Polyhedra tsis yog tsuas yog nyob hauv qhov chaw tseem ceeb hauv geometry, tab sis kuj tshwm sim hauv lub neej niaj hnub ntawm txhua tus neeg. Tsis txhob hais txog cov khoom tsim hauv tsev nyob rau hauv daim ntawv ntawm ntau yam polygons, pib nrog lub matchbox thiab xaus nrog architectural ntsiab, muaju nyob rau hauv daim ntawv ntawm lub voos xwmfab (ntsev), prism (crystal), pyramid (scheelite), octahedron (pob zeb diamond), e.
Lub tswvyim ntawm polyhedron, hom polyhedra hauv geometry
Geometry raws li kev tshawb fawb muaj ib ntu ntawm stereometry uas kawm txog cov yam ntxwv thiab cov khoom ntawm cov duab peb sab. Geometric lub cev, ob sab uas nyob rau hauv peb-dimensional qhov chaw yog tsim los ntawm tsawg dav hlau (lub ntsej muag), hu ua "polyhedra". Hom polyhedra suav nrog ntau dua kaum tus neeg sawv cev, sib txawv ntawm tus lej thiab cov duab ntawm lub ntsej muag.
Txawm li cas los xij, txhua tus polyhedra muaj cov khoom sib xws:
- Lawv txhua tus muaj 3 yam tseem ceeb: ntsej muag(nto ntawm ib tug polygon), vertex (cov ces kaum tsim ntawm qhov sib txuas ntawm lub ntsej muag), ntug (sab ntawm ib daim duab los yog ib ntu tsim ntawm qhov sib tshuam ntawm ob lub ntsej muag).
- Txhua lub kaum sab xis txuas ob thiab tsuas yog ob lub ntsej muag uas nyob ib sab.
- Convexity txhais tau hais tias lub cev tag nrho nyob ntawm ib sab ntawm lub dav hlau uas ib lub ntsej muag dag. Txoj cai siv rau txhua lub ntsej muag ntawm polyhedron. Xws li cov duab geometric hauv stereometry yog hu ua convex polyhedra. Qhov tshwj xeeb yog lub hnub qub zoo li polyhedra, uas yog derivatives ntawm ib txwm polyhedral geometric solids.
Polyhedra tuaj yeem muab faib ua:
- Hom convex polyhedra, muaj cov chav kawm hauv qab no: zoo tib yam lossis classical (prism, pyramid, parallelepiped), tsis tu ncua (tseem hu ua Platonic solids), semi-regular (thib ob lub npe - Archimedean solids).
- Non-convex polyhedra (lub hnub qub zoo li).
Prism thiab nws cov khoom
Stereometry raws li ib ceg ntawm geometry kawm txog cov khoom ntawm peb-dimensional duab, hom polyhedra (lub prism yog ib qho ntawm lawv). Lub prism yog ib lub cev geometric uas yuav tsum muaj ob lub ntsej muag zoo ib yam (lawv tseem hu ua bases) pw hauv cov dav hlau sib npaug, thiab cov n-th tus naj npawb ntawm sab ntsej muag nyob rau hauv daim ntawv ntawm parallelograms. Nyob rau hauv lem, lub prism kuj muaj ob peb ntau yam, nrog rau xws li hom ntawm polyhedra li:
- Parallelepiped - tsim yog tias lub hauv paus yog parallelogram -polygon nrog 2 khub ntawm qhov sib npaug ntawm lub kaum sab xis thiab 2 khub ntawm qhov sib txawv ntawm sab.
- Ib tug ncaj prism muaj ntug perpendicular rau lub hauv paus.
- Tilted prism yog tus cwj pwm los ntawm qhov muaj cov kaum sab xis tsis yog (tshwj tsis yog 90) ntawm lub ntsej muag thiab lub hauv paus.
- Ib lub prism tsis tu ncua yog tus cwj pwm los ntawm cov hauv paus hauv daim ntawv ntawm ib qho polygon tsis tu ncua nrog lub ntsej muag sib npaug.
Basic properties of a prism:
- Nyob hauv paus.
- Tag nrho cov npoo ntawm lub prism yog sib npaug thiab sib npaug rau ib leeg.
- Tag nrho cov ntsej muag yog parallelogram-puab.
Pyramid
Pyramid yog lub cev geometric, uas muaj ib lub hauv paus thiab n-th tus naj npawb ntawm daim duab peb sab, txuas ntawm ib qho taw tes - saum toj. Nws yuav tsum raug sau tseg tias yog lub ntsej muag ntawm lub pyramid yuav tsum tau sawv cev los ntawm daim duab peb sab, ces lub hauv paus tuaj yeem yog daim duab peb sab polygon, lossis plaub fab, lossis pentagon, thiab lwm yam ad infinitum. Nyob rau hauv rooj plaub no, lub npe ntawm lub pyramid yuav sib haum mus rau lub polygon ntawm lub hauv paus. Piv txwv li, yog tias daim duab peb sab nyob ntawm lub hauv paus ntawm lub pyramid, nws yog daim duab peb sab, lub quadrilateral yog plaub fab, thiab lwm yam.
Pyramids yog lub khob zoo li polyhedra. Cov hom polyhedra ntawm pab pawg no, ntxiv rau cov teev saum toj no, kuj suav nrog cov neeg sawv cev hauv qab no:
- Ib lub pyramid tsis tu ncua muaj ib qho polygon tsis tu ncua ntawm nws lub hauv paus, thiab nws qhov siab yog npaj rau qhov chawib lub voj voog sau nyob rau hauv lub hauv paus los yog circumscribed nyob ib ncig ntawm nws.
- Ib lub duab plaub yog tsim thaum ib qho ntawm sab ntug sib tshuam nrog lub hauv paus ntawm lub kaum sab xis. Nyob rau hauv cov ntaub ntawv no, nws kuj yog ib qho ncaj ncees hu rau lub ntug qhov siab ntawm lub pyramid.
Pyramid thaj chaw:
- Yog tias tag nrho cov ntug ntawm lub pyramid yog congruent (ntawm tib qhov siab), ces lawv txhua tus sib tshuam nrog lub hauv paus ntawm tib lub kaum sab xis, thiab nyob ib ncig ntawm lub hauv paus koj tuaj yeem kos ib lub voj voog nrog ib qho chaw coinciding nrog qhov projection. sab saum toj ntawm lub pyramid.
- Yog tias lub hauv paus ntawm lub pyramid yog ib qho polygon tsis tu ncua, ces tag nrho cov npoo yog congruent, thiab lub ntsej muag yog daim duab peb sab isosceles.
Regular polyhedron: hom thiab cov khoom ntawm polyhedra
Hauv stereometry, ib qho chaw tshwj xeeb yog nyob ntawm lub cev geometric nrog lub ntsej muag sib npaug, ntawm qhov chaw uas tib tus lej ntawm cov npoo txuas nrog. Cov khib nyiab no hu ua Platonic solids, lossis polyhedra tsis tu ncua. Hom polyhedra nrog cov khoom zoo li no tsuas muaj tsib daim duab:
- Tetrahedron.
- Hexahedron.
- Octahedron.
- Dodecahedron.
- Icosahedron.
Tsis tu ncua polyhedra tshuav lawv lub npe rau tus kws tshawb fawb Greek thaum ub Plato, uas tau piav qhia txog lub cev geometric hauv nws cov ntawv sau thiab txuas lawv nrog cov ntsiab lus ntuj: ntiaj teb, dej, hluav taws, huab cua. Daim duab thib tsib tau txais qhov zoo sib xws nrog cov qauv ntawm lub ntiaj teb. Nyob rau hauv nws lub tswv yim, lub atoms ntawm tej yam ntuj tso ntsiab nyob rau hauv cov duab zoo li cov hom ntawm tsis tu ncua polyhedra. Vim nws cov khoom zoo siab tshaj plaws -symmetry, cov geometric lub cev muaj kev txaus siab tsis yog rau cov neeg ua lej thaum ub thiab cov kws tshawb fawb, tab sis kuj rau architects, artists thiab sculptors txhua lub sijhawm. Lub xub ntiag ntawm tsuas yog 5 hom polyhedra nrog kev sib xyaw ua ke tau suav tias yog qhov kev tshawb pom tseem ceeb, lawv tau txais txiaj ntsig kev sib txuas nrog txoj cai los saum ntuj los.
Hexahedron thiab nws cov khoom
Nyob rau hauv daim ntawv ntawm hexagon, Plato tus successors assumed ib tug zoo xws li cov qauv ntawm lub atoms ntawm lub ntiaj teb. Tau kawg, tam sim no, qhov kev xav no tau raug lees paub tag nrho, uas, txawm li cas los xij, tsis tiv thaiv cov duab los ntawm kev nyiam lub siab ntawm cov duab nto moo nrog lawv cov kev zoo nkauj niaj hnub no.
Hauv geometry, hexahedron, tseem hu ua lub voos xwmfab, suav hais tias yog qhov tshwj xeeb ntawm parallelepiped, uas, dhau los, yog hom prism. Raws li, cov khoom ntawm lub voos xwmfab muaj feem xyuam rau cov khoom ntawm lub prism, nrog rau qhov sib txawv tsuas yog tias txhua lub ntsej muag thiab cov ces kaum ntawm lub voos xwmfab yog sib npaug. Cov khoom nram qab no ua raws li qhov no:
- Tag nrho cov npoo ntawm lub voos xwmfab yog congruent thiab pw nyob rau hauv parallel dav hlau nrog hwm rau ib leeg.
- Tag nrho cov ntsej muag yog congruent squares (muaj 6 nyob rau hauv tag nrho cov nyob rau hauv ib lub voos xwmfab), ib yam ntawm cov uas yuav muab tau raws li lub hauv paus.
- Txhua lub kaum sab xis yog 90.
- Ib tus lej sib npaug ntawm cov npoo emanates los ntawm txhua qhov vertex, uas yog 3.
- Lub voos xwmfab muaj 9 axes ntawm symmetry, uas txhua qhov sib tshuam ntawm qhov sib tshuam ntawm kab pheeb ces kaum ntawm hexahedron, hu ua qhov chaw ntawm symmetry.
Tetrahedron
Ib lub tetrahedron yog tetrahedron nrog lub ntsej muag sib npaug hauv daim duab peb sab, txhua qhov vertices uasyog qhov sib txuas ntawm peb lub ntsej muag.
Khoom ntawm tetrahedron li niaj zaus:
- Txhua lub ntsej muag ntawm tetrahedron yog daim duab peb sab sib npaug, uas txhais tau hais tias txhua lub ntsej muag ntawm lub tetrahedron yog concruent.
- Vim lub hauv paus yog sawv cev los ntawm cov duab geometric li niaj zaus, uas yog, nws muaj ob sab sib npaug, lub ntsej muag ntawm tetrahedron converge ntawm tib lub kaum sab xis, uas yog, txhua lub kaum sab xis yog sib npaug.
- Cov sum ntawm cov kaum sab xis ntawm txhua qhov vertices yog 180, txij li txhua lub kaum sab xis yog sib npaug, ces txhua lub kaum sab xis ntawm tetrahedron li niaj zaus yog 60.
- Txhua qhov chaw yog kwv yees mus rau qhov kev sib tshuam ntawm qhov siab ntawm qhov ntxeev siab (orthocenter) ntsej muag.
Lub octahedron thiab nws cov khoom
Piav txog hom polyhedra li niaj zaus, ib tus tsis tuaj yeem tsis nco qab xws li cov khoom xws li octahedron, uas tuaj yeem pom qhov muag pom tau tias yog ob lub quadrangular li niaj zaus pyramids glued ua ke los ntawm hauv paus.
Cov khoom ntawm octahedron:
- Lub npe ntawm lub cev geometric qhia tus lej ntawm nws lub ntsej muag. Lub octahedron muaj 8 congruent equilateral daim duab peb sab, nyob rau hauv txhua lub vertices uas muaj ib tug sib npaug ntawm lub ntsej muag converge, uas yog 4.
- Vim tag nrho cov ntsej muag ntawm ib qho octahedron sib npaug, nws cov ces kaum sib txuas kuj sib npaug, txhua qhov sib npaug rau 60, thiab cov lej ntawm lub dav hlau kaum ntawm ib qho ntawm cov vertices yog li 240.
Dodecahedron
Yog tias peb xav txog tias txhua lub ntsej muag ntawm lub cev geometric yog lub pentagon tsis tu ncua, ces peb tau txais dodecahedron -daim duab ntawm 12 polygons.
Cov khoom ntawm dodecahedron:
- Peb lub ntsej muag sib tshuam ntawm txhua qhov vertex.
- Txhua lub ntsej muag sib npaug thiab muaj tib ntug ntev thiab thaj tsam sib npaug.
- Lub dodecahedron muaj 15 axes thiab dav hlau ntawm symmetry, thiab ib qho ntawm lawv dhau los ntawm lub vertex ntawm lub ntsej muag thiab nruab nrab ntawm ntug ntug.
Icosahedron
Tsis muaj qhov nthuav ntau dua li dodecahedron, daim duab ntawm icosahedron yog peb-seem geometric lub cev nrog 20 lub ntsej muag sib npaug. Ntawm cov khoom ntawm nees nkaum-hedron tsis tu ncua, cov hauv qab no tuaj yeem raug sau tseg:
- Txhua lub ntsej muag ntawm icosahedron yog daim duab peb sab isosceles.
- Tsib lub ntsej muag sib sau ua ke ntawm txhua qhov vertex ntawm polyhedron, thiab cov lej ntawm cov kaum sab xis ntawm lub vertex yog 300.
- Lub icosahedron, zoo li lub dodecahedron, muaj 15 axes thiab dav hlau ntawm symmetry hla lub midpoints ntawm opposite ntsej muag.
Semi-regular polygons
Dhau li ntawm Platonic cov khib nyiab, cov pab pawg ntawm convex polyhedra kuj suav nrog Archimedean cov khib nyiab, uas tau txiav tawm tsis tu ncua polyhedra. Cov hom polyhedra ntawm pawg no muaj cov khoom nram qab no:
- Geometric lub cev muaj ob lub ntsej muag sib npaug ntawm ntau hom, piv txwv li, tetrahedron truncated muaj 8 lub ntsej muag, zoo li tetrahedron li niaj zaus, tab sis nyob rau hauv cov ntaub ntawv ntawm Archimedean khoom, 4 lub ntsej muag yuav yog daim duab peb sab thiab 4 yuav yog hexagonal.
- Txhua lub kaum ntawm ib lub vertex yog congruent.
Star polyhedra
Cov neeg sawv cev ntawm cov tsis-volumetric hom geometric lub cev yog stellate polyhedra uas nws lub ntsej muag sib cuam tshuam. Lawv tuaj yeem tsim los ntawm kev sib koom ua ke ob lub 3D cov khoom tsis tu ncua lossis los ntawm kev nthuav lawv lub ntsej muag.
Yog li, xws li cov stellated polyhedra yog lub npe hu ua: cov duab kos duab ntawm octahedron, dodecahedron, icosahedron, cuboctahedron, icosododecahedron.
