– Dakatar da bincike! Mun same shi a ƙarshe,' in ji Kwamishinan MacCarnigan.
– Wanene yallabai? ya tambayi Laftanar na biyu Pierron.
"Zuwa ɗaya daga cikin mafi girman 'yan damfara da za ku taɓa tunanin. Kusan shekaru 50 nake nema.
– Ba ni da wani tunani, Kwamishinan. Game da wanene?
– Lambarsa Ein Stein ce kuma ta ɗauki kusan tsawon rayuwata don gano ta.
- Wanene game da shi? Kuna da wasu hotunan ku a wajen?
– Eh, ina da ita a nan, wannan shine yadda yake kama, amma kar a yaudare shi da bayyanarsa marar laifi, wannan mai martaba a nan ya sa mu cikin shakka har kusan shekaru goma.
Don haka MacCarnigan ya nuna wa Agent Pierron hoton Ein Stein, wannan hoton:
In Stein.
Wannan taƙaitaccen tarihin ƴan sanda na iya zama kamar wasa, amma idan muka canza masu bincike na masana lissafi, ya zama ɗaya daga cikin mafi ban mamaki binciken ilimin lissafi da ya faru a cikin 'yan shekarun nan. Amma don fahimtar iyakar wannan labarin, da farko dole ne mu yi magana game da ɗaya daga cikin fannonin da ilimin lissafi da fasaha suka haɗu: mosaics.
mosaic jaridu
Dukanmu mun ga mosaic a wani lokaci a rayuwarmu. Waɗannan ƙananan ayyukan fasaha ne ko kayan ado waɗanda aka yi ta amfani da ƙananan ƙananan da suka dace tare.
Wasu misalai na mosaics
Lokacin da muke magana game da mosaics a lissafin lissafi, yawanci muna komawa ga abin da aka sani da tessellations, wanda shine hanyar tsara guntu ko tayal ta yadda waɗannan sassan suna da gefuna na gama-gari kuma kada su bar ramuka.
Tun da dadewa masana lissafi da lissafi sun yi tambaya mai zuwa
Wani nau'i ne zan iya tile jirgin da su?
Wato, wane nau'i nau'i ne zan iya amfani da shi don haka, sanya su ta yadda fale-falen sun taɓa juna a bangarorin gama gari, babu rata a cikin jirgin. A bayyane yake cewa da'irori ba su cikin wannan rukunin da aka zaɓa, tunda idan ina so in yi tile jirgin ta amfani da da'ira kawai za su bar ni da ramuka. Taho, zan jefa kafaffen gyale.
da'irori suna barin gibba
Duk da haka, akwai wasu siffofi da yawa waɗanda za mu iya taya jirgin da su, kamar triangles, murabba'ai ko hexagons.
Tessellation tare da polygon guda ɗaya na yau da kullun
Ko kuma za mu iya tile jirgin tare da haɗuwa da waɗannan ko wasu adadi.
Tessellation tare da polygons na yau da kullun
Ko kuma kuna iya har ma da tayal jirgin tare da ƙarin haɗe-haɗe masu yawa:
Sauran yuwuwar tilings
Amma kun yi la'akari da nau'in tilings iri-iri da kuka gabatar, duk suna da wani abu guda ɗaya, wato, na lokaci-lokaci. Kalmar lokaci-lokaci tana nufin gaskiyar cewa akwai fassarar, ban da sifili, wanda ya bar dukan mosaic iri ɗaya. Daga abin da muka fahimta, yana daidai da gaskiyar cewa idan muka yi tile a saman, yumbura idanu kuma wani ya motsa dukan mosaic a cikin wani takamaiman shugabanci sannan kuma ya rufe idanu, ba za mu iya fahimtar bambanci tsakanin mosaic na asali da wanda aka yi hijira ba.
mosaics ba tare da jaridu ba
Ya bambanta da tilings na lokaci-lokaci muna samun tilings waɗanda ba na lokaci-lokaci ba, waɗanda ba a fassara su ba, ba sifili ba, waɗanda ke barin mosaic da kamanni iri ɗaya. Ba shi da wahala a sami mosaics ba na lokaci-lokaci ba, ya isa, alal misali, ɗaukar tiling lokaci-lokaci, bari mu yi tunanin, alal misali, wanda aka kafa ta hanyar murabba'i kawai, kuma murabba'in murabba'in gabaɗayan mosaic ya kasu kashi biyu triangles. A bayyane yake har yanzu tessellation na jirgin sama ne, amma ba za a sami wata fassarar da za ta bar dukan tesserae iri ɗaya ba tun da za mu iya bambanta tsakanin mosaic na asali da wanda aka raba shi kawai ta hanyar lura da yanayin da aka gyara na triangles biyu.
tiling na lokaci-lokaci
Amma yanzu ne lokacin da abubuwa suke da ban sha'awa, domin shi ne lokacin da ra'ayin mosaic na lokaci-lokaci ya bayyana, waɗanda, yayin da ba na lokaci-lokaci ba, suna gamsar da ƙarin yanayin cewa ba su da manyan yankuna masu yawa na lokaci-lokaci. Hakanan ana iya jin wannan ra'ayin kamar a cikin mosaic na aperiodic, idan muka ɗauki babban isaccen yanki, ba ya maimaita sauran mosaic. Tabbatar cewa samfurin mosaic wanda babu wani lokaci da aka kwatanta a baya ba na ɗan lokaci ba tun da za mu iya samun manyan yankuna masu kama da juna waɗanda suke lokaci-lokaci, kawai ɗauki manyan ɓangarorin da ba su haɗa da triangle ba.
Don haka, tambayar da a zahiri take tasowa ita ce:
Akwai mosaics aperiodic?
Wannan tambaya, wacce aka fara yin nazari a cikin rabin na biyu na karni na karshe, ba da daɗewa ba ta sami amsa mai gamsarwa kuma ɗaya daga cikin na farko da ya samo tessellation na lokaci-lokaci shine Raphael M. Robinson. Mosaic da Robinson ya kwatanta a cikin 1971 ya ƙunshi tesserae guda 6 a jere.
robinson tiles
Bayan 'yan shekaru, kuma a cikin 70s, Roger Penrose ya sami fale-falen buraka guda biyu waɗanda za a iya gina su, kowanne yana amfani da tayal guda biyu kawai. Na farko daga cikin waɗannan tessellation an samo su ta hanyar rhombuses daban-daban guda biyu:
Tiles na Penrose (rhombuses)
Kuna iya yin mosaics kamar haka:
Penrose tiling
Na biyu na waɗannan tilings na aperiodic an bayar da su ta hanyar guda biyu da aka sani da kite da kibiya, saboda dalilai masu ma'ana:
Tiles Penrose (comet da kibiya)
To, akwai shakka cewa shuka zai iya zama kamar haka:
Shin akwai mosaics na lokaci-lokaci ta hanyar tayal guda ɗaya?
An san wannan matsala da matsalar Ein Stein (daga Jamusanci don "dutse") kuma kusan shekaru 50 ba a warware ta ba. Har zuwa Maris da ya gabata!
Binciken Ein Stein
A ranar 20 ga Maris, masanan kimiyya David Smith, Joseph Samuel Myers, Craig S. Kaplan da Chaim Goodman-Strauss daga Jami'o'in Cambridge, Waterloo da Arkansas sun buga aikin 'An aperiodic monotile' a cikin abin da suka bayyana yiwuwar siffar tayal da ake nema wanda ke ba da haɓakar mosaic aperiodic tare da guda ɗaya.
Tile da Smith, Myers, Kaplan da Goodman-Strauss suka bayyana
Tare da wannan tayal guda ɗaya, wanda a gare ni yana kama da T-shirt, yana nuna cewa ana iya gina mosaics na zamani kamar haka:
Aperiodic mosaic na tayal
Idan sha'awarku ta kasance mai hankali game da batun, zaku iya zurfafa zurfin bincike kan wannan binciken a cikin bidiyon da ke gaba,
wanda masu bincikensa ke magana da sauran mutanen da suka dace a yankin, gami da lambar yabo ta Nobel a Physics Roger Penrose.
ABCdario de las Matemáticas wani sashe ne wanda ya taso daga haɗin gwiwa tare da Hukumar Watsawa ta Royal Spanish Mathematical Society (RSME).
GAME DA MARUBUCI
Victor M. Manero
