Polyhedra. Hom polyhedra thiab lawv lub zog

Polyhedra tsis tau tsuas yog nyob ib tug muaj koob qhov chaw nyob rau hauv geometry, tab sis kuj tshwm sim nyob rau hauv lub neej txhua hnub ntawm txhua tus neeg. Tsis hais tus txiv neej-ua yam khoom hauv tsev nyob rau hauv ib tug ntau yam ntawm polygons, pib nrog ib tug matchbox thiab xaus nrog architectural ntsiab nyob rau hauv cov xwm raws li muaj muaju nyob rau hauv tus duab ntawm ib lub voos xwmfab (ntsev), prisms (crystal), pyramids (scheelite), lub octahedron (pob zeb diamond), etc. . d.

Lub tswvyim ntawm ib tug polyhedron, nyob rau hauv geometry hom polyhedrons

Geometry science comprises stereometry seem uas ua hauj lwm nrog tus yam ntxwv thiab thaj chaw ntawm tej daim. Geometric lub cev sab yog tsim nyob rau hauv peb-dimensional chaw bounded by dav hlau (seevcev) yog hu ua "polytopes". Hom polyhedra muaj ntau tshaj ib tug teb cov neeg sawv cev ntawm lub differing tooj thiab zoo ntawm ib sab.

Cuaj kaum, tag nrho cov polyhedra muaj ib qho zog:

1. Lawv tag nrho cov muaj peb ib yam: lub ntsej muag (polygonal nto), cov sab saum toj (cov ces kaum tsim nyob rau hauv lub hauv av seb compound), ib tug ntug (sab los yog txiav nrhiav tsim ntawm lub hlws ris ntawm ob lub ntsej muag).
2. Txhua polygon ntug txuas lub ob, thiab tsuas yog ob lub ntsej muag uas yog nyob rau hauv kev sib raug zoo rau txhua tus lwm yam yog ib sab.
3. Cov su txhais tau tias lub cev yog kiag li cov txheej txheem ntawm nyob rau hauv tsuas ib sab ntawm lub dav hlau uas rests ib tug ntawm cov ntsej muag. Cov cai siv rau tag nrho cov sab ntawm cov polyhedron. Cov geometric duab nyob rau hauv cov khoom geometry lub sij hawm hu ua convex polyhedra. Tsuav txhob yog cov stellate polyhedra uas yog derived ntawm kev polygonal geometric lub cev.

Polyhedra yuav tau muab faib mus rau hauv:

1. Hom ntawm convex polyhedra, muaj raws ntawm cov nram qab no cov chav kawm: cov pa los yog classic (a plaub, ib tug pyramid, ib lub thawv), txoj cai (kuj hu ua Platonic cov khib nyiab), semiregular (thib ob lub npe - Archimedean khib nyiab).
2. Tsis-convex polyhedrons (stellate).

Prism thiab nws cov khoom

Geometry raws li ib tug faib geometry kawm lub thaj chaw ntawm peb-dimensional shapes, hom polyhedra (plaub ntawm lawv). Prism hu ua geometric lub cev uas tau yuav tsum tau ob tug zoo tib lub ntsej muag (kuj hu ua bases) dag nyob rau hauv thaum uas tig mus dav hlau, thiab n-th ntawm lub sab ntsib nyob rau hauv daim ntawv ntawm cov parallelograms. Nyob rau hauv lem, tus plaub kuj muaj ob peb ntau yam, xws li xws hom polyhedra, xws li:

1. Parallelepiped - tsim thaum lub hauv paus yog ib tug parallelogram - ib tug polygon nrog officers ntawm ob tawm tsam sib npaug zos ces kaum thiab ob khub ntawm opposite sab zoo sib xws.
2. Prism yog perpendicular mus rau lub npoo ntawm lub hauv paus.
3. Lub inclined plaub yus indirect lub (lwm yam tshaj li 90) ntawm ib sab thiab puag.
4. Kom yus plaub bases nyob rau hauv daim ntawv ntawm ib tug tsis tu ncua polygon nrog sib npaug sab sab.

Lub ntsiab thaj chaw ntawm lub plaub:

• Congruent bases.
• Tag nrho cov npoo ntawm cov plaub yog muaj sib npaug thiab thaum uas tig mus rau txhua lwm yam.
• Tag nrho cov sab ntsej muag muaj ib tug duab ntawm ib parallelogram.

pyramid

Hauv hu ua geometric lub cev uas muaj ib lub hauv paus thiab ib tug ntawm cov n-th ntawm lub voos ntsej muag uas txuas rau ntawm ib tug taw tes - rau sab saum toj. Nws yuav tsum tau muab sau tseg tias yog hais tias lub sab ntsej muag ntawm lub pyramid yog sawv cev los ntawm cov vajvoos peb sab yog yuav tsum tau, ces yog lub hauv paus yuav ua tau zoo li ib tug voos polygon los yog quadrilateral thiab pentagonal, thiab thiaj li nyob ad infinitum. Nyob rau hauv cov ntaub ntawv no, lub npe ntawm lub pyramid raws li ib tug polygon nyob puag. Piv txwv li, yog tias lub hauv paus yog ib daim duab peb sab hauv - ib tug voos pyramid, quadrilateral - quadrangular, etc ...

Pyramids - nws konusopodobnye polyhedra. Hom polyhedra ntawm no pab pawg neeg, nyob rau hauv tas li ntawd mus rau saum toj, kuj muaj xws li cov nram qab no cov neeg sawv cev:

1. Tsis tu ncua hauv muaj hauv paus ntawm ib tug tsis tu ncua polygon, thiab nws qhov siab yog npaj rau ntawm qhov chaw ntawm ib lub voj voog inscribed nyob rau hauv rau lub hauv paus los yog circumscribed nyob ib ncig ntawm nws.
2. A lub pyramid yog tsim thaum ib tug ntawm sab npoo tshuam rau lub hauv paus ntawm ib txoj cai kaum sab xis. Nyob rau hauv xws li ib tug cov ntaub ntawv, qhov no ntug yeej muaj tseeb kuj hu ua pyramid qhov siab.

Hauv khoom:

• Nyob rau hauv lub rooj plaub uas tag nrho cov sab sawv congruent pyramids (lub qhov siab tib yam), lawv tag nrho sib tshooj nrog ib lub hauv paus ntawm ib tug lub, thiab nyob ib ncig ntawm puag yuav kos ib lub voj voog nrog qhov chaw coinciding nrog lub projection ntawm lub kab los sib ntsib ntawm lub pyramid.
• Yog hais tias lub hauv paus ntawm lub pyramid yog ib tug tsis tu ncua polygon, tag nrho cov sab npoo yog congruent, thiab rau ib sab yog ceskaum cov vajvoos peb sab.

Tsis tu ncua polyhedron: hom thiab thaj chaw ntawm polyhedra

Nyob rau hauv stereometrical nyob ib tug tshwj xeeb qhov chaw lub geometric lub cev nrog ib tug kiag li sib npaug zos rau txhua lwm yam kev seevcev lub vertices ntawm uas yog kev cob cog rua rau tib lub xov tooj ntawm ribs. Cov lub cev yog hu ua Platonic cov khib nyiab, los yog tsis tu ncua polyhedra. Hom polyhedra nrog xws li lub zog, muaj tsuas yog tsib nuj nqis:

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

Nws lub npe tsis tu ncua polyhedra yuav tsum ancient Greek philosopher Plato piav cov geometric lub cev nyob rau hauv lawv cov chaw ua hauj lwm thiab mus cuag lawv nrog lub ntsiab ntawm cov xwm: lub ntiaj teb, dej, hluav taws kub, huab cua. Fifth daim duab muab sib thooj nrog cov qauv ntawm cov ntug. Raws li nws, natural disasters atoms li cov ntawm tu ncua polyhedra. Tsaug rau nws cov feem ntau spectacular feature - symmetry, cov geometric duab uas zoo txaus siab xwb tsis rau tus ancient daim ntawv thom nug thiab philosophers, tab sis kuj rau architects, painters thiab sculptors ntawm txhua lub sij hawm. Lub xub ntiag ntawm xwb 5 hom nrog tsis symmetry polyhedra xam tau tias yog ib tug yuav tsum muaj discovery, lawv txawm muab tsub kev twb kev txuas nrog cov divine.

Hexahedron thiab nws cov khoom

Nyob rau hauv daim ntawv ntawm cov hexahedron successors Plato assumed kev zoo sib thooj nrog cov qauv ntawm lub ntiaj teb atoms. Ntawm cov hoob kawm, tam sim no kiag refuted no hypothesis, uas, li cas los xij, tsis cuam tshuam nrog cov drawings thiab modernity nyiam cov minds ntawm zoo-paub cov nuj nqis ntawm nws kom zoo nkauj.

Nyob rau hauv geometry, ib tug hexahedron, nws lub thawv muaj rau sab yog suav tias yog ib tug tshwj xeeb cov ntaub ntawv ntawm lub thawv, uas, nyob rau hauv lem, yog ib yam ntawm cov plaub. Raws li, qhov chaw uas txuam nrog lub voos xwmfab plaub zog nrog qhov txawv tsuas hais tias tag nrho cov npoo thiab cov ces kaum ntawm lub thawv muaj rau sab yog muaj sib npaug. Los ntawm no lub nram qab no cov khoom:

1. Tag nrho cov npoo ntawm ib lub voos xwmfab yog congruent thiab pw nyob rau hauv thaum uas tig mus dav hlau nrog rau kev hwm rau txhua lwm yam.
2. Tag nrho cov ntsej muag - congruent squares (ntawm lub voos xwmfab ntawm 6), ib yam ntawm cov uas yuav tsum tau coj raws li lub hauv paus.
3. Tag nrho cov ces kaum yog muaj sib npaug intergranal 90.
4. Los ntawm txhua tus kab los sib ntsib muaj vaj huam sib luag naj npawb ntawm cov tav, namely 3.
5. Lub thawv muaj rau sab muaj cuaj axes ntawm symmetry, uas tag nrho cov tshuam ntawm tus taw tes ntawm kev tshuam ntawm lub diagonals ntawm lub hexahedron, xa mus rau raws li ib tug chaw ntawm symmetry.

tetrahedron

Tetrahedron - ib tug tetrahedron nrog sawv sib npaug zos rau hauv cov duab ntawm cov vajvoos peb sab, txhua kab los sib ntsib ntawm uas yog lub hlws ris taw tes rau ntawm peb sawv.

Cov khoom ntawm ib tug tsis tu ncua tetrahedron:

1. Tag nrho rau ib sab ntawm tetrahedron - ib tug equilateral daim duab peb sab, uas txhais tau tias tag nrho cov ntsej muag ntawm ib tug tetrahedron yog sib xws.
2. Txij li thaum lub hauv paus yog ib txwm geometric duab, uas yog, nws muaj sib npaug zos sab, rau ib sab ntawm lub tetrahedron thiab sib nyob rau tib lub kaum ntse ntse, i.e. tag nrho cov ces kaum muaj sib npaug.
3. Nyiaj planar ces kaum ntawm txhua lub vertices yog sib npaug zos rau 180, txij li thaum tag nrho cov ces kaum yog muaj sib npaug, tej kaum sab xis ntawm ib tug tsis tu ncua tetrahedron 60.
4. Txhua lub vertices npaj kev tshuam taw tes ntawm lub heights ntawm qhov opposite (orthocenter) lub ntsej muag.

Octahedron thiab nws cov khoom

Piav txog hom kev polyhedra, nws yuav tsum tau muab sau tseg hais tias kwv li ib tug octahedron, uas yuav tsum tau pom muaj tuaj raws li ob glued quadrilateral bases ntawm kev pyramids.

Lub zog ntawm cov octahedron:

1. Tus heev lub npe ntawm tus duab lub cev qhia rau tus xov tooj ntawm nws ib sab. Octahedron muaj li ntawm 8 congruent equilateral cov vajvoos peb sab, txhua tus uas yog sib npaug zos rau cov xov tooj ntawm vertices convergent ntsej muag, namely 4.
2. Txij li thaum tag nrho cov sab ntawm cov octahedron yog muaj sib npaug thiab nws fab intergranal, txhua yam uas yog 60, thiab lub sum ntawm planar ces kaum tej yam vertices yog li no 240.

dodecahedron

Yog hais tias peb xav txog tej yam uas tag nrho cov sab ntawm cov duab lub cev yog ib tug tsis tu ncua pentagon, koj tau txais ib dodecahedron - ib daim duab ntawm 12 polygons.

Properties dodecahedron:

1. Nyob rau txhua kab los sib ntsib tshuam nrog peb sab.
2. Tag nrho cov ntsej muag yog muaj sib npaug thiab muaj tib lub ntev ntawm tav, thiab sib npaug zos cheeb tsam.
3. Thaum lub dodecahedron 15 axes thiab dav hlau ntawm symmetry, nrog ib tug ntawm lawv kis tau los ntawm nruab nrab ntawm sab saum toj lub ntsej muag thiab ib tug opposite ntug.

icosahedron

Attendance nthuav tshaj dodecahedron, icosahedron daim duab nruab nrab yog cov peb-dimensional geometric lub cev 20 sib npaug zos sab. Ntawm cov zog cai icosahedron yog cov nram qab no:

1. Tag nrho cov ntsej muag ntawm lub icosahedron - ceskaum cov vajvoos peb sab.
2. Nyob rau txhua kab los sib ntsib ntawm lub polyhedron sib tsib ntsej muag, thiab lub sum ntawm ib sab ces kaum yog 300 saum.
3. Icosahedron yog tib yam li thiab dodecahedron, 15 axes thiab dav hlau ntawm symmetry dua los ntawm nruab nrab ntsiab lus ntawm opposite sab.

semiregular uas

Tsis tas li ntawd platonic khib nyiab, polyhedrons convex pab pawg neeg kuj muaj xws li Archimedean khib nyiab, uas yog truncated kev polyhedrons. Hom polyhedra nyob rau hauv pab pawg neeg no muaj cov nram qab zog:

1. Geometric lub cev yog pairwise sib npaug zos ib sab ntawm ob peb hom, piv txwv li, truncated tetrahedron yog tib yam li ib txwm tetrahedron, 8 ntsej muag, tab sis nyob rau hauv rooj plaub lub cev 4 Archimedean sab yog voos zoo li tus thiab 4 - hexagonal.
2. Tag nrho cov ces kaum yog congruent rau ib tug kab los sib ntsib.

stellate polyhedra

Cov neeg sawv cev hom neobomnyh geometrical lub cev - stellate polyhedrons, lub ntsej muag uas tshuam nrog txhua lwm yam. Lawv muaj peev xwm yuav tsim nyob rau los ntawm ib tug merger ntawm ob tsis tu ncua peb-dimensional lub cev los yog raws li ib tug tshwm sim ntawm lub txuas ntxiv rau lawv cov ntsej muag.

Yog li, xws paub stellate polyhedra li: stellate zoo ntawm ib tug octahedron, dodecahedron, icosahedron, cuboctahedral, icosidodecahedron.