Polyhedra tau nyiam cov neeg ua lej thiab cov kws tshawb fawb txawm nyob rau lub sijhawm qub. Cov neeg Iyiv tau tsim cov pyramids. Thiab cov Greeks tau kawm "polyhedra tsis tu ncua". Lawv qee zaum hu ua Platonic solids. "Traditional polyhedra" muaj cov ntsej muag tiaj tus, cov npoo ncaj, thiab ntsug. Tab sis cov lus nug tseem ceeb yeej ib txwm yog cov cai tswj hwm qhov kev cais no yuav tsum ua kom tiav, nrog rau cov xwm txheej hauv ntiaj teb ntxiv yuav tsum tau ua kom tau raws li cov khoom kom tsim nyog raws li polyhedron. Cov lus teb rau lo lus nug no yuav nthuav tawm hauv kab lus.
Problems in definition
Daim duab no yog dab tsi? Lub polyhedron yog ib qho khoom kaw uas muaj lub ntsej muag tiaj tus thiab cov npoo ncaj. Yog li ntawd, thawj qhov teeb meem ntawm nws cov ntsiab lus tuaj yeem raug hu ua qhov tseeb ntawm ob sab ntawm daim duab. Tsis yog txhua lub ntsej muag dag hauv dav hlau yeej ib txwm yog lub cim ntawm polyhedron. Cia peb coj tus "peb lub tog raj kheej" ua piv txwv. Nws muaj dab tsi? Ib feem ntawm nws nto peb ua khubkev sib tshuam ntsug dav hlau tsis tuaj yeem suav tias yog polygons. Qhov laj thawj yog tias nws tsis muaj vertices. Qhov saum npoo ntawm daim duab no yog tsim los ntawm peb lub rays uas sib ntsib ntawm ib qho taw tes.
Ib qho teeb meem ntxiv - dav hlau. Nyob rau hauv cov ntaub ntawv ntawm "peb lub tog raj kheej" nws nyob rau hauv lawv unlimited qhov chaw. Ib daim duab raug suav hais tias yog convex yog tias kab ntu txuas ob lub ntsiab lus hauv lub teeb kuj tseem nyob hauv nws. Cia peb nthuav tawm ib qho ntawm lawv cov khoom tseem ceeb. Rau convex sets, nws yog hais tias cov ntsiab lus ntawm cov ntsiab lus ntawm lub teeb yog tib yam. Muaj lwm hom duab. Cov no yog cov uas tsis yog-convex 2D polyhedra uas muaj qhov tsis zoo lossis qhov.
Cov duab uas tsis yog polyhedra
Ib lub ntsiab lus tiaj tus tuaj yeem sib txawv (piv txwv li, tsis yog convex) thiab tsis ua raws li cov lus txhais ib txwm muaj ntawm polyhedron. Txawm tias dhau los ntawm nws, nws raug txwv los ntawm ntu ntawm kab. Cov kab ntawm lub convex polyhedron muaj cov duab convex. Txawm li cas los xij, txoj hauv kev no rau kev txhais tsis suav nrog ib daim duab mus rau infinity. Ib qho piv txwv ntawm qhov no yuav yog peb lub rays uas tsis sib ntsib ntawm tib lub ntsiab lus. Tab sis nyob rau tib lub sij hawm, lawv txuas nrog lub vertices ntawm lwm daim duab. Kev lig kev cai, nws yog ib qho tseem ceeb rau ib lub polyhedron uas nws muaj cov chaw tiaj tus. Tab sis dhau sij hawm, lub tswv yim nthuav dav, uas ua rau muaj kev txhim kho tseem ceeb hauv kev nkag siab tus thawj "qis" chav kawm ntawm polyhedra, nrog rau qhov tshwm sim ntawm qhov tshiab, dav dua.
Kab
Cia peb qhia ib qho ntxiv. Ib qho polyhedron tsis tu ncua yog ib qho uas txhua lub ntsej muag yog qhov sib txuam tsis tu ncuaconvex polygons, thiab tag nrho cov vertices yog "zoo ib yam". Qhov no txhais tau hais tias txhua qhov vertex muaj tib tus lej ntawm cov polygons tsis tu ncua. Siv qhov kev txhais no. Yog li koj tuaj yeem pom tsib polyhedra tsis tu ncua.
Thawj kauj ruam mus rau Euler lub theorem rau polyhedra
Cov Greeks paub txog lub polygon, uas niaj hnub no hu ua pentagram. Lub polygon no tuaj yeem hu ua tsis tu ncua vim tias tag nrho nws sab yog sib npaug ntev. Kuj tseem muaj lwm daim ntawv tseem ceeb. Lub kaum sab xis ntawm ob sab sib law liag yog ib txwm zoo ib yam. Txawm li cas los xij, thaum kos rau hauv ib lub dav hlau, nws tsis txhais hais tias lub teeb convex, thiab ob sab ntawm polyhedron sib tshuam. Txawm li cas los xij, qhov no tsis yog ib txwm muaj. Mathematicians tau ntev xav txog lub tswv yim ntawm "non-convex" tsis tu ncua polyhedra. Lub pentagram yog ib tug ntawm lawv. "Star polygons" kuj tau tso cai. Ntau qhov piv txwv tshiab ntawm "polyhedra tsis tu ncua" tau pom. Tam sim no lawv hu ua Kepler-Poinsot polyhedra. Tom qab ntawd, G. S. M. Coxeter thiab Branko Grünbaum txuas ntxiv cov kev cai thiab pom lwm yam "polyhedra tsis tu ncua".
Polyhedral formula
Kev kawm txog cov lej no tau pib ntxov hauv keeb kwm ntawm lej. Leonhard Euler yog thawj zaug pom tias cov qauv qhia txog cov naj npawb ntawm lawv cov vertices, ntsej muag thiab cov npoo tuav rau convex 3D polyhedra.
Nws zoo li no:
V + F - E=2, qhov twg V yog tus naj npawb ntawm polyhedral vertices, F yog tus naj npawb ntawm ntug ntawm polyhedra, thiab E yog tus naj npawb ntawm lub ntsej muag.
Leonhard Euler yog Swisslej uas suav hais tias yog ib tus kws tshawb fawb loj tshaj plaws thiab muaj txiaj ntsig tshaj plaws ntawm txhua lub sijhawm. Nws tau qhov muag tsis pom rau feem ntau ntawm nws lub neej, tab sis qhov poob ntawm nws qhov pom tau ua rau nws ua rau nws muaj txiaj ntsig ntau dua. Muaj ob peb lub npe hu ua nws, thiab ib qho uas peb nyuam qhuav saib yog qee zaum hu ua Euler polyhedra formula.
Muaj ib qho kev qhia meej. Euler's formula, txawm li cas los xij, tsuas yog ua haujlwm rau polyhedra uas ua raws qee cov cai. Lawv dag nyob rau hauv qhov tseeb hais tias daim ntawv yuav tsum tsis txhob muaj tej qhov. Thiab nws yog qhov tsim nyog rau nws hla nws tus kheej. Ib lub polyhedron kuj tsis tuaj yeem ua los ntawm ob qhov sib koom ua ke, xws li ob lub voos xwmfab nrog tib lub vertex. Euler tau hais txog qhov tshwm sim ntawm nws qhov kev tshawb fawb hauv tsab ntawv mus rau Christian Goldbach xyoo 1750. Tom qab ntawd, nws tau luam tawm ob daim ntawv uas nws piav qhia txog qhov nws tau sim nrhiav pov thawj ntawm nws qhov kev tshawb pom tshiab. Qhov tseeb, muaj cov ntawv uas muab cov lus teb sib txawv rau V + F - E. Cov lus teb rau qhov suav F + V - E=X yog hu ua Euler tus yam ntxwv. Nws muaj lwm yam. Qee cov duab kuj tseem muaj tus yam ntxwv Euler uas tsis zoo
Graph Theory
Qee zaum nws tau lees tias Descartes muab Euler lub theorem ua ntej. Txawm hais tias tus kws tshawb fawb no tau tshawb pom qhov tseeb txog peb-seem polyhedra uas yuav tso cai rau nws kom tau txais cov qauv xav tau, nws tsis tau ua cov kauj ruam ntxiv no. Niaj hnub no, Euler tau txais txiaj ntsig nrog "leej txiv" ntawm txoj kev xav graph. Nws daws qhov teeb meem ntawm tus choj Konigsberg siv nws lub tswv yim. Tab sis tus kws tshawb fawb tsis tau saib ntawm polyhedron hauv cov ntsiab lusgraph txoj kev xav. Euler sim muab cov ntaub ntawv pov thawj ntawm cov mis raws li qhov decomposition ntawm polyhedron rau hauv qhov yooj yim. Qhov kev sim no poob qis ntawm cov qauv niaj hnub los ua pov thawj. Txawm hais tias Euler tsis tau muab thawj qhov tseeb rau nws cov qauv, ib tus tsis tuaj yeem ua pov thawj qhov kev xav uas tsis tau ua. Txawm li cas los xij, cov txiaj ntsig, uas tau lees paub tom qab, ua rau nws muaj peev xwm siv Euler's theorem nyob rau lub sijhawm tam sim no thiab. Thawj qhov pov thawj tau txais los ntawm tus lej lej Adrian Marie Legendre.
Pov thawj ntawm Euler tus qauv
Euler thawj zaug tsim cov qauv polyhedral raws li kev xav ntawm polyhedra. Niaj hnub no nws yog feem ntau kho nyob rau hauv cov ntsiab lus dav dav ntawm cov duab sib txuas. Piv txwv li, raws li cov qauv uas muaj cov ntsiab lus thiab kab ntu txuas rau lawv, uas yog nyob rau hauv tib seem. Augustin Louis Cauchy yog thawj tus neeg nrhiav kev sib txuas tseem ceeb no. Nws tau ua pov thawj ntawm Euler lub theorem. Nws, nyob rau hauv qhov tseeb, pom tias daim duab ntawm ib tug convex polyhedron (los yog niaj hnub no hu ua xws li) yog topologically homeomorphic rau ib tug kheej kheej, muaj ib tug planar txuas graph. Nws yog dab tsi? Ib daim duab kos duab yog ib qho uas tau kos rau hauv lub dav hlau nyob rau hauv ib txoj kev uas nws cov npoo sib ntsib los yog sib tshuam tsuas yog ntawm ib qho vertex. Qhov no yog qhov kev sib txuas ntawm Euler lub theorem thiab cov duab pom.
Ib qho qhia txog qhov tseem ceeb ntawm qhov tshwm sim yog tias David Epstein muaj peev xwm sau tau kaum xya daim ntawv pov thawj sib txawv. Muaj ntau txoj hauv kev los ua pov thawj Euler's polyhedral formula. Hauv kev nkag siab, cov pov thawj pom tseeb tshaj plaws yog cov txheej txheem uas siv lej induction. Qhov tshwm sim tuaj yeem ua pov thawjkos nws raws tus naj npawb ntawm cov npoo, ntsej muag lossis vertices ntawm daim duab.
Pov thawj ntawm Rademacher thiab Toeplitz
Tshwj xeeb txaus nyiam yog cov pov thawj hauv qab no ntawm Rademacher thiab Toeplitz, raws li txoj hauv kev ntawm Von Staudt. Txhawm rau kom pom tseeb Euler's theorem, xav tias G yog daim duab txuas nrog rau hauv lub dav hlau. Yog hais tias nws muaj schemas, nws muaj peev xwm tshem tawm ib tug ntug ntawm txhua tus ntawm lawv nyob rau hauv xws li ib txoj kev xws li khaws cia cov cuab yeej uas nws tseem txuas nrog. Muaj kev sib txuas lus ib-rau-ib qhov nruab nrab ntawm cov khoom tshem tawm rau kev mus rau cov duab txuas nrog tsis tau kaw thiab cov uas tsis yog qhov kawg tsis kawg. Qhov kev tshawb fawb no tau coj mus rau kev faib tawm ntawm "qhov chaw uas tau hais tseg" raws li lub npe hu ua Euler yam ntxwv.
Jordan nkhaus. Theorem
Qhov tseem ceeb thesis, uas yog siv ncaj qha lossis tsis ncaj rau hauv cov ntaub ntawv pov thawj ntawm cov qauv polyhedra ntawm Euler theorem rau cov duab, nyob ntawm Jordan nkhaus. Lub tswv yim no muaj feem xyuam rau kev nthuav dav. Nws hais tias ib qho yooj yim kaw nkhaus faib lub dav hlau ua peb pawg: cov ntsiab lus ntawm nws, sab hauv thiab sab nraud. Raws li kev txaus siab rau Euler's polyhedral formula tsim nyob rau hauv lub xyoo pua puv 19, ntau qhov kev sim ua kom dav dav rau nws. Qhov kev tshawb fawb no tau tsim lub hauv paus rau kev txhim kho ntawm algebraic topology thiab txuas nrog algebra thiab lej theory.
Moebius group
Nws tau tshawb pom sai sai tias qee qhov chaw tsuas yog "qhia" hauv ib qho chaw hauv zos, tsis yog thoob ntiaj teb. Pab pawg neeg paub zoo Möbius ua ib qho piv txwv ntawm xws linto. Nws tau pom me ntsis ua ntej los ntawm Johann Listing. Lub tswv yim no suav nrog qhov kev xav ntawm lub genus ntawm ib daim duab: tsawg kawg ntawm tus piav qhia g. Nws yuav tsum tau muab ntxiv rau saum npoo ntawm tus kheej, thiab nws tuaj yeem kos rau ntawm qhov chaw txuas ntxiv nyob rau hauv txoj kev uas cov npoo tsuas yog sib ntsib ntawm qhov chaw. Nws hloov tawm hais tias txhua qhov chaw orientable nyob rau hauv Euclidean qhov chaw tuaj yeem suav hais tias yog ib qho chaw nrog ib tus lej ntawm tes.
Euler daim duab
Tus kws tshawb fawb tau tshawb pom lwm qhov, uas tseem siv niaj hnub no. Qhov no hu ua Euler daim duab yog ib daim duab sawv cev ntawm lub voj voog, feem ntau yog siv los qhia txog kev sib raug zoo ntawm pawg lossis pawg. Cov kab kos feem ntau suav nrog cov xim uas sib xyaw rau hauv qhov chaw uas lub voj voog sib tshooj. Cov teeb yog sawv cev los ntawm lub voj voog lossis ovals, txawm hais tias lwm cov duab tuaj yeem siv rau lawv. Ib qho kev suav nrog yog sawv cev los ntawm kev sib tshooj ntawm ellipses hu ua Euler voj voog.
Lawv sawv cev rau pawg thiab subsets. Qhov tshwj xeeb yog cov voj voog tsis sib tshooj. Daim duab Euler muaj feem cuam tshuam nrog lwm cov duab sawv cev. Lawv feem ntau tsis meej pem. Daim duab sawv cev no hu ua Venn diagrams. Nyob ntawm cov lus nug, ob lub versions yuav zoo tib yam. Txawm li cas los xij, hauv daim duab Venn, cov voj voog sib tshooj tsis tas qhia qhov sib xws ntawm cov teeb, tab sis tsuas yog muaj kev sib raug zoo yog tias lawv cov ntawv tsis nyob hauv.kev sib tshuam lub voj voog. Ob qho kev xaiv tau raug coj los siv rau kev qhia cov kev xav ua ib feem ntawm kev ua lej tshiab ntawm xyoo 1960s.
Fermat thiab Euler's theorems
Euler tshuav ib qho cim pom hauv kev kawm lej. Algebraic tooj txoj kev xav tau enriched los ntawm ib tug theorem npe tom qab nws. Nws kuj yog qhov tshwm sim ntawm lwm qhov kev tshawb pom tseem ceeb. Qhov no yog qhov hu ua general algebraic Lagrange theorem. Euler lub npe kuj tseem cuam tshuam nrog Fermat lub theorem me me. Nws hais tias yog p yog tus lej tseem ceeb thiab a yog tus lej tsis faib los ntawm p, ces:
ap-1 - 1 divisible by p.
Qee zaum tib qhov kev tshawb pom muaj lub npe sib txawv, feem ntau pom muaj nyob rau hauv cov ntaub ntawv txawv teb chaws. Nws suab zoo li Fermat's Christmas theorem. Qhov no yog qhov kev tshawb pom tau paub ua tsaug rau ib tsab ntawv los ntawm tus kws tshawb fawb xa tuaj rau lub Kaum Ob Hlis 25, 1640. Tab sis cov lus nws tus kheej tau ntsib ua ntej. Nws tau siv los ntawm lwm tus kws tshawb fawb hu ua Albert Girard. Fermat tsuas yog sim ua pov thawj nws txoj kev xav. Tus kws sau ntawv qhia hauv lwm tsab ntawv tias nws tau txais kev tshoov siab los ntawm txoj hauv kev uas tsis muaj qhov kawg. Tab sis nws tsis tau muab pov thawj. Tom qab ntawd, Eider kuj tig mus rau tib txoj kev. Thiab tom qab nws - ntau lwm tus kws tshawb fawb nto moo, suav nrog Lagrange, Gauss thiab Minkosky.
Txoj kev qhia tus kheej
Fermat's Little Theorem kuj tseem hu ua qhov tshwj xeeb ntawm lub theorem los ntawm tus lej txoj kev xav vim Euler. Hauv qhov kev xav no, Euler tus kheej muaj nuj nqi suav cov lej zoo mus txog rau tus lej muab n. Lawv yog coprime nrog hwmn. Euler's theorem nyob rau hauv tus lej txoj kev xav yog sau siv Greek tsab ntawv φ thiab zoo li φ(n). Nws tuaj yeem txhais tau ntau dua raws li tus lej ntawm cov lej k hauv thaj tsam 1 ≦ k ≦ n uas qhov loj tshaj plaws divisor gcd(n, k) yog 1. Cov ntawv sau φ(n) tuaj yeem hu ua Euler's phi function. Cov lej k ntawm daim ntawv no qee zaum hu ua totative. Nyob rau hauv lub plawv ntawm tus lej txoj kev xav, Euler tus kheej muaj nuj nqi yog ntau yam, txhais tau hais tias yog ob tus lej m thiab n yog coprime, ces φ(mn)=φ(m) φ(n). Nws kuj tseem ua lub luag haujlwm tseem ceeb hauv kev txhais RSA encryption system.
Lub luag haujlwm Euler tau qhia hauv 1763. Txawm li cas los xij, lub sijhawm ntawd tus lej tsis tau xaiv ib lub cim tshwj xeeb rau nws. Hauv kev tshaj tawm xyoo 1784, Euler tau kawm txog cov haujlwm no kom ntxaws thiab xaiv tsab ntawv Greek π los sawv cev rau nws. James Sylvester tau sau lo lus "tag nrho" rau qhov no. Yog li ntawd, nws tseem hu ua Euler tag nrho. Tag nrho φ(n) ntawm ib tug zoo integer n ntau dua 1 yog tus naj npawb ntawm zoo integers tsawg tshaj n uas kuj tseem ceeb mus txog n.φ(1) yog txhais raws li 1. Euler muaj nuj nqi los yog phi(φ) muaj nuj nqi yog tus lej tseem ceeb heev-theoretic muaj nuj nqi cuam tshuam nrog cov lej tseem ceeb thiab qhov kev txiav txim ntawm cov lej.
