Krooožnica
5. julij 2009 ob 7:08 | Borut Levart |Gruča slik čisto spodaj je zanimiva sama po sebi. Recimo. Obenem imajo slike preprost fizikalni oziroma geometrijski izvor, na katerega sam hitro pozabim, ko jih še enkrat pregledujem.
Veliko kolo ima enotski polmer in se vrti z enotsko frekvenco, v levi risanki gre na primer naokrog v petih sekundah. Na njegovem obodu je pritrjeno pol manjše kolo, ki se vrti trikrat hitreje, v petih obratih velikega se torej zavrti 15-krat. Tudi na njegovem obodu je kolo, trikrat manjše od velikega, vrti pa se devetkrat hitreje. Točka na njegovem obodu popiše sivo krivuljo. Njen podpis, vrednosti frekvenc, zapišimo v seznam Ω = (1, 3, 9). Podobno so naslovljene slike v gruči.
Tu se spomnim enega Calvin and Hobbes stripa, v njem je sicer samo Gašper s svojim fotrom, a se strip temu trenutku lepo poda.
Še nekaj. Če dajo frekvence pri deljenju s številom n isti ostanek (pravimo, da so med seboj kongruentne), ima krivulja n-tero simetrijo.
Tretji primer zgoraj: -3 ≡ 12 ≡ 7 (mod 5), -3 je kongruentno 12 je kongruentno 7 modulo 5, ostanek je zmeraj 2, krivulja ima peterno simetrijo, kot je označeno na sliki.
5. julij, 2009 ob 11:19
Odličen prispevek!
5. julij, 2009 ob 11:39
Hej!
Lepo da takole popularizirate matematiko.
Tudi slikice so lepe.
5. julij, 2009 ob 11:47
S podobno idejo (epicikli) je že Ptolomej utemeljil svoj geocentrični sistem našega osončja. S tem je za dolga stoletja pokopal Aristarhov heliocentrični model, vse do prihoda Kopernika in končno Galilea.
Tudi tako se da uporabljati matematika :)
5. julij, 2009 ob 12:06
@BOTA
Hočeš reči, da se se da matematiko uporabiti tudi za dokaz neresničnih stvari? In to za tak, temeljni dokaz, dokaz ki zavre razvoj za tisoč let?
Se strinjam, matematika ni “dobra” že kar na sebi, je le orodje, ki se ga lahko uporabi tako ali drugače.
Sodoben zgled za to so predpostavke ekonomskih teorij, ki se jih “dokaže” oz. zacementira njihovo veljavnost, njihovo oblast, tako da se “matematizira” ekonomijo. Če hoče kdo ugovarjati, npr. nasprotuje”finančnemu poglabljanju” zavarovalnice Vzajemna, se mu takoj zabrusi v obraz, da mora biti tako, saj je veljavnost temeljno dokazala matematika.
5. julij, 2009 ob 12:13
@aha
Drži, saj narava ne rešuje enačb :)
Povedano drugače matematika je samo orodje za opis modelov. Modeli so lahko pravi ali napačni. Tako to je :)
p.s. Tudi slavna Newtonova enačba gravitacije F=k*(m1*m2)/r^2 je napačna, pa se z njeno pomočjo vseeno da poleteti in ostati v vesolju :)
5. julij, 2009 ob 19:23
Calvin & Hobbes sta carja :D
5. julij, 2009 ob 22:21
>aha:
Seveda, jasno. Zakaj pa misliš, da je obveljal geocentrični model. Zato ker se ni skladal z opazovanji? Geocentrični model je prav lepo napovedoval običajne astronomske dogodke tistega časa. Seveda je pa vseeno napačen. Pravzaprav heliocentrični model na začetku sploh ni ničesar napovedoval, Kopernik pa je bolj kot kaj drugega pisal filozofijo. Heliocentrični model dobi moč napovedovanja šele z Keplerjem in Newtonom…
6. julij, 2009 ob 7:25
@nik
Nič ni tako zelo jasno. Prehod iz enega v drug sistem je predvsem ideološki in ne toliko znanstven. Tista travmatična točka prehoda se pozabi, in se v novem sistemu trdi, kako da je vse jasno-samoumevno, z enako vehemenco, kot je bilo vse jasno v prejšnjem sistemu. Moj poudarek ni toliko na znanstvenih revolucijah, kot na današnjem stanju ekonomske ideologije, neoliberalizma, kjer se pridobitve kontrarevolucije zaščiti tako, da se ji da avro “objektivne” matematike, da se matematizira celo ekonomsko znanost, in se tako prepreči, da bi se razvijale nove ekonomske teorije, ki bi lahko nadomestile sedanji neoliberlaizem.
Zgled:ko so ljudje hoteli preprečiti nadaljno krajo zavarovalnice Vzajemna, so se takoj oglasili glasovi, češ da zagovorniki podružbljajna nimajo dovolj ekonomskih izkušenj, matematičnih izkušenj: “kaj pa vi veste o razpršitvi tveganj”, skratka: “you cant argue with mathematics”.
6. julij, 2009 ob 8:42
>aha:
>Nič ni tako zelo jasno. Prehod iz enega v drug sistem je predvsem ideološki in ne toliko znanstven.
Zato pa pravim, da je Kopernik pisal filozofijo (Hm, ideologijo? Ajde, naj ti bo).
>Tista travmatična točka prehoda se pozabi, in se v novem sistemu trdi, kako da je vse
>jasno-samoumevno, z enako vehemenco, kot je bilo vse jasno v prejšnjem sistemu.
Točno, novi sistem se hvali kot samoumeven, o starem sistemu pa se govori omalovažujoče in se uporablja kot kritika cerkve. V resnici stvari sploh niso tako jasne, to je ideološki pogled. Jaz bi rekel da je prehod filozofski, danes pa se razlaga skozi materialistično (ateistično) ideologijo.
6. julij, 2009 ob 9:33
Spet se vračamo na znanstveno revolucijo, čeprav jaz ves čas hočem opozoriti na ekonomsko kontrarevolucijo. Ideologija je zelo širok pojem in religija spada vanjo.
Cerkev je sama po vsej sili vezala-utemeljevala svojo oblast na geocentrizmu, imela je možnost izvesti mehki prehod na heliocentirzem, Gallileo bi bil potem le še en svetnik, imel je ta potencial, pa je cerkev raje trmasto vztrajala, zato ni napačno, da se kritizira cerkev ob omembi heliocentirzma.
Modernost, francoska revolucija prenese vir suverenosti iz boga na ljudstvo, od takrat dalje se oblasti ne sme več utemeljevati v bogu.
Čeprav se v modernosti ukine boga in suverenost-oblast utemeljeno v bogu, se pri tem ne gre dovolj daleč, le tega teorija ne ukine dovolj, ne zato ker bi bog zares obstajal, ampak ker arhaične silnice “pračlovekovih” religij nevidno in neovirano delujejo dalje tudi v modernosti. Vrača se Realno in ne Bog.
Descartes je moderno znanost rešil vpetosti v teološko navlako, po njem je znanost sedaj samostojna, ni ji več treba zalagati teologije z dokazi o bogu - a pri Descartesu hkrati znanost tudi ne sme kritizirati boga, saj je na njem utemeljena kot na dobrem garantu ujemanja naših pojmov z zunanjimi stvarmi.
6. julij, 2009 ob 9:55
Pa, o ekonomiji ne morem govoriti. Edino poanto vidim v tem - če se lahko že v fiziki (astronomiji) zgodi, da imaš model, ki pojasnjuje (izbrana) opazovanja, je pa napačen (kot si se na začetku čudil geocentričnemu modelu), potem je v ekonomji to še toliko bolj verjetno, saj so opazovanja dosti bolj fleksibilna in “prilagodljiva”. Torej lahko s primerno matematiko v ekonomiji dokažeš “kar hočeš”…
Zakaj pa bi kdo prešel na heliocentrični model, če pa geocentrični model daje dobre napovedi in ni nobenega dokaza, da je heliocentrični pravilen in pa sploh ničesar ne napoveduje. Ravno vztrajanje na heliocentričnem modelu brez dokazov je ideološko.
(sicer pa vidim v prejšnjem sporočilu en kup ideologije, da).
6. julij, 2009 ob 12:45
Teorija je sestavljena iz idej, ideje pa so zelo ideološke.
8. julij, 2009 ob 10:22
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