Fibonacci tooj thiab cov golden piv: lub kev sib raug zoo

Nyob rau hauv lub qab ntuj khwb yog tseem muaj ntau unsolved mysteries, ib co kev uas zaum tau paub thiab piav qhia txog. Fibonacci tooj thiab cov golden piv yog lub hauv paus ntawm peb lub ntiaj teb, qhov kev siv ntawm nws lub cev thiab zoo tib neeg pom tau qhov sib xaav, uas nws muaj peev xwm xav tias txoj kev zoo nkauj thiab sib raug zoo.

Golden Section

Lub hauv paus ntsiab lus ntawm cov golden seem ntau thiab tsawg pab yog lub hauv paus ntawm zoo tag nrho ntawm tag nrho cov ntiaj teb thiab nws qhov chaw nyob rau hauv nws cov qauv thiab muaj nuj nqi, nws manifestation yuav pom nyob rau hauv cov xwm, kos duab thiab technology. Cov lus qhuab qhia ntawm lub golden piv tau siv raws li ib tug tshwm sim ntawm cov kev tshawb fawb ntawm ancient tej lus qhia muaj xwm tus xov tooj.

Nws yog raws li nyob rau hauv cov kev tshawb xav ntawm proportions thiab ratios ntawm lengths ntawm kev sib cais uas tau ua rau tus ancient philosopher thiab mathematician Pythagoras. Nws muaj pov thawj hais tias qhov kev sib cais ntawm lub ya mus rau hauv ob qhov chaw: lub X (me me) thiab Y (loj loj), qhov ratio ntawm loj me me yog sib npaug zos rau cov piv ntawm cov sum (tag nrho cov ntev):

X: Y = Y: X + Y.

Cov no kab zauv: x ² - x - 1 = 0, uas yog solved li x = (1 ± √5) / 2.

Yog hais tias peb saib lub ratio ntawm 1 / x, ces nws yog sib npaug zos rau 1.618 ...

Pov thawj ntawm txoj kev siv ntawm cov ancient thinkers ntawm lub golden piv yog muab nyob rau hauv phau ntawv ntawm Euclid lub "hais", sau raws li thaum ntxov raws li 3. BC, uas muaj ntaub ntawv txoj cai no los tsim kom tau qhov tseeb 5-gon. Lub Pythagoreans, qhov no xam yog dawb no dawb ceev vim hais tias nws yog ob qho tib si symmetrical thiab asymmetrical. Pentagram symbolizes lub neej thiab noj qab haus huv.

Fibonacci tooj

Cov naas ej phau ntawv liber abaci mathematician Leonardo Pizanskogo nyob rau hauv ltalis, uas tom qab ua lub npe hu ua Fibonacci, twb luam tawm nyob rau hauv 1202. Nyob rau hauv nws tus paub txog thawj lead qauv ntawm tus xov tooj nyob rau hauv uas txhua tus xov tooj yog lub sum ntawm tus xov tooj ntawm 2 yav dhau los cov xov tooj. Fibonacci ib theem zuj zus yog raws li nram no:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, thiab lwm yam

Tsis tas li ntawd, tus paub txog tau coj ib tug xov tooj ntawm cov kev cai:

• Tej xov tooj ntawm kab muab faib los ntawm lub tom ntej, yuav muaj sib npaug zos rau ib tug nqi uas nyhav 0.618. Thiab thawj Fibonacci tooj tsis muab xws li ib tug xov tooj, tab sis raws li koj tau zoo los ntawm thaum pib ntawm ib theem zuj zus, qhov ratio no yuav muaj ntau leej.
• Yog hais tias peb tau faib cov xov tooj ntawm kab nyob rau hauv lub yav dhau los ib tug, ces cov yuav maj mus 1.618.
• Ib tug xov tooj muab faib los ntawm lub tom ntej no ib tug, yuav qhia cov nqi uas nyhav 0,382.

Kev siv cov kev sib txuas lus thiab cov qauv ntawm cov golden seem, Fibonacci xov tooj (0.618) yuav pom tau tias muaj nyob rau hauv kev kawm txog zauv, tab sis kuj nyob rau hauv cov xwm, keeb kwm, architecture thiab kev tsim kho, thiab ntau lwm yam sciences.

Archimedean kauv thiab cov golden duab plaub

Qwj yog heev nyob rau hauv cov xwm, nws tau raug ntsuam xyuas los ntawm Archimedes, uas txawm coj nws kab zauv. Muab Kauv zoo yog raws li nyob rau hauv cov kev cai ntawm lub golden seem. Nyob rau hauv nws cov unwinding ntev tau, uas yuav tsum tau ua ntawv thov thiab cov proportions ntawm Fibonacci tooj, cov kauj ruam nce tshwm sim uniformly.

Parallel ntawm lub Fibonacci tooj thiab cov golden seem, koj yuav saib tau thiab tsim kom muaj ib tug "golden duab plaub", uas nws sab yog proportional raws li 1.618: 1. Nws yog lub tsev, kev mus los ntawm me me mus rau loj duab plaub li ntawd qhov ntev ntawm lub sab yuav ua ncaj rau cov xov tooj ntawm cov series. Cov kev siv yuav ua tau nyob rau hauv rov qab txiav txim, pib nrog lub square "1". Thaum txuas kab, fab ntawm lub duab plaub nyob rau hauv lub center ntawm cov kev tshuam tau Fibonacci los yog logarithmic kauv.

Lub keeb kwm ntawm kev siv ntawm kub proportions

Muaj ntau ancient Egypt architectural monuments tau ua siv kub proportions: nto moo Great Hauv lwm yam Architects Ancient tim Nkij teb chaws ispolzoval lawv dav nyob rau hauv kev tsim kho ntawm architectural khoom, xws li cov tuam tsev, siv lub amphitheatre, stadiums .. Piv txwv li, xws proportions tau siv nyob rau hauv kev tsim kho ntawm lub ancient Parthenon, ua yeeb yam Dionysos (Athens), thiab lwm yam khoom uas ua Masterpiece qub architecture, qhia kev sib raug zoo, raws li nyob rau hauv cov zauv regularity.

Pua xyoo tom qab, txaus siab nyob rau hauv lub Golden seem subsided, thiab tej kev cai tau raug tsis nco qab lawm, tab sis rov dua nyob rau hauv lub Renaissance nrog rau hauv phau ntawv Franciscan monk L. Pacioli Di Borgo "Divine kev faib ua feem" (1509). Nws Illustrations los ntawm Leonardo da Vinci tau raug coj, thiab cov uas ruaj lub npe tshiab ntawm "Golden seem". Muaj kuj tau pov thawj ncav 12 thaj chaw ntawm lub golden piv thiab tus sau tham txog yuav ua li cas nws manifests nws tus kheej nyob rau hauv cov xwm, nyob rau hauv kos duab thiab hu ua nws "lub hauv paus ntsiab lus ntawm lub tsev thaj yeeb nyab xeeb thiab cov xwm."

Leonardo lub Vitruvian txiv neej

Daim duab uas Leonardo da Vinci nyob rau hauv 1492 illustrated rau hauv phau ntawv ntawm Vitruvius, nws depicts ib tug tib neeg daim duab nyob rau hauv lub 2-txoj hauj lwm nrog ob txhais tes sib nrauj nyob rau hauv lub sab. Tus naj npawb inscribed nyob rau hauv ib lub voj voog thiab ib tug square. Qhov no xam yog xav tau cov canonical proportions ntawm tib neeg lub cev (txiv neej), tau piav los ntawm Leonardo rau lub hauv paus ntawm lawv txoj kev kawm nyob rau hauv lub treatises ntawm lub Roman kes duab vajtse Vitruvius.

hub lub cev raws li ib tug taw tes equidistant los ntawm lub kawg ntawm lub caj npab thiab txhais ceg xam tau tias yog mob plab, caj npab ntev sib npaug zos rau ib tug neeg lub siab, lub xub pwg dav tshaj plaws = 1/8 qhov siab, qhov kev ncua deb ntawm lub sab saum toj ntawm lub hauv siab rau cov plaub hau = 1/7, los ntawm lub hauv siab mus rau lub sab saum toj ntawm lub taub hau sab saum toj = 1/6 thiab lwm yam

Txij thaum ntawd los, cov duab no yog siv raws li ib tug cim, uas qhia cov internal symmetry ntawm tib neeg lub cev.

Lub sij hawm "Golden Section" Leonardo siv los piav proportional kev sib raug zoo nyob rau hauv tib neeg daim duab. Piv txwv li, qhov kev ncua deb ntawm lub duav rov sauv mus rau lub ko taw ntawm ob txhais ceg hnov mus rau tib deb ntawm lub puj ntaws mus rau lub sab saum toj raws li zoo raws li lub kev loj hlob ntawm cov thawj ntev (los ntawm lub duav rov sauv down). Cov suav yog ua nyob rau hauv tib lub piv ntawm cov feem nyob rau hauv lub xam ntawm lub golden piv thiab nyhav rau 1,618.

Tag nrho cov harmonious proportions yog feem ntau siv ntxias mus tsim kev zoo nkauj thiab impressive tej hauj lwm.

golden seem kev tshawb fawb nyob rau hauv 16-19 centuries

Siv cov golden piv thiab Fibonacci tooj, cov kev tshawb fawb ua hauj lwm rau lub proportions ntxiv rau centuries. Nyob rau hauv thaum uas tig mus nrog cov Leonardo da Vinci German artist Albrecht Dürer nws kuj yog muab kev koom tes nyob rau hauv txoj kev loj hlob ntawm cov kev tshawb xav ntawm qhov tseeb proportions ntawm tib neeg lub cev. Rau qhov no, lawv txawm tshwj xeeb compass twb tsim.

Nyob rau hauv lub xyoo pua 16th. nyob rau hauv cov kev sib raug zoo ntawm Fibonacci tooj thiab cov golden seem twb mob siab rau tus ua hauj lwm ntawm lub astronomer Kepler, uas thawj thov cov kev cai kom botany.

Tshiab "foundations" yuav tsum nyob rau hauv lub Golden seem 19. nrog rau cov ntawv ntawm "puas zoo nkauj Studies" German tus paub txog cov xibfwb Tseyziga. Nws tsa cov kev faib ua feem rau lub meej thiab tshaj tawm hais tias lawv yog universal rau txhua yam tshwm sim. Lawv tau kawm loj tus naj npawb ntawm cov neeg, los yog es lawv lub cev proportions (txog 2 txhiab.), Nyob rau uas cov lus xaus ntawm qhov kev tshwm sim ntawm lub statistical regularities paub tseeb hais tias nyob rau hauv lub proportions ntawm txawv lub cev qhov chaw: caj npab ntev, caj npab, txhais tes, ntiv tes, thiab lwm yam

kuj kos duab khoom (vases, architectural lug), suab paj nruas tones tau soj ntsuam, qhov ntev nyob rau hauv lub sau ntawv ntawm cov kwv huam - tag nrho cov Tseyzig tau tso tawm los ntawm qhov ntev ntawm cov kab thiab cov nuj nqis, nws kuj coined lub sij hawm "zauv kom zoo nkauj." Tom qab tau txais cov kev tshwm sim qhia tias cov Fibonacci series yog tau.

Fibonacci tooj thiab cov golden seem nyob rau hauv cov xwm

Nyob rau hauv lub zaub thiab tsiaj ntiaj teb no, muaj ib tug qauv ntawm shaping nyob rau hauv daim ntawv ntawm symmetry, uas yog pom nyob rau hauv cov kev taw qhia ntawm kev loj hlob thiab lub zog. Lub division rau hauv symmetrical qhov chaw, uas complied nrog lub golden kev faib ua feem - yog ib tug qauv ntau rau ntau yam nroj tsuag thiab tsiaj txhu.

Xwm nyob ib ncig ntawm peb tej zaum yuav tau piav los ntawm ib tug Fibonacci zauv, piv txwv li:

• qhov chaw ntawm tej ceg ntoo los yog nplooj ntawm cov nroj tsuag, raws li zoo raws li lub mus sib haum mus rau ib tug xov tooj ntawm muab tus xov tooj 1, 1, 2, 3, 5, 8, 13 thiab ntxiv;
• sunflower noob (rau ib tug scale cones, txiv puv luj cell), dag nyob rau hauv ob kab ntawm cov qwj nyob rau hauv txawv kev qhia;
• tus piv ntawm ntev ntawm tus Tsov tus tw thiab lub cev nabqa tsiav;
• qe zoo lawm, yog tias ib tug kab tshem tawm thoob plaws ib tug ntau ib feem ntawm nws;
• cov nam piv ntawm cov ntiv tes nyob rau hauv tib txhais tes.

Thiab, ntawm chav kawm, feem ntau nthuav ntaub ntawv yog qwj qwj plhaub qauv rau lub Web site, lub zog ntawm cov cua nyob rau hauv ib tug nag xob nag cua, lub ob helix DNA qauv, thiab galaxies - tag nrho cov ntawm lawv muaj xws li cov Fibonacci ib theem zuj zus.

Siv cov golden seem nyob rau hauv kos duab

Soj ntsuam kev koom tes nyob rau hauv lub kos duab ntawm nrhiav piv txwv ntawm cov kev siv ntawm cov golden seem, nthuav dav tshawb txawv architectural khoom thiab tej hauj lwm ntawm cov kos duab. Paub rau tus naas ej sculptures, lub creators ntawm uas ua raws li cov golden proportions, - tus pej thuam uas Olympian Zeus, Apollona Belvederskogo thiab Athena Parthenos.

Ib tug ntawm cov tej hauj lwm ntawm Leonardo da Vinci - "Portrait ntawm Mona Lisa" - tau ntev lawm ib tug kawm ntawm kev tshawb fawb tshawb fawb. Lawv nrhiav tau hais tias cov muaj pes tsawg leeg ntawm lub chaw ua hauj lwm muaj nkaus ntawm lub "Golden daim duab peb sab", tuaj koom ua ke rau hauv ib ncua pentagon lub hnub qub. Tag nrho cov ua hauj lwm da Vinci yog ib tug tim khawv yuav ua li cas sib sib zog nqus yog nws kev txawj ntse ntawm cov qauv thiab cov proportions ntawm tib neeg lub cev, thiaj li hais tias nws yuav ntes tus incredibly enigmatic luag ntawm lub Mona Lisa.

Golden Section architecture

Raws li ib qho piv txwv, zaum tau kawm masterpieces ntawm architecture, tsim los ntawm cov kev cai ntawm lub "golden seem": cov pyramids Iyiv, lub Pantheon, lub Parthenon, Notre-Dame de Paris, St. Vasiliya Blazhennogo thiab lwm tus neeg.

Parthenon - ib qho ntawm feem zoo nkauj vaj tse nyob rau hauv Ancient tim Nkij teb chaws (5 xyoo pua BC.) - muaj 8 kab thiab 17 rau ob tog, qhov ratio ntawm nws qhov siab rau qhov ntev ntawm lub sab yog sib npaug zos rau 0,618. Lub projections rau nws façade ua los ntawm "Golden seem" (yees duab hauv qab no).

Ib tug ntawm cov zaum uas invented thiab ntse thov kev txhim kho ntawm lub modular system rau cov proportions ntawm architectural khoom (thiaj li hu ua "Modulor") - yog tus Fabkis kes duab vajtse Le Korbyuze. Lub hauv paus ntawm lub Modulor muab tso rau lub xab system txuam nrog ib tug nyob ntawm division rau hauv qhov chaw ntawm lub cev tib neeg.

Lavxias teb sab kws kes duab vajtse Mikhail Kazakov, leej twg los ua ob peb thaj vaj tse nyob rau hauv Moscow, raws li zoo raws li lub Senate tsev nyob rau hauv lub Kremlin, thiab cov Golitsyn tsev kho mob (tam sim no tus 1 Soj ntsuam Pirogov.) - yog ib tug ntawm cov architects uas siv cov qauv siv thiab kev tsim kho kev cai lij choj lub golden seem.

Daim ntawv thov proportions nyob rau hauv kev tsim

Tus tsim ntawm tag nrho cov tsoos tsho designers ua tshiab dluab thiab cov qauv noj mus rau hauv tus account lub proportions ntawm tib neeg lub cev thiab cov kev cai ntawm lub golden seem, txawm hais tias los ntawm qhov, tsis yog txhua txhua tus neeg muaj lub zoo tagnrho proportions.

Thaum npaj ib tug toj roob hauv pes tsim thiab creation ntawm volumetric park compositions siv nroj tsuag (ntoo thiab lov tas vau), chaw thiab me me architectural khoom thiab cov qauv yuav siv tau "divine proportions". Tom qab tag nrho, tus yeees ntawm lub tiaj ua si yuav tsum tau tswj tsim lub tswv yim rau cov qhua, uas yuav dawb do next nws thiab nrhiav tau ib puas center.

Tag nrho cov ntsiab ntawm lub tiaj ua si yog nyob rau hauv xws proportions uas los ntawm txoj kev lub geometric qauv, txheeb ze txoj hauj lwm, teeb pom kev zoo, lub teeb, ua ib tug neeg lub tswv yim ntawm kev sib raug zoo thiab zoo tag nrho.

Kev siv cov golden seem nyob rau hauv cybernetics thiab technology

Cov cai ntawm lub Golden Section thiab Fibonacci tooj kuj tshwm sim nyob rau hauv lub zog zaus nyob rau hauv lub dab noj qhov chaw nrog cov elementary hais constituting cov tshuaj compound, nyob rau hauv qhov chaw nruab nyob rau hauv noob qauv ntawm DNA.

Zoo li cov dab tshwm sim nyob rau hauv tib neeg lub cev, uas manifests nws tus kheej nyob rau hauv lub biorhythms ntawm nws lub neej, nyob rau hauv qhov kev txiav txim hauv nruab nrog cev, xws li lub hlwb los yog tsis pom kev.

Algorithms thiab cov qauv golden proportions yog lug siv nyob rau hauv niaj hnub cybernetics thiab Informatics. Ib tug ntawm cov yooj yim paub tab, uas muab novice programmers los daws kom tau - thiab sau ib tug formula los mus txiav txim lub sum ntawm Fibonacci tooj mus rau ib tug tej yam muaj pes tsawg tus, siv programming lus.

Niaj hnub nimno cov kev tshawb fawb nyob rau hauv cov kev tshawb xav ntawm Golden piv

Txij li thaum lub nruab nrab ntawm lub xyoo pua 20th, txaus siab nyob rau hauv cov teeb meem thiab tej yam uas cov kev cai ntawm lub golden proportions ntawm ib tug neeg lub neej nce dramatically, thiab los ntawm ntau zaum los ntawm ntau yam professions: mathematicians, haiv neeg ntawm kev soj ntsuam, biologists, philosophers, kev kho mob cov tub txawg, economists, musicians thiab lwm tus neeg.

Nyob rau hauv lub US txij thaum 1970-hgodov pib luam tawm phau ntawv journal Lub Fibonacci txhuab peb lub hlis, uas luam tawm cov ntaub ntawv rau cov kawm. Nyob rau hauv lub xovxwm muaj tej hauj lwm nyob rau hauv uas lub generalized txoj cai ntawm lub golden seem thiab cov Fibonacci series yog siv nyob rau hauv ntau yam teb ntawm kev txawj ntse. Piv txwv li, rau encode ntaub ntawv, kev tshawb fawb tshuaj, lom, thiab lwm yam

Tag nrho cov no yuav qhia cov uas nrhiav tau ntawm ancient thiab niaj hnub kws tshawb fawb hais tias cov golden kev faib ua feem comprehensively txuas rau siv cov lus nug ntawm science thiab symmetry tshwm nyob rau hauv ntau tej hauj lwm, thiab tshwm sim ntawm lub ntiaj teb no nyob ib ncig ntawm peb.