N\u011bkter\u00e9 frakt\u00e1ly jsou abstraktn\u00ed objekty vytvo\u0159en\u00e9 fantazi\u00ed matematik\u016f, jin\u00e9 maj\u00ed vzory p\u0159\u00edmo v p\u0159\u00edrod\u011b: b\u0159ehy potok\u016f, \u0159ek, pob\u0159e\u017e\u00ed ostrov\u016f, plocha mozku, v\u011btven\u00ed strom\u016f, v\u011btven\u00ed c\u00e9v a \u017eil v t\u011ble, hromad\u011bn\u00ed bakteri\u00ed v koloni\u00edch, hory, mraky, elektrick\u00e9 v\u00fdboje, \u0161pinav\u00e9 skvrny, apod. Podle toho u\u017e vid\u00edme, \u017ee se frakt\u00e1ln\u00ed geometrie zab\u00fdv\u00e1 nepravidelnost\u00ed objekt\u016f.<\/p>\n\n\n\n
P\u0159ibl\u00ed\u017een\u00ed frakt\u00e1l\u016f<\/h3>\n\n\n\n
Abychom pochopili, o co vlastn\u011b jde, je nutn\u00e9 si p\u0159edstavit klasickou Euklidovskou geometrii<\/em>, kterou zn\u00e1me ze \u0161koly, a ve kter\u00e9 m\u00e1 ka\u017ed\u00fd objekt tzv. dimenzi hodnoty 1 (nap\u0159. p\u0159\u00edmka), 2 (nap\u0159. troj\u00faheln\u00edk) nebo 3 (nap\u0159. jehlan). Uveden\u00e1 hodnota je vlastn\u011b po\u010det parametr\u016f, kter\u00e9 jsou pot\u0159eba k jednozna\u010dn\u00e9mu ur\u010den\u00ed objektu. Tato dimenze se naz\u00fdv\u00e1 topologick\u00e1<\/strong>.<\/p>\n\n\n\n Ve frakt\u00e1ln\u00ed geometrii je objekt ur\u010dov\u00e1n podle tzv. dimenze frakt\u00e1ln\u00ed<\/strong> neboli Hausdorffovy – Besicovitchovy<\/em>, kter\u00e1 ur\u010duje m\u00edru nepravidelnosti objektu. Z \u010d\u00edseln\u00e9ho pohledu se jedn\u00e1 o to, \u017ee hodnota dan\u00e9 dimenze je v\u017edy o n\u011bjak\u00e9 to desetinn\u00e9 \u010d\u00edslo v\u011bt\u0161\u00ed ne\u017e v dimenzi topologick\u00e9. \u010c\u00edm je tato dimenze v\u011bt\u0161\u00ed v intervalu mezi dv\u011bma celo\u010d\u00edseln\u00fdmi hodnotami (0\u20131, 1-2, 2-3), t\u00edm je objekt \u010dlenit\u011bj\u0161\u00ed. A pr\u00e1v\u011b takov\u00fd \u00fatvar se naz\u00fdv\u00e1 frakt\u00e1l. \u0158\u00edk\u00e1 se tak\u00e9, \u017ee frakt\u00e1ln\u00ed objekty vznikl\u00e9 vytvo\u0159en\u00edm kopi\u00ed p\u016fvodn\u00edho objektu jsou statisticky sob\u011bpodobn\u00e9<\/strong>.<\/p>\n\n\n\n V podstat\u011b m\u016f\u017eeme \u0159\u00edct, \u017ee nap\u0159. zv\u011bt\u0161en\u00edm \u010di zmen\u0161en\u00edm objektu se d\u00edky teorii chaosu<\/em> (n\u00e1hodn\u00e9mu p\u016fsoben\u00ed vn\u011bj\u0161\u00edch vliv\u016f) nikdy nedos\u00e1hne dokonale v\u011brn\u00e9 kopie. Vzhled t\u011bchto objekt\u016f zaujme p\u0159edev\u0161\u00edm svou harmoni\u00ed mezi ur\u010ditou pravidelnost\u00ed a nahodilost\u00ed, podobn\u011b jako p\u0159\u00edrodn\u00ed \u00fatvary, kter\u00e9 mo\u017en\u00e1 p\u016fsob\u00ed chaoticky, p\u0159esto u nich vn\u00edm\u00e1me ur\u010dit\u00fd \u0159\u00e1d.<\/p>\n\n\n\n Teorie chaosu vlastn\u011b \u0159\u00edk\u00e1, \u017ee cosi jako zcela \u00fasp\u011b\u0161n\u00e1 p\u0159edpov\u011b\u010f nem\u016f\u017ee ve slo\u017eit\u00fdch jevech v\u016fbec existovat. Pokud by tomu tak toti\u017e bylo, dok\u00e1zali bychom bez probl\u00e9m\u016f \u0159\u00edci, \u017ee v \u010cesku na \u00fapat\u00ed Kru\u0161n\u00fdch hor bude v\u011btrn\u00e1 smr\u0161\u0165, nebo\u0165 v Braz\u00edlii m\u00e1vnul mot\u00fdl k\u0159\u00eddly, \u010d\u00edm\u017e dal p\u0159\u00ed\u010dinu vzniku jevu o n\u011bkolik t\u00fddn\u016f zpo\u017ed\u011bn\u00e9mu. Mo\u017en\u00e1 si pr\u00e1v\u011b nyn\u00ed uv\u011bdomujeme, \u017ee p\u0159\u00edroda se sv\u00fdm systematicko-chaotick\u00fdm p\u0159\u00edstupem neust\u00e1le br\u00e1n\u00ed lidsk\u00e9 snaze o vytvo\u0159en\u00ed co nejdokonalej\u0161\u00ed pravidelnosti pr\u00e1v\u011b t\u00edm, \u017ee jej\u00ed objekty jsou pouze „podobn\u00e9“.<\/p>\n\n\n\n V\u00fdchoz\u00ed rovnice pro ur\u010den\u00ed dimenze m\u00e1 tvar N.rD<\/sup>=1<\/strong>. Pokud chceme vyj\u00e1d\u0159it dimenzi D<\/em>, pou\u017eijeme logaritmus. Vzhledem k tomu, \u017ee logaritmick\u00e9 rovnice p\u0159i v\u00fduce p\u016fsob\u00ed pom\u011brn\u011b samo\u00fa\u010deln\u011b, kone\u010dn\u011b je k dispozici aplikace do konkr\u00e9tn\u00ed situace, nav\u00edc po\u010detn\u011b velmi jednoduch\u00e1. Kr\u00e1sa rovnice je i ve skute\u010dnosti, \u017ee je pou\u017eiteln\u00e1 pro topologickou dimenzi (doporu\u010duji jednoduch\u00e9 cvi\u010den\u00ed).<\/p>\n\n\n\n P\u0159edstavme si, \u017ee m\u00e1me \u010dtverec (resp. jeho obvod), ve t\u0159etin\u011b ka\u017ed\u00e9 jeho strany vytvo\u0159\u00edme nov\u00fd \u010dtverec (bez jedn\u00e9 strany – vznikne tak spojit\u00e1 lomen\u00e1 \u010d\u00e1ra). V podstat\u011b jsme p\u016fvodn\u00ed d\u00e9lku strany p\u0159etransformovali na 1\/3 (r=1\/3) p\u016fvodn\u00ed hodnoty, po\u010det podobn\u00fdch \u00fasek\u016f bude 5 (N=5).<\/p>\n\n\n\n N . rD<\/sup> = 1 Tato hodnota \u0159ad\u00ed nov\u011b vznikl\u00fd \u00fatvar n\u011bkam mezi p\u0159\u00edmku (D=1) a plochu (D=2). Nov\u011b vznikl\u00fd \u00fatvar bude z\u0159ejm\u011b neust\u00e1lou transformac\u00ed p\u0159eveden v objekt, kter\u00fd bude m\u00edt ve frakt\u00e1lov\u00e9m pojet\u00ed nekone\u010dn\u00fd obvod a j\u00edm vymezen\u00fd obrazec bude m\u00edt kone\u010dnou plochu. \u010cerven\u00e1 transformace odpov\u00edd\u00e1 prvn\u00edmu p\u0159ibl\u00ed\u017een\u00ed a nahrazuje \u017elutou \u010d\u00e1st. M\u00edsto jedn\u00e9 t\u0159etiny p\u016fvodn\u00ed d\u00e9lky byly p\u0159id\u00e1ny t\u0159i.<\/p>\n\n\n\n Jednoduch\u00e9 lomen\u00e9 \u010d\u00e1ry typu „Kochova vlo\u010dka“ jsou velmi snadno programovateln\u00e9 v n\u011bjak\u00e9m grafick\u00e9m prost\u0159ed\u00ed. Nab\u00edz\u00ed se program KTurtle, neboli \u017eelv\u00ed programov\u00e1n\u00ed. Tato linuxov\u00e1 platforma je pro algoritmizaci a v\u00fduku vhodn\u00e1 ji\u017e na z\u00e1kladn\u00ed \u0161kole a v podob\u011b podobn\u00fdch v\u00fdstup\u016f opravdu motivuj\u00edc\u00ed. Z pohledu geometrie jde pouze o systematickou volbu vhodn\u00fdch \u00fahl\u016f a d\u00e9lek, pop\u0159. vyu\u017eit\u00ed cykl\u016f. Vzhledem k jednoduch\u00e9 syntaxi a okam\u017eit\u00e9 zp\u011btn\u00e9 vazb\u011b je mo\u017en\u00e9 p\u0159ipravovat snadno programovateln\u00e9 \u00falohy.<\/p>\n\n\n\n Krom\u011b t\u011bchto jednoduch\u00fdch a snadno matematicky od\u016fvoditeln\u00fdch frakt\u00e1l\u016f je mo\u017en\u00e9 p\u0159istoupit k jejich aplikov\u00e1n\u00ed do v\u00fdtvarn\u00fdch \u010dinnost\u00ed. Po\u010d\u00ednaje Mandelbrotovou mno\u017einou m\u00e1me k dispozici r\u016fzn\u00e9 modifikace. I kdy\u017e se jedn\u00e1 o slo\u017eit\u00e9 struktury vznikaj\u00edc\u00ed na mno\u017ein\u011b komplexn\u00edch \u010d\u00edsel, maj\u00ed rovn\u011b\u017e sv\u00e9 z\u00e1vislosti. Na Internetu existuje \u0159ada program\u016f zdarma, kter\u00e9 umo\u017e\u0148uj\u00ed frakt\u00e1ly tvo\u0159it a transformovat.<\/p>\n\n\n\n Cantorovo diskontinuum<\/strong> –<\/strong> <\/em>vznikne postupn\u00fdm d\u011blen\u00edm \u00fase\u010dky na 3 shodn\u00e9 \u010d\u00e1sti. Prost\u0159edn\u00ed vypust\u00edme, a tak postupujeme d\u00e1l. D\u00e9lka „nekone\u010dn\u011b“ mnoha \u00fase\u010dek konvertuje k nule, ale po\u010det bod\u016f jde k nekone\u010dnu. Pomoc\u00ed v\u00fdpo\u010dtu lze ov\u011b\u0159it, \u017ee jde o frakt\u00e1l s dimenz\u00ed 0,63 (mezi bodem a line\u00e1rn\u00edm \u00fatvarem).<\/p>\n\n\n\n Kochova k\u0159ivka<\/strong> – na obr\u00e1zku je vid\u011bt, \u017ee postupn\u00e1 transformace d\u00e9lku k\u0159ivky neust\u00e1le prodlu\u017euje.<\/p>\n\n\n\n Kochova vlo\u010dka<\/strong> – k z\u00e1kladn\u00edmu troj\u00faheln\u00edku p\u0159id\u00e1me k prost\u0159edn\u00ed t\u0159etin\u011b ka\u017ed\u00e9 strany troj\u00faheln\u00edk o t\u0159etinu men\u0161\u00ed. Hranic\u00ed tohoto vznikl\u00e9ho \u00fatvaru je l\u00e1man\u00e1 k\u0159ivka, kter\u00e1 nikdy neprotne sebe sama. Plocha konvertuje ke konkr\u00e9tn\u00ed hodnot\u011b, obvod je nekone\u010dn\u011b dlouh\u00fd. V\u00fdpo\u010dtem dojdeme k frakt\u00e1lu s dimenz\u00ed 1,26 (mezi line\u00e1rn\u00edm a plo\u0161n\u00fdm \u00fatvarem). Kochova vlo\u010dka pat\u0159\u00ed k nejkr\u00e1sn\u011bj\u0161\u00edm frakt\u00e1l\u016fm, a to p\u0159edev\u0161\u00edm d\u00edky jednoduch\u00e9 linii, kter\u00e1 se d\u00edky podrobn\u011bj\u0161\u00edmu m\u011b\u0159\u00edtku m\u011bn\u00ed v zaj\u00edmavou „ostrovn\u00ed“ hranici. Ta se s ka\u017ed\u00fdm zjemn\u011bn\u00edm prodlu\u017euje, i kdy\u017e m\u00e1me pocit kone\u010dnosti.<\/p>\n\n\n\n Sierpinsk\u00e9ho troj\u00faheln\u00edk – <\/em><\/strong>z troj\u00faheln\u00edka vyjmeme troj\u00faheln\u00edk vytvo\u0159en\u00fd st\u0159edn\u00edmi p\u0159\u00ed\u010dkami. Toto opakujeme st\u00e1le u ka\u017ed\u00e9ho nov\u011b vznikl\u00e9ho \u00fatvaru, a\u017e dostaneme nekone\u010dn\u011b mnoho troj\u00faheln\u00edk\u016f s plochou konvertuj\u00edc\u00ed k nule.<\/p>\n\n\n\n Mengerova houba – <\/em><\/strong>trojrozm\u011brn\u00e1 m\u0159\u00ed\u017eka, kter\u00e1 m\u00e1 nekone\u010dn\u011b velk\u00fd povrch a jej\u00ed\u017e objem konvertuje k nule. P\u0159edstavme si krychli, kter\u00e1 se skl\u00e1d\u00e1 z 27 mal\u00fdch krychli\u010dek. Vyjmeme 7 z nich tak, aby zmizely prost\u0159edn\u00ed ve stran\u00e1ch a ta uprost\u0159ed krychle. V dal\u0161\u00edm kroku cel\u00fd postup opakujeme na v\u0161ech 20 zbyl\u00fdch krychli\u010dk\u00e1ch.<\/p>\n\n\n\n Barnsleyho kapradina <\/strong>– n\u00e1dhern\u00fd geometrick\u00fd \u00fatvar, kdy se ukazuje, \u017ee ka\u017ed\u00fd l\u00edstek kapradiny je t\u00e9m\u011b\u0159 kopi\u00ed sebe sama. V podstat\u011b vznikl tak, \u017ee p\u016fvodn\u00ed \u00fatvar byl postupn\u011b dopl\u0148ov\u00e1n dal\u0161\u00edmi \u00fatvary, kter\u00e9 byly zmen\u0161enou kopi\u00ed p\u016fvodn\u00edho. Tyto men\u0161\u00ed \u00fatvary byly pokl\u00e1d\u00e1ny tak, \u017ee mohly i p\u0159ekr\u00fdvat p\u016fvodn\u00ed objekt. Drobn\u00e9 zkreslen\u00ed je naprosto zanedbateln\u00e9, \u0159ekl bych, \u017ee a\u017e \u017e\u00e1douc\u00ed. Takto vznikl\u00fd \u00fatvar teprve vedl k pokusu o „zmatematizov\u00e1n\u00ed“.<\/p>\n\n\n\n Newtonova mno\u017eina<\/strong> – \u00fatvar vznikl grafick\u00fdm \u0159e\u0161en\u00edm Newtonovy rovnice x3<\/sup>-1=0 v oboru komplexn\u00edch \u010d\u00edsel. Body (modely \u0159e\u0161en\u00ed) v\u017edy v grafick\u00e9m zobrazen\u00ed konvertuj\u00ed k ur\u010dit\u00e9mu ko\u0159enu. P\u0159i grafick\u00e9m \u0159e\u0161en\u00ed se na rozhran\u00ed ploch ur\u010duj\u00edc\u00edch \u0159e\u0161en\u00ed objevily dal\u0161\u00ed nov\u00e9 obrazce.<\/p>\n\n\n\n Juliova mno\u017eina<\/strong> – n\u00e1hodn\u011b zvolen\u00e9 komplexn\u00ed \u010d\u00edslo c bude charakterizovat mno\u017einu. Ka\u017ed\u00fd bod z roviny v komplexn\u00edm oboru umocn\u00edme a p\u0159i\u010dteme k n\u011bmu c. V p\u0159\u00edpad\u011b, \u017ee v\u00fdsledn\u00e9 \u010d\u00edslo konvertuje k nule, pat\u0159\u00ed do Juliovy mno\u017einy. Velmi zaj\u00edmav\u00e9 je zbarven\u00ed mno\u017ein. Z\u00e1vis\u00ed toti\u017e na na po\u010dtu iterac\u00ed pot\u0159ebn\u00fdch ke zji\u0161t\u011bn\u00ed, zda \u010d\u00edslo pat\u0159\u00ed do Juliovy mno\u017einy.<\/p>\n\n\n\n Mandelbrotova mno\u017eina<\/strong> –<\/strong> <\/em>tato mno\u017eina v komplexn\u00ed rovin\u011b vznikne tak, \u017ee v ka\u017ed\u00e9m sv\u00e9m bod\u011b ur\u010duje vzhled Juliovy mno\u017einy. K ur\u010dit\u00e9mu komplexn\u00edmu \u010d\u00edslu p\u0159i\u010dteme jeho druhou mocninu. Tento v\u00fdsledek umocn\u00edme a p\u0159i\u010dteme k n\u011bmu p\u016fvodn\u00ed \u010d\u00edslo. Pokud v\u00fdsledek nep\u0159es\u00e1hne hodnotu 2, pat\u0159\u00ed bod do mno\u017einy. Mohou zde b\u00fdt r\u016fzn\u00e9 typy mno\u017ein z\u00e1visl\u00e9 na zvolen\u00e9m exponentu, co\u017e vede a\u017e k neuv\u011b\u0159iteln\u011b kr\u00e1sn\u00fdm \u00fatvar\u016fm. Jin\u00fdmi slovy, tato mno\u017eina je propojena s Juliovou mno\u017einou tak, \u017ee ka\u017ed\u00fd bod Mandelbrotovy mno\u017einy ur\u010duje vzhled mno\u017einy Juliovy ve vztahu k ur\u010dit\u00e9mu bodu. Po\u010det iterac\u00ed ur\u010duje barvu \u010d\u00edsla.<\/p>\n\n\n\n Atraktor<\/strong> – je definov\u00e1n jako kone\u010dn\u00fd stav syst\u00e9mu. Charakterizuje c\u00edl, do kter\u00e9ho sm\u011b\u0159uje pohyb, nap\u0159. u kyvadla. Uveden\u00e9 obr\u00e1zky t\u00e9to p\u0159edstav\u011b p\u0159\u00edli\u0161 neodpov\u00eddaj\u00ed, ale specifick\u00e9 frakt\u00e1ly ji\u017e dynamick\u00fd pohyb modeluj\u00ed.<\/p>\n\n\n\n Lorenz\u016fv atraktor<\/strong> – popisuje chov\u00e1n\u00ed vodn\u00edho kola, kdy v\u00fdsledkem je nekone\u010dn\u00e1 k\u0159ivka, kter\u00e1 nikdy neprotne sebe sama. P\u0159ipom\u00edn\u00e1 mot\u00fdl\u00ed k\u0159\u00eddla, kter\u00e1 se mo\u017en\u00e1 stala symbolem chaosu.<\/p>\n\n\n\n H\u00e9non\u016fv atraktor<\/strong> – vznik\u00e1 neust\u00e1l\u00fdm natahov\u00e1n\u00edm a oh\u00fdb\u00e1n\u00edm f\u00e1zov\u00e9ho prostoru. P\u0159es svou jednoduchost je pro matematiky st\u00e1le „z\u00e1hadn\u00fd“, nebo\u0165 jednotliv\u00e9 vznikl\u00e9 k\u0159ivky jsou vlastn\u011b p\u00e1ry k\u0159ivek vedle sebe. Bez v\u00fdpo\u010dtu nejde ur\u010dit, kde se objev\u00ed n\u00e1sleduj\u00edc\u00ed bod k\u0159ivky.<\/p>\n\n\n\n King\u00b4s dream<\/strong> – frakt\u00e1l vytvo\u0159en\u00fd opravdu jen pro radost.<\/p>\n\n\n\n Existuje mnoho aplikac\u00ed pro mobiln\u00ed opera\u010dn\u00ed syst\u00e9my, ve kter\u00fdch lze frakt\u00e1ly bu\u010f editovat, nebo prohl\u00ed\u017eet v podob\u011b galeri\u00ed. D\u00edky kvalit\u011b rozli\u0161en\u00ed a mo\u017enosti p\u0159ibl\u00ed\u017een\u00ed se tak nab\u00edz\u00ed zaj\u00edmav\u00e1 platforma nap\u0159\u00edklad pro v\u00fdtvarn\u00e9 hr\u00e1tky. P\u0159i\u010dem\u017e jde st\u00e1le o matematiku.<\/p>\n\n\n\n N\u011bkter\u00e9 vyzkou\u0161en\u00e9 aplikace: Fractile Plus, FractalTree, MandelPad, Fast Fractal (iOS), Fractals (Android), IFS Fractal (Windows Phone), jejich po\u010det st\u00e1le nar\u016fst\u00e1. V\u00fdhodou je jednozna\u010dn\u011b nez\u00e1vislost na zvolen\u00e9m syst\u00e9mu, pokud se bude hledat n\u011bjak\u00fd konkr\u00e9tn\u00ed frakt\u00e1l, v r\u00e1mci BYOD nen\u00ed t\u0159eba \u0159e\u0161it konkr\u00e9tn\u00ed aplikaci, ale c\u00edl.<\/p>\n\n\n\n Autor: Petr Chlebek<\/strong> <\/p>\n","protected":false},"excerpt":{"rendered":" Pokud hled\u00e1me motiva\u010dn\u00ed t\u00e9mata, kter\u00e1 mohou popularizovat matematiku, nem\u011bli bychom zapomenout na frakt\u00e1ly a jejich geometrick\u00e9 modelov\u00e1n\u00ed. Frakt\u00e1ln\u00ed geometrie vych\u00e1z\u00ed z pojmu frakt\u00e1l. Ten m\u016f\u017eeme definovat jako nekone\u010dn\u011b \u010dlenit\u00fd \u00fatvar, kter\u00fd je geometricky nepravideln\u00fd a z n\u011bho\u017e po rozd\u011blen\u00ed vznikne n\u011bkolik pseudokopi\u00ed p\u016fvodn\u00edho celku. Tyto \u00fatvary jsou nez\u00e1visl\u00e9 na m\u011b\u0159\u00edtku, z matematick\u00e9ho pohledu maj\u00ed nekone\u010dn\u011b dlouh\u00fd…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,7],"tags":[108,32,15,34],"class_list":["post-132","post","type-post","status-publish","format-standard","hentry","category-sekce-ict","category-sekce-matematiky-a-informatiky","tag-digitalni-kompetence-2","tag-grafika","tag-ict","tag-matematika"],"_links":{"self":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=132"}],"version-history":[{"count":2,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/132\/revisions"}],"predecessor-version":[{"id":809,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=\/wp\/v2\/posts\/132\/revisions\/809"}],"wp:attachment":[{"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ucitel.kvcso.cz\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Frakt\u00e1ly ve v\u00fduce<\/h3>\n\n\n\n
A) V\u00fdpo\u010det dimenze<\/strong><\/em><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/dimenze2.jpg\")
<\/figure><\/div>\n\n\n\n
rD<\/sup> = 1\/N
D . log r = log (1\/N)
D = log (1\/N) \/ log (r) \u2013 po \u00faprav\u011b
D = log 5 \/ log 3 = 1,46<\/strong><\/p>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/ctverec11.jpg\")
<\/figure><\/div>\n\n\n\nB) Frakt\u00e1ly v informatice<\/strong><\/em><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/vlocky1.jpg\")
<\/figure><\/div>\n\n\n\nC) V\u00fdtvarn\u00e9 n\u00e1pady<\/strong><\/em><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/fraktal6a.jpg\")
<\/figure><\/div>\n\n\n\nN\u011bkolik zn\u00e1m\u00fdch frakt\u00e1l\u016f<\/h3>\n\n\n\n
L – syst\u00e9my<\/strong><\/em><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/cantor1.jpg\")
<\/figure><\/div>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/kochova_hranice.jpg\")
<\/figure><\/div>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/sierpinski.jpg\")
<\/figure><\/div>\n\n\n\nSyst\u00e9my iterovan\u00fdch funkc\u00ed (polynomick\u00e9 frakt\u00e1ly)<\/strong><\/em><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/barnsley2.jpg\")
<\/figure><\/div>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/newton.jpg\")
<\/figure><\/div>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/julia.jpg\")
<\/figure><\/div>\n\n\n\n![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/mandelbrot.jpg\")
<\/figure><\/div>\n\n\n\nDynamick\u00e9 syst\u00e9my<\/em><\/strong><\/h4>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/atractor.jpg\")
<\/figure><\/div>\n\n\n\nFrakt\u00e1ly v mobiln\u00edch za\u0159\u00edzen\u00edch<\/strong><\/h3>\n\n\n\n
![\"Frakt\u00e1ln\u00ed](\"http:\/\/metodik.kvcso.cz\/storage\/obrazky\/fraktaly\/osfraktaly2.jpg\")
<\/figure>\n\n\n\n