Polhedra li niaj zaus: ntsiab, symmetry thiab thaj chaw

Geometry zoo nkauj li ntawd, tsis zoo li algebra, qhov twg nws tsis yog ib txwm meej li cas thiab vim li cas koj xav tias, muab cov khoom pom kev. Qhov no amazing ntiaj teb ntawm lub cev sib txawv yog decorated nrog li niaj zaus polyhedra.

Cov ntaub ntawv tseem ceeb ntawm cov cai lij choj ntawm polyhedra

Raws li ntau, li niaj zaus polyhedra, los yog raws li lawv hu ua Platonic lub cev, muaj cov cim tshwj xeeb. Muaj ntau cov kev tshawb nrhiav pom zoo nrog cov khoom no. Thaum koj pib kawm cov kab mob hauv lub cev, koj to taub tias koj paub tsis paub dab tsi txog lub tswv yim li niaj zaus polyhedra. Kev nthuav qhia ntawm cov khoom hauv tsev kawm ntawv tsis yog ib txwm nyiam, muaj coob tus tsis nco qab lawv tau hu li cas. Hauv lub cim xeeb ntawm cov neeg feem coob, tsuas yog cov thawv muaj nyob xwb. Tsis muaj lub cev nyob hauv geometry muaj kev zoo li kev cai li niaj zaus polyhedra. Tag nrho cov npe ntawm cov kab zauv hauv lub cev yog los ntawm Ancient Greece. Lawv txhais tias muaj pes tsawg leej: tetrahedron - tetrahedral, hexahedron - hexahedral, octahedron - octahedral, dodecahedron - kaum ob-tog, icosahedron - nees nkaum-tog. Tag nrho cov qauv ntawm cov qauv hauv thaj tsam no yog qhov chaw tseem ceeb hauv Plato qhov kev xav ntawm lub ntiaj teb. Plaub ntawm lawv sawv cev cov ntsiab lus lossis essences: tetrahedron - hluav taws, icosahedron - dej, lub voos xwmfab - lub ntiaj teb, octahedron - pa. Lub dodecahedron embodied tag nrho cov uas tshwm sim. Nws suav hais tias yog lub ntsiab, vim nws yog lub cim ntawm lub qab ntuj khwb.

Generalization ntawm lub tswvyim ntawm ib tug polyhedron

Lub polyhedron yog ib phau ntawm ib tug xov tooj ntawm cov xov tooj uas:

• Txhua sab ntawm ib qho ntawm cov duab sib npaug ntawm ib sab ntawm tsuas yog ib qho sib txawv raws tib sab nkaus xwb;
• Los ntawm txhua tus ntawm cov duab, koj tuaj yeem mus rau lwm tus uas dhau los ntawm cov uas nyob ib sab.

Cov duab uas tsim cov polyhedron yog nws cov ntsej muag, thiab lawv cov sab yog cov sawv. Lub vertices ntawm cov voj yog cov vertices ntawm cov duab. Yog hais tias lub tswv yim ntawm ib tug polygon yog to taub tias yuav raug kaw cov kab hauv plhom, ces ib qho los txog tib lub ntsiab lus ntawm lub tshuab polyhedron. Nyob rau hauv cov ntaub ntawv thaum lub sij hawm no txhais tau tias yog ib feem ntawm lub dav hlau uas khi los ntawm cov kab tawg, nws yog ib qho tsim nyog to taub ib qhov chaw uas muaj ntau yam ntawm cov ntawv polygonal. Lub rooj sib npaug zos polyhedron yog lub cev pw ntawm ib sab ntawm lub dav hlau uas nyob ib sab rau nws lub ntsej muag.

Lwm lub ntsiab lus ntawm lub tshuab cuav thiab lub ntsiab lus

Lub polyhedron yog ib qhov sib npaug ntawm cov duab uas ua tiav ib lub cev hauv lub cev. Lawv yog:

• Nonconvex;
• Convex (txoj cai thiab tsis yog lawm).

Ib txwm siv polyhedron yog ib txoj kab uas muaj siab polyhedron nrog ntau yam symmetry. Cov ntsiab lus ntawm cov polyhedra:

• Tetrahedron: 6 npoo, 4 ntsej muag, 5 ib ntus;
• Hexahedron (lub voos xwmfab): 12, 6, 8;
• Dodecahedron: 30, 12, 20;
• Octahedron: 12, 8, 6;
• Icosahedron: 30, 20, 12.

Euler txoj theorem

Nws tsim kom muaj kev sib txuas ntawm cov npoo ntawm cov npoo, qhov muag, thiab ntsej muag zoo ib yam nkaus. Ntxiv cov vertices thiab cov ntsej muag (B + D) rau txawv polyhedra thiab sib piv lawv nrog cov npoo, ib qho peev xwm tsim ib qho raws li: qhov sib npaug ntawm cov ntsej muag thiab ib qhov sib npaug sib npaug ntawm cov npoo (P) tau nce los ntawm 2. Ib tus tuaj yeem ua ib qho yooj yim formula:

• B + F = P + 2.

Cov mis mos no muaj tseeb rau txhua qhov kev sib tham hauv polyhedra.

Cov Ntsiab Lus Txhais

Lub tswvyim ntawm ib txwm siv polyhedron tsis tuaj yeem piav los ntawm ib kab lus. Nws yog ntau dua polysemantic thiab voluminous. Rau ib lub cev yuav tsum pom zoo xws li, nws yog ib qho tseem ceeb uas nws ua raws ib qho ntawm cov ntsiab lus. Yog li, ib lub cev hauv lub cev yuav tsum yog ib txwm siv cov polyhedron thaum muaj raws li nram qab no:

• Nws yog convex;
• Tib cov npoo ntawm ib qho ntawm ib qho zuj zus;
• Tag nrho nws cov ntsej muag yog cov sib npaug zos sib npaug sib npaug sib luag;
• Tag nrho nws cov dihedral cov ces kaum yog sib npaug.

Cov khoom ntawm li niaj zaus polyhedra

Muaj 5 hom sib txawv ntawm cov polyhedra:

1. Lub voos xwmfab (hexahedron) - nws muaj ib lub tiaj kaum ntawm lub kaum ntawm 90 °. Nws muaj 3-angled angle. Qhov sib npaug ntawm cov npuaj kaum ntawm lub apex yog 270 °.
2. Lub tetrahedron yog ib lub tiaj tiaj tus nyob ntawm lub apex ntawm 60 °. Nws muaj 3-angled angle. Qhov sib npaug ntawm cov npuaj kaum ntawm lub vertex yog 180 °.
3. Lub octahedron yog lub kaum sab xis ntawm lub kaum ntawm 60 °. Nws muaj 4-kaum sab xis. Qhov sib ntxiv ntawm cov seem kaum ntawm sab saum toj yog 240 °.
4. Lub dodecahedron yog lub kaum sab xis ntawm lub kaum ntawm 108 °. Nws muaj 3-angled angle. Qhov sib npaug ntawm cov npuaj kaum ntawm qhov chaw nyob yog 324 °.
5. Icosahedron - nws muaj ib lub tiaj tus nyob rau sab saum toj - 60 °. Nws muaj 5-angled angle. Qhov sib npaug ntawm cov npuaj kaum ntawm lub apex yog 300 °.

Thaj tsam ntawm cov polyhedra

Qhov thaj tsam ntawm cov kab mob hauv ntiaj teb (S) no yog xam raws li thaj tsam ntawm ib qho polygon ntau zuj zus los ntawm tus npoo ntawm nws cov ntsej muag (G):

• S = (ib: 2) x 2G ctg π / p.

Lub ntim ntawm ib txwm polyhedron

Tus nqi no yog muab xam los ntawm kev muab cov ntim ntawm lub cev hauv qhov nruab nrab, ntawm lub hauv paus uas yog tus xwm txheej polygon, los ntawm cov ntsej muag, thiab nws qhov siab yog lub vojvoog ntawm cov kab sau kab (r):

• V = 1: 3rS.

Cov tagnrho ntawm cov cai lij choj polyhedra

Ib yam li lwm lub cev hauv lub cev, txoj cai polyhedra muaj ntau qhov sib txawv. Hauv qab no yog cov qauv uas lawv muaj peev xwm muab xam tau:

• Tetrahedron: α kh 3√: 12;
• Octahedron: α kh 3√q: 3;
• Icosahedron; Α kh 3;
• Hexahedron (lub voos xwmfab): 5 x α kh 3 h (3 + √5): 12;
• Dodecahedron: α Kh 3 (15 + 7 + 5): 4.

Cov ntsiab lus ntawm li niaj zaus polyhedra

Lub hexahedron thiab octahedron yog ob lub cev lub cev. Ua lwm yam lus, lawv tuaj yeem tau txais los ntawm txhua lwm qhov kev tshwm sim uas qhov chaw ntawm lub ntiajteb txawj nqus ntawm lub ntsej muag ntawm ib qho yog coj los ua tus vertex ntawm lwm tus, thiab lwm yam. Ntxiv thiab, cov icosahedron thiab cov dodecahedron yog dual. Tsuas yog ib tug tetrahedron yog dual rau nws tus kheej. Los ntawm txoj kev ntawm Euclid, ib tus tuaj yeem tau txais lub dodecahedron ntawm tus hexahedron los ntawm kev tsim "ru tsev" ntawm lub ntsej muag ntawm lub voos xwmfab. Lub vuas ntawm lub tetrahedron yog 4 yam ntawm lub voos xwmfab tsis nyob ib sab nyob rau hauv ob tog ua ke. Los ntawm lub hexahedron (lub voos xwmfab) ib qho peev xwm tau lwm yam li niaj zaus polyhedra. Txawm tias muaj tseeb hais tias kev uas muaj suav tsis tau, tu ncua polyhedra, muaj tsuas yog 5.

Lawm ntawm ib txwm lub cev

Nrog txhua yam ntawm cov kabmob geometric, 3 concentric spheres muaj kev cob cog rua:

• Piav, dhau ntawm nws cov vertices;
• Nws yog sau, kov txhua lub ntsej muag ntawm qhov chaw ntawm nws;
• Nruab nrab, kov tag nrho cov qhab nia nyob nruab nrab.

Lub vojvoog ntawm tus kheej piav qhia yog xam los ntawm cov qauv nram qab no:

• R = a: 2 x tg π / g x tg θ: 2.

Lub vojvoog ntawm tus kheej ntawm cov ntawv sau yog xam los ntawm cov qauv mis:

• R = a: 2 × ctg π / p × tg θ: 2,

Qhov twg θ yog ob sab-sided uas hais lus ntawm cov ntsej muag uas nyob ib sab.

Lub vojvoog ntawm qhov nruab nrab ntawm qhov nruab nrab tuaj yeem raug xam los ntawm cov qauv nram qab no:

• Ρ = cos cos π / p: 2 kev txhaum π / h,

Qhov twg h yog qhov tseem ceeb ntawm 4.6, 6.10 lossis 10. Qhov sib piv ntawm cov lus piav thiab sau kab lus yog qhov zoo rau p thiab q. Nws yog xam los ntawm cov mis:

• R / r = tg π / p × tg π / q.

Symmetry ntawm polyhedra

Lub symmetry ntawm polyhedra li niaj zaus ua rau lub ntsiab paj hauv cov kab mob hauv lub cev. Nws to taub hais tias lub zog ntawm ib lub cev nyob rau hauv qhov chaw, uas tawm tib lub xov tooj ntawm cov vertices, ntsej muag thiab sawv. Ua lwm yam lus, nyob rau hauv qhov kev txiav txim ntawm symmetry transformation, ntug, vertex, los yog lub ntsej muag tseem khaws nws txoj haujlwm qub, los yog tsiv mus rau qhov chaw tseem ceeb ntawm lwm qhov ntug, ntxiv vertex lossis ntsej muag.

Cov ntsiab lus ntawm symmetry ntawm cov polyhedra li niaj zaus nyob rau hauv txhua hom ntawm cov qauv hauv lub cev. Ntawm no peb tab tom tham txog tus neeg hloov siab los ntseeg, uas yoojyim cov ntsiab lus hauv qhov chaw tseem ceeb. Yog li, thaum tig lub polygonal prism, ob peb symmetries yuav tau. Ib qho ntawm lawv tuaj yeem sawv cev tau khoom raws li cov khoom siv ntawm kev xav. Symmetry, uas yog qhov khoom ntawm ib tug xov tooj ntawm cov reflections, yog hu ua ib txoj kab. Yog hais tias nws yog qhov khoom ntawm ib qho xov tsawg ntawm reflections, ces nws yog hu ua tus inverse. Yog li, txhua txoj kev ncig ntawm txoj kab ncaj ncaj yog sawv cev rau cov kev coj ncaj. Ib qho kev xav ntawm lub polyhedron yog qhov zoo nkaus li symmetry.

Yuav kom nkag siab zoo txog cov khoom symmetry ntawm polyhedra, peb tuaj yeem ua ib qho piv txwv ntawm lub tetrahedron. Tej kab uas yuav kis tau los ntawm ib qho ntawm lub vertices thiab qhov chaw ntawm lub geometric zoo, yuav muab qhov chaw, thiab los ntawm qhov chaw ntawm lub ntug opposite rau nws. Txhua qhov twists ntawm 120 thiab 240 ° nyob ib ncig ntawm txoj kab yog rau ntau tus zauv ntawm cov khoom tetrahedron. Txij li thaum nws muaj 4 ib sab thiab sab ntsej muag, muaj tsuas yog yim qhov ncaj qha cov kabmob. Ib qho ntawm cov kab ncaj nraim mus txog nruab nrab ntawm cov tav thiab qhov nruab nrab ntawm lub cev no dhau los ntawm qhov nruab nrab ntawm nws cov lus sib tw. Cov tig ntawm 180 °, hu ua ib nrab-tig, nyob ib ncig ntawm txoj kab yog symmetry. Txij li thaum ib tug tetrahedron muaj peb khub ntawm sawv, ces muaj peb ntau ncaj qha symmetries. Tawm los ntawm cov ntawv teev tseg, nws muaj peev xwm yuav xaus lus tias tag nrho cov nyiaj ncaj qha, nrog rau cov kev hloov zoo tib yam, yuav tuaj txog kaum ob. Muaj tsis muaj lwm txoj kab ke rau cov tetrahedron, tab sis nws muaj 12 rov qab cov kab symmetries. Thiaj li, lub tetrahedron yog tus cwj pwm los ntawm tsuas yog 24 symmetries. Rau clarity, koj yuav tsim ib tug qauv ntawm ib txwm tetrahedron los ntawm cardboard thiab nco ntsoov tias lub cev geometric tiag tiag muaj 24 symmetries.

Lub dodecahedron thiab icosahedron yog lub cev uas ze tshaj rau tus kheej. Lub icosahedron muaj ntau tshaj plaws ntawm cov ntsej muag, qhov loj tshaj dihedral lub kaum ntse ntse, thiab qhov ze nws tuaj yeem mus rau nested kheej. Lub dodecahedron muaj lub me tshaj angular defect, qhov loj tshaj plaws lub kaum sab xis ntawm lub apex. Nws tuaj yeem sau nws tus kheej ntau li ntau tau.

Deploying polyhedra

Qhov tseeb polyhedrons ntawm lub sweep, uas peb txhua tus tau ua ke ua ke thaum yau, muaj ntau lub ntsiab lus. Yog hais tias muaj ib phau ntawm cov duab, txhua sab uas tau txheeb xyuas nrog tsuas yog ib sab ntawm lub polyhedron, ces qhov kev qhia ntawm ob tog yuav tsum sib haum mus rau ob tug mob:

• Los ntawm txhua lub polygon nws yog ib qho ua tau los ntawm cov duab uas muaj qhov pom ntawm sab nraud;
• Cov tog neeg yuav tsum muaj tib lub sijhawm.

Nws yog qhov sau los ntawm cov kabmob uas txaus siab rau cov haujlwm no hu ua unfolding ntawm lub polyhedron. Txhua tus ntawm lub cev muaj ntau yam ntawm lawv. Piv txwv, lub voos xwmfab muaj 11 daim.