Polyhedra. Iri polyhedra da dũkiyõyinsu

Polyhedra ba kawai zauna wani shahararren wuri a cikin lissafi, amma kuma da faruwa a cikin rayuwar yau da kullum na kowane mutum. Ba a ma maganar da ta wucin gadi alaka abubuwa a da dama polygons, fara daga matchbox da kuma kawo karshen gine-gine da abubuwa a yanayi ma faruwa lu'ulu'u a cikin nau'i na wani shigen sukari (gishiri), prisms (crystal), dala (scheelite), octahedra (lu'u-lu'u), da dai sauransu . d.

A ra'ayi na mai polyhedron, a lissafi iri polyhedrons

Lissafi kimiyya qunshi stereometry sashe cewa hulda da halaye da kuma kaddarorin girma siffofi. Lissafi jiki bangarorin kafa a uku-girma sarari a daure da jirage (fuskoki dabam-dabam) da aka sani da "polytopes". Iri polyhedra yana da fiye da dozin wakilan da iri dabam-dabam da lambar da siffar fuskoki.

Duk da haka, duk polyhedra da kowa Properties:

1. Su duk da uku na game aka gyara: fuskar (polygonal surface), saman (da kusassari kafa a cikin ƙasa fuskoki dabam-dabam fili), wani gefe (gefen ko yanke siffofi kafa a jamsin na biyu fuskoki).
2. Kowane polygon baki haɗu da biyu, da kuma kawai fuskoki biyu da suke a dangane da juna ne m.
3. The kumbura yana nufin cewa jiki ne gaba daya shirya a kan kawai a gefe daya daga cikin jirgin sama a kan wanda ya ginu ne daya daga cikin fuskoki. The mulki ya shafi dukan fuskoki da polyhedron. Wadannan lissafi siffofi a m lissafi lokaci kira convex polyhedra. Ban ne stellate polyhedra da aka samu daga na yau da kullum polygonal lissafi jikinsu.

Polyhedra za a iya raba:

1. Iri convex polyhedra, kunsha na da wadannan azuzuwan: al'ada ko classic (a Prism, a dala, a akwatin), dama (wanda kuma ake kira Platonic daskararru), semiregular (na biyu sunan - Archimedean daskararru).
2. Non-convex polyhedrons (stellate).

Prism kuma da kaddarorin

Lissafi a matsayin rabo lissafi karatu da kaddarorin uku-girma siffofi, iri polyhedra (Prism daga gare su). Prism kira na lissafi jiki wanda ya bukata biyu m fuskoki (wanda kuma ake kira da kwasfansu) kwance a layi daya jirage, kuma n-th na gefen fuskantar a cikin nau'i na parallelograms. Bi da bi, da Prism ma yana da dama iri, ciki har da irin wannan nau'i na polyhedra, kamar:

1. Parallelepiped - kafa a lokacin da tushe ne parallelogram - a polygon tare da nau'i-nau'i daga abokan gaba biyu daidai kusassari da kuma nau'i-nau'i biyu daga sãɓãni, congruent.
2. Prism ne perpendicular zuwa gefuna da tushe.
3. A karkata Prism halin da kaikaitaccen kwana (wanin 90) tsakanin fuskoki da tushe.
4. Proper halin Prism sansanonin a cikin wani nau'i na yau da kullum da polygon tare da daidaita kaikaice bangarorin.

Babban kaddarorin da Prism:

• Congruent sansanonin.
• All gefuna da Prism ne daidai da layi daya da juna.
• All gefen fuskoki da siffar wani parallelogram.

dala

Dala kira na lissafi jiki da cewa qunshi wani tushe da kuma daya daga cikin n-th na triangular fuskoki da cewa gama a wata guda batu - saman. Ya kamata a lura da cewa idan da gefen fuskoki na dala aka wakilta triangles ake bukata, sa'an nan da tushe na iya zama kamar wani triangular polygon ko quadrilateral da pentagonal, kuma haka a ad infinitum. A wannan yanayin, sunan da dala yayi dace a polygon a tushe. Alal misali, idan tushe ne alwatika dala - a triangular dala, quadrilateral - quadrangular, da dai sauransu ...

Pyramids - shi konusopodobnye polyhedra. Iri polyhedra na wannan kungiya, a cikin Baya ga sama, kuma sun hada da wadannan wakilan:

1. Regular dala yana da akai na yau da kullum polygon, kuma tsayinsa an kimanta to a tsakiyar wata da'irar rubũtacce a gindi ko circumscribed a kusa da shi.
2. A rectangular dala da aka kafa a lokacin da daya daga cikin gefen gefuna rarraba da tushe a wata dama kwana. A irin wannan yanayi, wannan gefe gaskiya ma kira dala tsawo.

Dala Properties:

• A cikin akwati inda duk gefen gefuna congruent pyramids (wannan tsawo), su duka zoba da wani tushe a daya kwana, kuma a kusa da barikin iya zana a da'irar da cibiyar zo daidai da lokacin aikin na kokuwa na dala.
• Idan tushe na dala ne na yau da kullum polygon, duk a kaikaice gefuna ne congruent, da kuma fuskokin su isosceles triangles.

Regular polyhedron:-daban da kuma kaddarorin polyhedra

A stereometrical zauna a wuri na musamman da na lissafi da jiki da gaba daya daidai da juna fuskoki dabam-dabam da vertices na wanda aka haɗa zuwa wannan yawan hakarkarinsa. Waɗannan jikuna an kira Platonic daskararru, ko yau da kullum polyhedra. Iri polyhedra da irin wannan Properties, akwai kawai biyar Figures:

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

Sunansa na yau da kullum polyhedra da ake bukata don tsoho Greek Falsafa Plato ya bayyana wadannan lissafi jikuna a aikinsu da kuma a haɗa su tare da abubuwa a yanayi: ƙasa, da ruwa, wuta, iska. Biyar adadi bayar da kamance da tsarin da sararin samaniya. A cewar shi, bala'o'i atoms kama da iri na yau da kullum polyhedra. Godiya ta zuwa ga mafi m fasalin - fasali, wadannan lissafi siffofi na mai ban sha'awa ba kawai ga tsoho lissafi da kuma falsafa, amma kuma ga gine-ginen, painters da sculptors dukan lokaci. A gaban kawai 5 jinsin tare da cikakken fasali polyhedra dauke da wani muhimman hakkokin samu, su ma bayar da dangane da allahntaka.

Hexahedron kuma da kaddarorin

A cikin irin hexahedron mãsu mayẽwa Plato zaci kama tare da tsarin da ƙasa kwayoyin halitta. Hakika, a yanzu gaba daya karyata wannan jarrabawa, wanda, duk da haka, ba ya tsoma baki tare da zane da kuma wayewar don jawo hankalin zukatan sanannun Figures ya ilmi.

A lissafi, wani hexahedron, ya ta Cube dauke da wani musamman idan akwai daga cikin akwatin, wanda, bi da bi, wani nau'i ne na Prism. Haka kuma, da kaddarorin dangantawa da shigen sukari Prism Properties tare da kawai bambancin da cewa duk gefuna da sasanninta na shigen sukari ne daidai. Daga wannan da wadannan Properties:

1. All gefuna da wani shigen sukari ne congruent da kuma karya a layi daya jirage game da juna.
2. All fuskoki - congruent murabba'ai (na shigen sukari na 6), da wani abin da za a iya dauka a matsayin tushen.
3. All kusassari ne daidai intergranal 90.
4. Daga kowane kokuwa yana da wani daidai yawan hakarkarinsa, wato 3.
5. A shigen sukari yana da tara da gatura da fasali, wanda duk rarraba a batu na rarrabawa da diagonals na hexahedron, kira a matsayin wata cibiyar da fasali.

tetrahedron

Tetrahedron - a tetrahedron da gefuna daidaita a siffar triangles, kowane kokuwa na wanda shi ne maha? Ar batu na uku gefuna.

A Properties na yau da kullum tetrahedron:

1. All fuskokin tetrahedron - a equilateral alwatika, wanda ke nufin cewa duk fuskokin wani tetrahedron ne congruent.
2. Tun da tushe ne na yau da kullum na lissafi adadi, cewa shi ne, shi yana daidai tarnaƙi, cikin fuskõkin tetrahedron da converge a wannan kwana, Ina nufin duk kusassari ne daidai.
3. Adadin planar kusassari a kowane daga cikin vertices ne daidai to 180, tun duk kusassari ne daidai, wani kwana na yau da kullum tetrahedron 60.
4. Kowace daga cikin vertices kimanta mahada aya daga cikin tuddan da m (orthocenter) fuska.

Octahedron kuma da kaddarorin

Describing iri na yau da kullum polyhedra, ya kamata a lura da cewa abu kamar wani octahedron, wanda za a iya gani wakilta a matsayin biyu gam quadrilateral kwasfa na yau da kullum da pyramids.

A Properties daga cikin octahedron:

1. A sosai sunan lissafi jiki ya gaya yawan ta fuskoki. Octahedron hada 8 congruent equilateral triangles, wanda kowannensu yana da daidai da adadin vertices convergent fuskoki, wato 4.
2. Tun da dukan fuskoki da octahedron ne daidai da sasanninta intergranal, wanda kowannensu yana da 60, da kuma Naira Miliyan Xari planar kusassari wani na vertices ne kamar haka 240.

dodecahedron

Idan muka kwatanta da cewa duk da fuskoki da lissafi jiki ne na yau da kullum Pentagon, ka samu wani dodecahedron - wani adadi na 12 polygons.

Kadarorin dodecahedron:

1. A kowane kokuwa rarraba tare uku bangarorin.
2. All fuskoki ne daidai kuma daidai da tsawon hakarkarinsa, kuma daidai yankin.
3. A dodecahedron 15 gatura da kuma jirage na fasali, tare da wani daya daga cikinsu ya wuce ta tsakiyar saman fuska da wani m baki.

icosahedron

Daidai da ban sha'awa fiye da dodecahedron, icosahedron adadi wakiltar uku girma na lissafi jiki 20 da daidaita bangarorin biyu. Daga cikin Properties dama icosahedron ne da wadannan:

1. All fuskoki da icosahedron - isosceles triangles.
2. A kowane kokuwa na polyhedron converge biyar fuskoki, da kuma Naira Miliyan Xari da m kusassari ne 300 fi.
3. Icosahedron ne guda kamar yadda kuma dodecahedron, 15 da gatura da kuma jirage na fasali wucewa ta cikin tsakiyar maki na tarnaƙi.

semiregular polygons

Bugu da ƙari platonic daskararru, polyhedrons convex kungiyar kuma ya hada da Archimedean daskararru, wanda suke truncated yau da kullum polyhedrons. Iri polyhedra a cikin wannan kungiyar da wadannan Properties:

1. Lissafi jiki ne pairwise daidai fuskoki da dama iri, misali, truncated tetrahedron ne guda a matsayin na yau da kullum tetrahedron, 8 fuskokin, amma a yanayin saukan jiki 4 Archimedean fuskoki ne triangular siffa da 4 - kyakkyawan.
2. All kusassari ne congruent zuwa daya kokuwa.

stellate polyhedra

Wakilai jinsunan neobomnyh geometrical gawarwakin - stellate polyhedrons, fuskokin wanda rarraba da juna. Su za a iya kafa ta a ci na biyu yau da kullum da uku-girma jikinsu ko a sakamakon ci gaba da fuskõkinsu.

Saboda haka, irin wannan a san stellate polyhedra kamar: stellate siffar wani octahedron, dodecahedron, icosahedron, cuboctahedral, icosidodecahedron.