Fractal kya hai? Har paimaane par wahi pattern

Swa-samroop aakritiyaan

Fern ko dhyaan se dekho. Har patti poore fern ki chhoti nakal hai, aur patti ka har hissa phir se chhoti nakal. Ped ka tana daaliyon mein, daaliyaan shaakhaaon mein aur shaakhaaein tehniyon mein bantti hain; baadal, tat-rekha, pahaad aur bijli bhi aise hi bane hain. Aisi aakritiyaan, jo apna pattern chhote se chhote size par dohraati hain, fractal kehlaati hain. Ganit bahut saade niyamon se fractal kadam dar kadam bana sakta hai.

Sierpinski carpet

Polish ganitagya Sierpinski ne ek varg liya, use 9 barabar vargon mein kaata aur beech wala hata diya. Phir bache 8 vargon mein se har ek ke saath yahi kiya, aur aage bhi. Maan lo kadam n ke baad Rₙ bache varg aur Hₙ chhed hain. Har bacha varg agle kadam par 8 varg deta hai, isliye Rₙ₊₁ = 8Rₙ aur Rₙ = 8ⁿ. Har bacha varg ek naya chhed bhi banaata hai, jabki puraane chhed rehte hain, isliye Hₙ₊₁ = Hₙ + Rₙ: chhed 0, 1, 9, 73, … hain, yaani 1 + 8 + 8² + … + 8ⁿ⁻¹. Har kadam 8/9 kshetrafal bachta hai, isliye n kadam baad pehle varg ka (8/9)ⁿ bachta hai.

Sierpinski triangle

Yahi vichaar samabahu tribhuj par bhi chalta hai. Bhujaaon ke madhya-bindu jodne se ye 4 ek jaise samabahu tribhujon mein bant jaata hai, kyunki har kone ke tribhuj mein 60° ke kon ke dono taraf barabar aadhi bhujaayein hain, isliye wo khud samabahu hai. Beech wala tribhuj hatao, phir bache 3 par dohraao. n kadam baad 3ⁿ tribhuj, 1 + 3 + … + 3ⁿ⁻¹ chhed, aur (3/4)ⁿ kshetrafal bachta hai.

Koch snowflake

Swedish ganitagya von Koch ne 1904 mein ise bataaya; ye bhi samabahu tribhuj se shuru hota hai. Har bhuja ko teen barabar hisson mein baanto, beech wale hisse par samabahu tribhuj banao aur wo beech ka hissa hata do, taaki har bhuja 4 chhoti bhujaaon ka ubhaar ban jaaye. Nayi aakriti ki har bhuja par dohraao. 1 ikai ki bhujaaon se shuru karne par kadam n par 3 × 4ⁿ bhujaayein hain, har ek (1/3)ⁿ lambi, isliye parimaap 3 × (4/3)ⁿ hai: 3, 4, 16/3, 64/9, … Ye har kadam ke saath badhta rehta hai.

Kala mein fractal

Shaayad insaan ke banaaye sabse puraane fractal Bharat ke mandiron mein hain. Lagbhag 1025 CE mein poora hua Khajuraho ka Kandariya Mahadev Mandir aise shikhar mein uthta hai jo poore shikhar ki chhoti nakalon se bana hai, jin par aur chhoti nakalein hain, aur Madurai, Hampi, Rameswaram aur Varanasi ke mandiron mein bhi aise pattern hain. Nigeria ke Fulani shaadi ke kambal heere ke andar heere ke pattern bunte hain, aur Dutch kalaakaar M.C. Escher ki kriti ‘Smaller and Smaller’ chhipkaliyon ko ghatte size mein dohraati hai.

Fractal kyon maayne rakhte hain

Fractal sirf sundar chitra nahi hain. Inki sateek ginti saade niyamon se aati hai, isliye ye ghaaton aur patternon ka achha abhyaas hain: 8ⁿ, 3ⁿ aur 4ⁿ bahut tezi se badhte hain, jabki (8/9)ⁿ aur (3/4)ⁿ shoonya ki or ghatte hain. Ye ajeeb vyavhaar bhi karte hain. Sierpinski carpet ke paas lagbhag koi kshetrafal nahi bachta par anginat chhed hote hain, aur Koch snowflake ek tay vritt ke andar rehte hue bhi aisi seema rakhta hai jo kisi bhi chuni lambaai se lambi ho jaati hai. Aisi aakritiyaan vaigyaanikon ko tat-rekha, rakt-vaahiniyon aur nadi-jaal jaisi khurdari, shaakhaon wali cheezein samjhaane mein madad karti hain.

Is chapter se aur: thos aakritiyon ko samajhna, thos aakritiyon ke net, chapter ke notes aur NCERT solutions.