Polyhedra. Mhando polyhedra vavo zvivako

Polyhedra kwete chete kugara nzvimbo akakurumbira pakati geometry, asiwo kuitika muupenyu hwezuva nezuva ose munhu. Tisingatauri hazvizivi ukama zvinhu zvakasiyana-siyana polygons, kutangira kubva matchbox uye kupera mapurani zvinhu zvakasikwa uyewo kuitika crystals ari muchimiro kuutio (munyu), prisms (sekiristaro), Pyramid (scheelite), octahedra (nedhaimani), etc. . d.

Pfungwa imwe polyhedron, mu geometry marudzi polyhedrons

Geometry sainzi unotora stereometry chikamu runotaura unhu uye zvivako zviri hombe zvitarisiko. Geometric muviri mativi vari achiumbwa-dimensional matatu nzvimbo rakatenderedzwa ndege (kunobatanidza) vanozivikanwa se "polytopes". Mhando polyhedra ane vanopfuura gumi nevaviri vamiririri zvakasiyana nhamba uye chimiro zviso.

Kunyange zvakadaro, polyhedra zvose dzinowanikwa ehupfumi:

1. Vose vane zvitatu chinokosha zvinoriumba: chiso (polygonal penyika), musoro (pamakona achiumbwa pasi kunobatanidza mukomboni), imwe unopinza (divi kana akatema zvitarisiko achiumbwa panguva panosangana zviso zviviri).
2. Mumwe polygon kumucheto rinobatanidza maviri, uye vaviri chete nezviso zviri ukama mumwe vari pedyo.
3. The kuzvimba zvinoreva kuti muviri hwakarongwa chose rimwe chete kurutivi Ndege iyo hunogara mumwe zviso. Mutemo unoshanda zviso zvose polyhedron. Izvi geometric Zvekare zvimiro zvemhashu mu hwakasimba geometry izwi inonzi convex polyhedra. Anosiyiwa vari stellate polyhedra izvo zvinobviswa dzose polygonal geometric miviri.

Polyhedra inogona kukamurwa:

1. Mhando convex polyhedra, zvinosanganisira zvinotevera zvidzidzo: kwakajairika kana chaiwo (a prism, chepiramidhi, bhokisi), zvakanaka (anonziwo Platonic nezvishoma), semiregular (wechipiri zita - Archimedean nezvishoma).
2. Non-convex polyhedrons (stellate).

Prism aro zvivako

Geometry sezvo kupesana geometry inodzidza ehupfumi ose-dimensional matatu zvitarisiko, mhando polyhedra (prism pakati pavo). Prism akadana muviri geometric iyo waitoda zviso zviviri akafanana (anonziwo zvigadziko) enhema yakafanana ndege, uye N-ndipo rutivi anotarisana dziri parallelograms. Uyewo prism anewo mhando dzakawanda, kusanganisira mhando dzakadai polyhedra, akadai:

1. Parallelepiped - akaumba apo pechingoro ari parallelogram - mumwe polygon necheni maviri anoshorana vakaenzana pamakona maviri yegwara akatarisana kumativi congruent.
2. Prism ndiyo perpendicular kuti micheto pechingoro.
3. The prism vachida dzinoratidzwa kunanga angled (kunze 90) pakati zviso nechigadziko.
4. Yakafanira dzinoratidzwa prism nezvigadziko muchimiro nguva dzose polygon zvakaenzana lateral kumativi.

The chikuru ehupfumi ari prism:

• Congruent zvigadziko.
• All micheto ose prism vakaenzana uye kufanana nomumwe.
• All zviso divi vane chimiro parallelogram.

piramidhi

Pyramid akadana muviri geometric kuti unotora pakadarowo uye mumwe N-Th pamusoro mativi zviso kuti batanidza pane nzvimbo imwe chete - kumusoro. Zvinofanira kucherechedzwa kuti kana kurutivi zviso chepiramidhi vanomirirwa zvokuridza kunodiwa, ipapo nechigadziko kunogona kuva akafanana mativi polygon kana quadrilateral uye pentagonal, uye zvichingodaro vaGiriyedhi infinitum. Panyaya iyi, zita chepiramidhi unoenderana polygon nechepasi. Somuenzaniso, kana pechingoro iri netriangle Pyramid - munhu mativi Pyramid, quadrilateral - quadrangular, etc ...

Pyramids - nayo konusopodobnye polyhedra. Mhando polyhedra boka iri, kuwedzera kune apa, zvinosanganisirawo vamiririri dzinotevera:

1. Kugara Pyramid ane hwaro nguva polygon, uye kukwirira kwayo kuri projected kuti musimboti denderedzwa akanyorwa pangoro kana circumscribed yakapoteredza.
2. A rectangular Pyramid aumbwa apo mumwe divi micheto piringisha pechingoro pane zvakanaka akachinjikana. Pakadaro, pamuuyo ichi ichokwadi anonziwo chepiramidhi yakakwirira.

Pyramid Properties:

• In Kana apo vose divi micheto congruent mapiramidhi (chete nokukwirira), vose bindepinde vaine chawo pane imwe yorusvingo, uye vakakomberedza chawo vanogona kuwana denderedzwa chete pakati coinciding pamwe fungidziro iri vertex yacho chepiramidhi.
• Kana mujinga chepiramidhi ndiyo nguva polygon, zvose lateral micheto vari congruent, uye zviso vari isosceles zvokuridza.

Kugara polyhedron: mhando uye zvivako zviri polyhedra

In stereometrical nenzvimbo nzvimbo chaiwo geometric muviri ane zvachose akaenzana mumwe zvokushanda kune vertices uko zvakabatana chete nhamba mbabvu. mitumbi izvi zvinonzi Platonic nezvishoma, kana kugara polyhedra. Mhando polyhedra pamwe zvivako zvakadaro, pane nhamba shanu chete:

1. Tetrahedron.
2. Hexahedron.
3. Octahedron.
4. Dodecahedron.
5. Icosahedron.

zita rake nguva polyhedra vanofanira muzivi wechiGiriki yekare Plato akatsanangura geometric mitumbi izvi mubasa ravo uye kubatanidza navo zvinhu zvakasikwa: pasi, mvura, moto, mhepo. Wechishanu mufananidzo paakavapa zvakafanana pamwe nemamiriro wezvinhu zvose. Maererano naye, njodzi dzinongoitika dzoga maatomu akafanana mhando dzose polyhedra. Nemhaka yayo zvikuru zvinoshamisa chikabviswa - symmetry, izvi geometric zvitarisiko zvikuru kwete chete nokuda mathematicians vekare uye vazivi, asiwo ezvivako, Painters uye vavezi vemabwe nguva dzose. Kuvapo kwemarudzi 5 chete chete chakakwana symmetry polyhedra vaiona chinokosha zvakawanikwa, ivo kunyange paakavapa chokuita naMwari.

Hexahedron aro zvivako

Muchimiro hexahedron vatsivi Plato akafunga Kufanana ne nemamiriro enyika maatomu. Chokwadi, zvino neizvi chose pfungwa iyi, iyo Zvisinei, asingadi kupindira vanodhirowa uye modernity kukwezva pfungwa inonyatsozivikanwa inofananidza aesthetics ake.

In geometry, mumwe hexahedron, iye wacho Cube anonzi nyaya inokosha bhokisi, izvo zvinozoita rudzi prism. Saka ehupfumi chokuita kuutio prism ehupfumi pamwe musiyano chete kuti micheto zvose uye makona kuutio vakaenzana. Kubva ichi zvinotevera ehupfumi:

1. micheto All ane kuutio vari congruent uye vanovandira zvakafanana ndege noruremekedzo kune mumwe.
2. zviso zvose - congruent dzinoungana (ari kuutio 6), chero rinogona kutorwa sehwaro.
3. pamakona All vakaenzana intergranal 90.
4. Kubva mumwe vertex ane wakaenzana nhamba yembabvu, kureva 3.
5. The kuutio ane pfumbamwe nematemo symmetry, izvo zvose piringisha kutoda mharadzano pakati diagonals ari hexahedron, anotaurwa somuzinda symmetry.

tetrahedron

Tetrahedron - mumwe tetrahedron pamwe micheto vakaenzana pezvakaita zvokuridza, mumwe vertex izvo ndiwo mhesano achitandadza matatu micheto.

The ehupfumi munhu dzose tetrahedron:

1. Zvose zviso tetrahedron - munhu equilateral, triangle, izvo zvinoreva kuti zvose zviso imwe tetrahedron vari congruent.
2. Sezvo mujinga ari dzose geometric mufananidzo, ndiko kuti, ane zvakaenzana mativi, zviso tetrahedron uye wekuita panguva iyoyo pakona yorusvingo, i.e. pamakona vose vakaenzana.
3. Amount planar pamakona panguva nechimwe vertices akaenzana 180, sezvo pamakona vose vakaenzana, chero pakona imwe nguva tetrahedron 60.
4. Imwe neimwe vertices projected mharadzano pfungwa dzakakwirira nevakomana (orthocenter) nechiso.

Octahedron aro zvivako

Richirondedzera mhando dzose polyhedra, zvinofanira kucherechedzwa kuti chinhu sechinhu octahedron, iyo inogona nemaziso achimiririrwa vaviri glued quadrilateral zvigadziko dzose mapiramidhi.

The ehupfumi of octahedron:

1. The zita geometric muviri anoudza nhamba zviso ayo. Octahedron anoumbwa 8 congruent equilateral zvokuridza, chimwe nechimwe akaenzana nokuwanda vertices convergent zviso, kureva 4.
2. Sezvo zviso zvose octahedron vakaenzana uye makona ayo intergranal, mumwe nomumwe riri 60, uye uwandu planar pamakona chero vertices Saka 240.

dodecahedron

Kana isu vanofunga kuti zvose zviso geometric muviri chinhu dzose nePentagon, iwe kuwana dodecahedron - dimikira 12 polygons.

Properties dodecahedron:

1. At mumwe vertex piringisha pamwe kumativi matatu.
2. zviso zvose vakaenzana uye vane kuenzana mbabvu, uye nharaunda zvakaenzana.
3. Panguva dodecahedron 15 namasanhu uye ndege kubva symmetry, pamwe umwe wavo ichipfuura pakati pamusoro chiso uye pakatarisana mucheto.

icosahedron

Zvakafanana dzinonakidza kupfuura dodecahedron, icosahedron nhamba anomiririra-dimensional matatu geometric muviri 20 zvakaenzana mativi. Pakati ehupfumi rworudyi icosahedron ndedzinotevera:

1. zviso zvose zviri icosahedron - isosceles zvokuridza.
2. At neimwe vertex ari polyhedron wekuita zviso mashanu, uye uwandu pedyo pamakona ndiyo misoro 300.
3. Icosahedron kwakafanana uye dodecahedron, 15 namasanhu uye ndege kuti symmetry achipfuura nepakati zvakatinakidza akatarisana kumativi.

semiregular polygons

Uyezve platonic nezvishoma, polyhedrons convex boka anosanganisirawo Archimedean nezvishoma, izvo truncated dzose polyhedrons. Mhando polyhedra muboka iri vane zvinotevera ehupfumi:

1. Geometric muviri zviri pairwise vakaenzana zviso pakubhadhara anoverengeka Somuenzaniso, truncated tetrahedron kwakafanana nguva tetrahedron, 8 zviso, asi pane nyaya muviri 4 zviso Archimedean vari mativi yakaumba 4 - hexagonal.
2. pamakona ose congruent imwe vertex.

stellate polyhedra

Representatives howa neobomnyh geometrical miviri - stellate polyhedrons, zviso iyo piringisha mumwe nomumwe. Vanogona inoumbwa kwemapazi maviri dzose-dimensional matatu miviri kana somugumisiro kuenderera mberi zvavo.

Saka, zvakadai kuvazivisa stellate polyhedra sezvo: stellate chimiro ane octahedron, dodecahedron, icosahedron, cuboctahedral, icosidodecahedron.