri nyore inonyatsoita:

• B + D = P + 2.

mutoo uyu chinoshanda vose convex polyhedra.

zvinhu tsanangudzo

Pfungwa nguva polyhedron zvisingaiti kutsanangura pamutsara mumwe. Inonyanya vanokosheswa uye vhoriyamu. A muviri kuonekwa sezvo vakadaro kuti asangana dzinoverengeka tsanangudzo, zvakakodzera. Saka, munhu geometric muviri richava dzose polyhedron kana izvi zvinhu zviri akasangana:

• zviri convex;
• chete nhamba mbabvu converges pane mumwe nomumwe vertices dzayo;
• zvose zviri zvake - dzose polygons, akaenzana nomumwe;
• dihedral pamakona All vakaenzana.

Properties dzinogara polyhedra

Kune 5 marudzi akasiyana dzose polyhedra:

1. Cube (hexahedron) - ine sandara chakaderedzwa mutengo angled 90 °. It ane 3-mativi angled. Amount chiso pamakona panguva chakaderedzwa mutengo kuti 270 °.
2. Tetrahedron - sandara chakaderedzwa mutengo angled pamusoro - 60 °. It ane 3-mativi angled. Amount chiso pamakona panguva chakaderedzwa mutengo - 180 °.
3. Octahedron - sandara chakaderedzwa mutengo angled pamusoro - 60 °. It ane mativi mana angled. Amount chiso pamakona panguva chakaderedzwa mutengo - 240 °.
4. Dodecahedron - a sandara chakaderedzwa mutengo pakona rimwe 108 °. It ane 3-mativi angled. Amount chiso pamakona panguva chakaderedzwa mutengo - 324 °.
5. Icosahedron - ine sandara chakaderedzwa mutengo angled pamusoro - 60 °. It ane shanu-mativi angled. Amount chiso pamakona panguva chakaderedzwa mutengo kuti 300 °.

The nharaunda dzose polyhedra

The pose nzvimbo geometrical mitumbi (S) rinowanikwa ari polygon dzose nzvimbo kane nokuwanda kunobatanidza (G):

• S = (a: 2) x 2G ctg π / p.

Bhuku nguva polyhedron

ukoshi ichi rinowanikwa nevanhu vachiwedzera bhuku nguva chepiramidhi ane garo iri nguva polygon, uwandu zviso, uye kukwirira kwayo ndiye akanyorwa nharaunda yacho nzvimbo (R):

• V = 1: 3rS.

ZvemaBhaibheri dzose polyhedra

Somumwe geometric zvikukutu, nguva polyhedra vane bhuku zvakasiyana. Pazasi pane zvokuimba nawo vanogona overenga:

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

Zvinhu dzose polyhedra

Hexahedron uye octahedron vari hunyambiri geometric miviri. Nemamwe mashoko, vanogona kubuda mumwe ari chiitiko kuti centroid mumwe achatorwa sezvo pamusoro pomumwe, uye vakutarisire. Uyewo vari Dual icosahedron uye dodecahedron. Pachake chete tetrahedron ndiye Dual. Maererano nzira Euclid kunogona kuwanikwa kubva dodecahedron hexahedron ne rokuvaka "matenga" riri zviso kuutio. The vertices of tetrahedron vari chero 4 vertices ari kuutio, kwete pedyo maviri pamwe mupendero. Kubva hexahedron (kuutio) kunogona kuwanikwa, uye dzimwe nguva polyhedra. Pasinei chokwadi kuti nguva polygons pane dzisingaverengeki, nguva dzose polyhedra, pane 5 chete.

The radii nguva dzose polygons

With nerimwe geometric mitumbi aya kunofambidzana concentric mabhora 3:

• akatsanangura achipfuura vertices;
• yakanyorwa pamusoro mumwe nomumwe zviso yaro pakati payo;
• evaMedhia pamusoro micheto pakati.

The nharaunda yacho bhora inotsanangurwa mutoo rinowanikwa:

• R = munhu: 2 X nkm π / d X nkm θ: 2.

The nharaunda yacho yakanyorwa bhora rinowanikwa sezvinotevera:

• R = munhu: 2 X ctg π / p X nkm θ: 2,

apo θ - dihedral angled riri pakati pedyo zvabatanidzwa.

The evaMedhia nharaunda yacho nzvimbo zvinogona rinowanikwa nokushandisa mutoo:

• ρ = mumwe Kosi π / p: 2 chivi π / h,

apo H = kwakakura 4,6, 6.10, kana 10. The reshiyo pamusoro radii of yakanyorwa rakarondedzerwa uye symmetrically pakureva peji uye q. It rinowanikwa sezvinotevera:

• R / R = c nkm π / p X nkm π / q.

The symmetry pamusoro polyhedra

The symmetry of dzose polyhedra iri chikuru kufarira kune geometric miviri iyi. Zvinonzwisiswa sezvo sangano muviri ari muchadenga, iyo inosiya chete nhamba vertices, zviso uye kumicheto. Nemamwe mashoko, achifuridzirwa symmetry transformations unopinza, vertex, kana chiso anoramba nzvimbo yaro yepakutanga, kana inofamba kumba nzvimbo imwe rumbabvu, vamwe vertices kana zviso.

Zvinhu symmetry of dzose polyhedra ndezvemamiriro mhando dzose geometric nezvishoma. Pano zviri kuitwa pamusoro nokuzivikanwa Kushandurwa, iyo inosiya mumwe pfungwa chepakutanga chinzvimbo. Saka, kana iwe Vanodzinga polygonal prism anogona kuwana mamwe symmetries. Chero navo zvinogona achimiririrwa kungoerekana Kufungisisa. Symmetry, inova chigadzirwa ane kunyange nhamba pfungwa, inonzi zvakananga. Kana ari chigadzirwa ane vasinganzwisisike nhamba pfungwa, zvino inonzi dzinovaka. Saka, vose ritenderere mutsetse zvinomiririra ngaatarire symmetry. Chero kufungisisa polyhedron - ndiro Ivan Kuchin symmetry.

Kuti tinzwisise symmetry zvinhu dzose polyhedra nani, unogona kutora muenzaniso tetrahedron. Chero mutsetse kuti achapfuura napakati mumwe vertices uye pakati geometric chimiro, kuchaitika, uye kuburikidza nechepakati pamupendero pakatarisana naro. Imwe neimwe majana 120 uye 240 ° akapoteredza mutsetse ndohwaJehovha muuzhinji tetrahedral symmetry. Sezvo 4 vertices uye zviso tinowana dzinokwana sere yakananga symmetries. Chero pemitsara achipfuura nepakati kumicheto uye pakati muviri, unopfuura nepakati nevakomana mucheto. Chero vachishandisa kutenderera 180 °, akadana nehafu-votendeuka achirurama symmetry. Sezvo tetrahedron ane maviri matatu yembabvu, wawana misara mitatu symmetry. Inobva kumusoro, tinogona kugumisa kuti uwandu yakananga symmetry, uye kusanganisira zvinocherechedzesa Kushandurwa, achava kusvika gumi. Other zvakananga symmetry tetrahedron hakuwaniki, asi ine 12 Ivan Kuchin symmetry. Somugumisiro, 24 chete inozivikanwa tetrahedron symmetries. Kuti zvijeke, tinogona kuvaka muenzaniso nguva tetrahedron yakagadzirwa makadhibhokisi uye nechokwadi ndicho geometric muviri chaizvoizvo ane 24 chete symmetry.

Dodecahedron uye icosahedron - pedyo nzvimbo muviri. Icosahedron ane Nhamba huru zviso, ari dihedral angled uye kupfuura zvose zvinogona zvakasimba kuomerera rakanyorwa bhora. Dodecahedron ane yakadereresa chisina kutsetseka urema rakakura chakasimba pakona panguva vertex. Zvinogona maximize kuti zadza circumscribed bhora.

kushanda pakuvheneka polyhedra

Kugara polyhedra Ongororo, izvo isu tose vakaramba vakabatana pauduku, vane yakawanda pfungwa. Kana paine mamwe polygons, mumwe kurutivi iyo yaizivikanwa chete rumwe polyhedron, ari kuziviswa vemapato vanofanira kutevedza maviri ezvinhu:

• mumwe polygon, unogona kuenda kune mumwe polygon ane kuziva kurutivi;
• kuvezwa divi vanofanira zvakafanana kureba.

Zviri chine polygons kuti asangane izvi zvinhu, uye inonzi polyhedron scan. Mumwe nemumwe muviri izvi ane zvakawanda kwavari. Somuenzaniso, imwe kuutio izvo pane zvidimbu 11.