Rotational symmetry kya hai?

Kaagaz ki windmill

Kaagaz ki windmill symmetrical dikhti hai, par use kaise bhi modo, dono hisse kabhi mel nahi khaate: usme koi line of symmetry nahi hai. Phir bhi use beech ki pin ke chaaron taraf 90° ghumao to wo bilkul waisi dikhti hai. Hum kehte hain ki windmill mein rotational symmetry hai.

• Jis fixed point ke chaaron taraf figure ghumaayi jaati hai, wo centre of rotation hai.
• Jitna ghumane pe figure bilkul waisi dikhe, wo angle of rotational symmetry, ya sirf angle of symmetry, hai.

Windmill ke liye 90° se kam ghumana kaam nahi karta. Uske angles of symmetry 90° (chauthai chakkar), 180° (aadha chakkar), 270° (teen-chauthai chakkar) aur 360° (poora chakkar) hain. Har figure 360° ke baad apni jagah wapas aa jaati hai, isliye 360° hamesha angle of symmetry hai. Figure mein rotational symmetry tabhi hai jab 0° aur 360° ke beech koi angle of symmetry ho. Angles of symmetry ki ginti, 360° milakar, order of rotational symmetry hai: windmill ka order 4 hai.

Square

Square bhi apne diagonals ke milne ke point ke chaaron taraf 90° ghumane pe apne aap pe aa jaata hai. Ye ghumaav kone A ko B ki jagah, B ko C ki, C ko D ki aur D ko A ki jagah le jaata hai. Iske angles of symmetry 90°, 180°, 270° aur 360° hain. Doosri taraf, trapezium jaisi patti aadhe chakkar ke baad ulti ho jaati hai, lambi side upar; wo sirf 360° ke baad mel khaati hai, isliye usme rotational symmetry nahi.

Radial arms wali figures

90° pe khadi 4 barabar radial arms wali figure ke angles of symmetry 90°, 180°, 270° aur 360° hain. Arms ke beech ke angles badlo to chauthai chakkar mel nahi khaata. Sirf do angles of symmetry, 180° aur 360°, rakhne ke liye arms ko mod sakte ho ya do aamne-saamne ki arms ko baaki do se lamba kar sakte ho.

3 radial arms wali figure ghumane pe tabhi mel khaati hai jab har arm agli arm ki jagah pahunche. Isliye arms ke beech ke teeno angles barabar hone chahiye, aur kyunki wo milkar 360° hain, har ek 360° ÷ 3 = 120° hona chahiye. Tab angles of symmetry 120°, 240° aur 360° hain. Isi tarah 5 barabar arms ke liye 72° ki doori, aur 6 arms ke liye 60° chahiye. 7 arms mein sabse chhota angle 360° ÷ 7 = 51 3/7° hoga, jo poora number nahi.

Angles mein pattern

Angles of symmetry ki ginti	Angles
2	180°, 360°
3	120°, 240°, 360°
4	90°, 180°, 270°, 360°
6	60°, 120°, 180°, 240°, 300°, 360°

Har row mein angles sabse chhote angle ke multiples hain, aur ye hamesha sach hai. Sabse chhote angle se do baar ghumana bhi mel khaata hai, isliye uska har multiple angle of symmetry hai. Aur koi aur nahi ho sakta: do multiples ke beech ka koi angle of symmetry, chhote multiple jitna wapas ghumane pe, sabse chhote se bhi chhota mel khaata ghumaav de dega. Isliye sabse chhota angle of symmetry poora number ho to wo 360 ko poora baantta hai: wo 360 ka factor hai. Isiliye 45° sabse chhota angle ho sakta hai (45 × 8 = 360), par 17° nahi.

Circle aur hamare aas-paas

Circle ko uske centre ke chaaron taraf kisi bhi angle se ghumao, wo apne aap pe aa jaata hai, isliye har angle angle of symmetry hai. Har diameter bhi line of symmetry hai. Chhat ke pankhe, pahiye, kai phool aur Ashoka Chakra, sab mein rotational symmetry hai. Kuch figures mein sirf lines of symmetry hoti hain, kuch mein sirf rotational symmetry, aur kuch mein, jaise square, dono; tulna ke liye dekho line of symmetry kya hai.