Tiling patterns kya hain?

Bina gap ke dhakna

Kisi jagah ko shapes se aise dhakna ki na koi gap rahe na koi overlap, tiling kehlaata hai. Square tiles ka farsh ek tiling hai; eenton ki deewar bhi. Chapter ka ye hissa poochhta hai ki kaun si shapes kaun si jagah dhak sakti hain, aur jab dhakna namumkin ho to ye pakka kaise jaanein.

Tangram

Tangram China mein shuru hui paheliyan hain. Ek square ko saat tukdon mein kaata jaata hai: do bade triangle, ek medium triangle, do chhote triangle, ek square aur ek parallelogram. Inhi saat tukdon ko alag tarah se jodkar teer, machhli, billi, daudta aadmi aur bahut kuch ban sakta hai. Agar square ki side 4 units ho, to tukdon ke area 4, 4, 2, 1, 1, 2 aur 2 hain, isliye saaton se bani har figure ka area 16 hai. Figure shuru karne ka achha tareeka hai pehle ye dhoondhna ki do bade triangle kahan fit honge, kyunki dono milkar aadha area dhakte hain.

2 × 1 tiles se grid dhakna

4 × 6 grid mein 4 rows aur 6 columns hain. Ise 2 × 1 tiles se dhak sakte hain, jinhe khada ya leta rakh sakte hain: jaise har column mein do khadi tiles. 4 × 7 grid bhi aise hi dhakta hai. 5 × 7 grid nahi dhak sakta, kyunki usme 35 squares hain aur har tile theek 2 dhakta hai, isliye kitni bhi tiles even ginti ke squares hi dhakengi. Aam taur pe m × n grid tab dhakta hai jab m ya n even ho, aur jab dono odd hon to kabhi nahi.

Chessboard rang ki chaal

5 × 3 grid se ek square hatao to 14 squares bachte hain, jo even hai. Kabhi grid phir dhak jaata hai, aur kabhi kitni bhi koshish karo, nahi dhakta. Pakka jaanne ke liye squares ko chessboard ki tarah rango. Har 2 × 1 tile do padosi squares dhakta hai, ek kaala aur ek safed, isliye dhake gaye har hisse mein dono rang barabar hote hain. Upar ka beech waala square hataane pe 8 safed aur 6 kaale squares bachte hain, isliye koi tiling nahi hai. Rangon se sawaal ko mushkil banaane ne hi use suljhaana aasaan kar diya.

Poora plane dhakna

Squares kabhi na khatm hone waala farsh dhak sakte hain, aur equilateral triangle aur regular hexagon bhi. Har case mein ek bindu pe milne waale angles ka jod theek 360° hai: chhe 60° angles, chaar 90° angles ya teen 120° angles. Regular pentagon ke 108° angles ye nahi kar sakte. Plane ek se zyaada shapes se, ya tedhi-medhi shapes se, bhi dhak sakta hai: Dutch kalakaar Escher (1898–1972) ne ise jaanwaron ki shapes se dhaka, aur nayi tilings aaj bhi mil rahi hain; book ki ek tiling to 2023 mein hi mili.

Kudrat bhi tiling karti hai. Madhumakkhi ke chhatton aur kuch tataiyon ke ghonslon ka saamne ka hissa hexagon khaanon se dhaka hota hai, jinme ande, chhote keede aur khaana rakha jaata hai, bina koi jagah barbaad kiye. Keede itne regular hexagon kaise banaate hain, ye scientists ke liye aaj bhi paheli hai.

Ye kyun zaroori hai

Tiling ke vichaar farsh, packing aur design mein kaam aate hain, aur geometry mein aaj bhi ispe research chal rahi hai. Hexagon kaise banta hai ye regular hexagon kaise banayein? mein dekho, aur geometric constructions kaise karein? dohraao. Textbook ke sawaalon ke jawab, saari tiling paheliyon samet, Part 2 Chapter 6 ke NCERT solutions mein hain; poore Part 2 Chapter 6 ke notes dekho.