Geometry yog qhov zoo nkauj vim hais tias, piv rau algebra, qhov twg nws tsis yog ib txwm paub meej tias koj xav li cas thiab vim li cas, nws ua rau pom qhov khoom. Lub ntiaj teb no zoo kawg nkaus ntawm ntau lub cev yog dai kom zoo nkauj nrog polyhedra tsis tu ncua.
Cov ntaub ntawv dav dav txog polyhedra tsis tu ncua
Raws li ntau, ib txwm polyhedra, lossis raws li lawv kuj hu ua Platonic solids, muaj cov khoom tshwj xeeb. Ntau qhov kev xav txog kev tshawb fawb tau cuam tshuam nrog cov khoom no. Thaum koj pib kawm cov geometric lub cev, koj nkag siab tias koj paub xyaum tsis muaj dab tsi txog lub tswv yim li niaj zaus polyhedra. Kev nthuav qhia ntawm cov khoom no hauv tsev kawm ntawv tsis yog ib txwm nthuav, yog li ntau tus tsis nco qab tias lawv hu ua dab tsi. Cov neeg feem coob nco txog lub voos xwmfab xwb. Tsis muaj ib lub cev nyob rau hauv geometry yog zoo meej raws li niaj zaus polyhedra. Tag nrho cov npe ntawm lub cev geometric no yog los ntawm Ancient Greece. Lawv txhais tau tias lub ntsej muag: tetrahedron - plaub-sided, hexahedron - rau-sided, octahedron - octahedral, dodecahedron - kaum ob sab, icosahedron - nees nkaum-sab. Tag nrho cov no geometric lub cevnyob hauv ib qho chaw tseem ceeb hauv Plato lub tswv yim ntawm lub ntiaj teb. Plaub ntawm lawv tus kheej lub ntsiab lus lossis cov koom haum: tetrahedron - hluav taws, icosahedron - dej, lub voos xwmfab - ntiaj teb, octahedron - huab cua. Lub dodecahedron embodied txhua yam uas muaj. Nws raug suav hais tias yog lub ntsiab, vim nws yog lub cim ntawm lub ntiaj teb.
Kev nthuav dav ntawm lub tswvyim ntawm polyhedron
Ib lub polyhedron yog ib phau ntawm ntau tus lej ntawm polygons xws li:
- Txhua sab ntawm ib qho ntawm cov polygons yog tib lub sijhawm sab ntawm tsuas yog ib sab ntawm ib sab ntawm tib sab;
- ntawm txhua tus polygons koj tuaj yeem mus rau lwm tus los ntawm kev hla cov polygons uas nyob ib sab mus.
Cov polygons uas ua ib lub polyhedron yog nws lub ntsej muag, thiab lawv ob sab yog ntug. Lub vertices ntawm polyhedra yog cov vertices ntawm polygons. Yog hais tias lub tswvyim ntawm ib tug polygon yog to taub raws li ca kaw cov kab tawg, ces ib tug tuaj txog ntawm ib tug txhais ntawm ib tug polyhedron. Nyob rau hauv rooj plaub thaum lub tswv yim no txhais tau hais tias ib feem ntawm lub dav hlau uas yog txwv los ntawm cov kab tawg, ces ib tug nto uas muaj polygonal pieces yuav tsum to taub. Convex polyhedron yog lub cev pw ntawm ib sab ntawm lub dav hlau uas nyob ib sab ntawm nws lub ntsej muag.
Lwm lub ntsiab lus ntawm polyhedron thiab nws cov ntsiab
Ib lub polyhedron yog ib qho chaw uas muaj cov polygons uas txwv lub cev geometric. Lawv yog:
- tsis-paj;
- convex (yog thiab tsis raug).
Ib lub polyhedron tsis tu ncua yog ib qho convex polyhedron nrog qhov siab tshaj plaws symmetry. Cov ntsiab lus ntawm ib txwm polyhedra:
- tetrahedron: 6 ntug, 4 ntsej muag, 5 vertices;
- hexahedron (lub voos xwmfab): 12, 6, 8;
- dodecahedron: 30, 12, 20;
- octahedron: 12, 8, 6;
- icosahedron: 30, 20, 12.
Euler's theorem
Nws tsim kev sib raug zoo ntawm tus lej ntawm cov npoo, ntsug thiab lub ntsej muag uas yog topologically sib npaug rau ib puag ncig. Los ntawm kev ntxiv tus naj npawb ntawm vertices thiab ntsej muag (B + D) ntawm ntau yam tsis tu ncua polyhedra thiab muab piv nrog cov npoo, ib qho qauv tuaj yeem tsim tau: cov lej ntawm cov ntsej muag thiab cov vertices sib npaug ntawm cov npoo (P) nce. los ntawm 2. Koj tuaj yeem muab cov qauv yooj yim:
B + D=R + 2
Cov mis no muaj tseeb rau txhua qhov convex polyhedra.
Basic definitions
Lub tswv yim ntawm ib lub polyhedron ib txwm tsis tuaj yeem piav qhia hauv ib kab lus. Nws yog ntau lub ntsiab lus thiab voluminous. Rau ib lub cev kom pom tau tias zoo li no, nws yuav tsum ua kom tau raws li ntau lub ntsiab lus. Yog li, lub cev geometric yuav yog polyhedron tsis tu ncua yog tias muaj raws li hauv qab no:
- nws yog convex;
- tib tus lej ntawm cov npoo sib sau ua ke ntawm txhua qhov ntawm nws qhov vertices;
- tag nrho nws lub ntsej muag yog cov polygons tsis tu ncua, sib npaug rau ib leeg;
- tag nrho nws cov kaum sab xis sib npaug.
Cov khoom ntawm ib txwm polyhedra
Muaj 5 yam sib txawv ntawm cov polyhedra tsis tu ncua:
- Lub voos xwmfab (hexahedron) - nws muaj lub kaum sab xis ntawm sab saum toj yog 90 °. Nws muaj 3 lub kaum sab xis. Tus lej ntawm cov kaum sab xis ntawm sab saum toj yog 270 °.
- Tetrahedron - lub kaum sab xis ntawm sab saum toj - 60 °. Nws muaj 3 lub kaum sab xis. Qhov sib npaug ntawm cov kaum sab xis saum toj yog 180 °.
- Octahedron - tiaj tus vertex lub kaum sab xis - 60 °. Nws muaj 4-sided kaum. Tus lej ntawm cov kaum sab xis ntawm sab saum toj yog 240 °.
- Dodecahedron - lub kaum sab xis ntawm vertex 108 °. Nws muaj 3 lub kaum sab xis. Qhov sib npaug ntawm cov kaum sab xis saum toj yog 324 °.
- Icosahedron - nws muaj lub kaum sab xis ntawm sab saum toj - 60 °. Nws muaj 5 lub kaum sab xis. Tus lej ntawm cov kaum sab xis saum toj yog 300 °.
Thaj chaw ntawm cov polyhedra tsis tu ncua
Qhov chaw saum npoo ntawm cov geometric lub cev (S) yog xam raws li thaj tsam ntawm ib lub polygon tsis tu ncua muab faib los ntawm tus lej ntawm nws lub ntsej muag (G):
S=(a: 2) x 2G ctg π/p
Lub ntim ntawm ib txwm polyhedron
Tus nqi no yog xam los ntawm kev sib npaug ntawm qhov ntim ntawm lub pyramid tsis tu ncua, ntawm lub hauv paus ntawm uas muaj ib lub polygon tsis tu ncua, los ntawm tus naj npawb ntawm lub ntsej muag, thiab nws qhov siab yog lub vojvoog ntawm lub inscribed sphere (r):
V=1: 3rS
Vim ntawm ib txwm polyhedra
Zoo li lwm lub cev geometric, ib txwm polyhedra muaj qhov sib txawv. Hauv qab no yog cov qauv uas koj tuaj yeem suav lawv:
- tetrahedron: α x 3√2: 12;
- octahedron: α x 3√2: 3;
- icosahedron; α x 3;
- hexahedron (cube): 5 x α x 3 x (3 + √5): 12;
- dodecahedron: α x 3 (15 + 7√5): 4.
Cov ntsiab lus ntawm cov polyhedra tsis tu ncua
Hexahedron thiab octahedron yog ob lub cev geometric. Hauv lwm lo lus, lawv tuaj yeem tau txais los ntawm ib leeg yog tias qhov nruab nrab ntawm lub ntiajteb txawj nqus ntawm lub ntsej muag ntawm ib tus raug coj los ua lub vertex ntawm lwm tus, thiab rov ua dua. Lub icosahedron thiab dodecahedron kuj yog ob. Tsuas yog lub tetrahedron yog dual rau nws tus kheej. Raws li txoj kev Euclid, koj tuaj yeem tau txais dodecahedron los ntawm hexahedron los ntawm kev tsim "ru tsev" ntawm lub ntsej muag ntawm lub voos xwmfab. Lub vertices ntawm ib tug tetrahedron yuav yog ib qho 4 vertices ntawm ib lub voos xwmfab uas tsis nyob ib sab ua khub raws ntug. Los ntawm hexahedron (lub voos xwmfab) koj tuaj yeem tau txais lwm yam polyhedra tsis tu ncua. Txawm tias muaj tseeb hais tias muaj ntau suav tsis tu ncua polygons, tsuas muaj 5 ib txwm polyhedra.
Radius ntawm ib txwm polygons
Muaj 3 concentric spheres txuam nrog txhua lub cev geometric:
- piav qhia, dhau ntawm nws lub ncov;
- sau, kov txhua lub ntsej muag hauv nws qhov chaw;
- nruab nrab, kov tag nrho cov npoo hauv nruab nrab.
Lub vojvoog ntawm tus kheej tau piav qhia yog suav los ntawm cov qauv hauv qab no:
R=a: 2 x tg π/g x tg θ: 2
Lub vojvoog ntawm ib qho chaw sau ntawv yog xam los ntawm tus qauv:
R=a: 2 x ctg π/p x tg θ: 2,
qhov twg θ yog lub kaum sab xis ntawm lub ntsej muag uas nyob ib sab.
Lub vojvoog ntawm qhov nruab nrab ntawm tus kheej tuaj yeem suav nrog siv cov qauv hauv qab no:
ρ=a cos π/p: 2 sin π/h,
qhov twg h tus nqi=4, 6, 6, 10 lossis 10. Qhov piv ntawm circumscribed thiab inscribed radii yog symmetrical nrog rau p thiab q. Nwsxam los ntawm tus qauv:
R/r=tg π/p x tg π/q
Symmetry ntawm polyhedra
Qhov sib luag ntawm cov polyhedra tsis tu ncua ua rau muaj kev txaus siab rau lub cev geometric no. Nws yog to taub raws li xws li ib tug txav ntawm lub cev nyob rau hauv qhov chaw, uas tawm ntawm tib tus naj npawb ntawm vertices, ntsej muag thiab ntug. Nyob rau hauv lwm yam lus, nyob rau hauv cov nyhuv ntawm ib tug symmetry transformation, ib tug ntug, vertex, lub ntsej muag yog tuav nws thawj txoj hauj lwm los yog txav mus rau qhov qub txoj hauj lwm ntawm lwm ntug, vertex, los yog lub ntsej muag.
Cov ntsiab lus ntawm symmetry ntawm ib txwm polyhedra yog cov yam ntxwv ntawm txhua yam ntawm lub cev geometric. Ntawm no peb tab tom tham txog kev hloov pauv zoo ib yam uas tawm ntawm ib qho ntawm cov ntsiab lus hauv nws txoj haujlwm qub. Yog li, thaum koj tig ib tug polygonal prism, koj tuaj yeem tau txais ob peb symmetries. Ib qho ntawm lawv tuaj yeem sawv cev raws li cov khoom ntawm kev xav. Ib tug symmetry uas yog cov khoom ntawm ib tug txawm tus naj npawb ntawm reflections yog hu ua ib tug ncaj kab. Yog hais tias nws yog cov khoom ntawm ib tug khib tooj ntawm reflections, ces nws yog hu ua inverse. Yog li, tag nrho cov kev sib hloov ntawm ib kab yog ncaj qha symmetry. Txhua qhov kev xav ntawm polyhedron yog qhov sib txawv ntawm qhov sib txawv.
Kom nkag siab zoo dua cov ntsiab lus ntawm cov polyhedra tsis tu ncua, peb tuaj yeem ua piv txwv ntawm tetrahedron. Txhua txoj kab ncaj nraim uas yuav dhau los ntawm ib qho ntawm cov vertices thiab qhov nruab nrab ntawm daim duab geometric no kuj yuav dhau los ntawm qhov nruab nrab ntawm lub ntsej muag rov qab rau nws. Txhua qhov ntawm 120 ° thiab 240 ° tig ib ncig ntawm kab yog ntau ntau.symmetry ntawm tetrahedron. Txij li thaum nws muaj 4 vertices thiab 4 lub ntsej muag, tsuas muaj yim symmetries ncaj qha. Ib qho ntawm cov kab dhau los ntawm nruab nrab ntawm ntug thiab qhov nruab nrab ntawm lub cev no dhau los ntawm qhov nruab nrab ntawm nws qhov opposite ntug. Ib qho kev sib hloov 180 °, hu ua ib nrab tig, ib ncig ntawm txoj kab ncaj nraim yog qhov sib luag. Txij li thaum lub tetrahedron muaj peb khub ntawm npoo, muaj peb symmetries ncaj qha. Raws li cov lus hais saum toj no, peb tuaj yeem xaus tias tag nrho cov naj npawb ntawm cov symmetries ncaj qha, suav nrog kev hloov pauv zoo ib yam, yuav ncav cuag kaum ob. Lub tetrahedron tsis muaj lwm yam symmetries ncaj qha, tab sis nws muaj 12 inverse symmetries. Yog li ntawd, tetrahedron yog tus yam ntxwv los ntawm tag nrho ntawm 24 symmetries. Kom meej meej, koj tuaj yeem tsim tus qauv ntawm tetrahedron li niaj zaus los ntawm cov duab los qhia thiab xyuas kom meej tias lub cev geometric no tsuas muaj 24 symmetries xwb.
Lub dodecahedron thiab lub icosahedron nyob ze tshaj plaws ntawm lub cev. Lub icosahedron muaj ntau lub ntsej muag, qhov loj tshaj plaws dihedral lub kaum sab xis, thiab tuaj yeem nruj nreem nias tawm tsam ib qho chaw inscribed. Lub dodecahedron muaj qhov me tshaj qhov tsis xws luag, lub kaum sab xis loj tshaj plaws ntawm lub vertex. Nws tuaj yeem sau nws tus kheej piav qhia txog qhov siab tshaj plaws.
Sweeps of polyhedra
Tsis tu ncua polyhedra, uas peb txhua tus tau txuas ua ke thaum yau, muaj ntau lub tswv yim. Yog hais tias muaj ib qho kev sau ntawm polygons, txhua sab uas tau txheeb xyuas nrog tsuas yog ib sab ntawm lub polyhedron, ces kev txheeb xyuas ntawm ob sab yuav tsum ua raws li ob yam xwm txheej:
- los ntawm txhua tus polygon, koj tuaj yeem hla cov polygon uas muajidentified sab;
- qhia sab yuav tsum muaj qhov ntev tib yam.
Nws yog cov txheej txheem polygons uas ua kom txaus siab rau cov xwm txheej no uas hu ua kev txhim kho ntawm polyhedron. Txhua lub cev muaj ob peb ntawm lawv. Yog li, piv txwv li, ib lub voos xwmfab muaj 11 ntawm lawv.
