Class 8 Maths Part 2 Chapter 4: Exploring Some Geometric Themes (Hinglish Notes)
Ye chapter 5 lines mein
Exploring Some Geometric Themes ke do hisse hain. Fractal apna pattern chhote se chhote paimaane par dohraate hain, jaise fern, Sierpinski carpet aur Koch snowflake. Thos aakritiyon ko samajhne mein profile, falak, kinaare aur sheersh, net, dabbe par sabse chhota raasta, saamne, upar aur bagal ke drishya, aur isometric chitra aate hain, jismein cube ek niyamit shatbhuj ban jaata hai.
Ye chapter kis baare mein hai?
Ye chapter do jyaamitiya vishay khojta hai. Pehla hai fractal: aisi aakritiyaan jo apna pattern chhote se chhote paimaane par dohraati hain. Doosra hai thos aakritiyon ko samajhna: aakritiyon ko man mein banaana, unhe net mein kholna, unki sateh par sabse chhota raasta dhoondhna, aur unhe prakshep aur isometric grid se samtal kaagaz par banaana.
Fractal
Fern ye vichaar achhi tarah dikhaata hai: har patti poore fern jaisi chhoti pattiyon se bani hai, aur ve aur chhoti pattiyon se. Ped, baadal, tat-rekha, pahaad aur bijli bhi aise hi khud ko dohraate hain. Ganitagya fractal kadam dar kadam banaate hain. Sierpinski carpet mein varg ko 9 mein kaatkar beech wala baar-baar hataaya jaata hai, isliye n kadam baad 8ⁿ varg bachte hain aur kshetrafal (8/9)ⁿ rehta hai. Sierpinski triangle mein madhya-bindu jodne se tribhuj 4 ek jaise tribhujon mein bantta hai aur beech wala hataaya jaata hai. Koch snowflake mein har bhuja ka beech ka tihaai hissa chhota tribhuj ban jaata hai, isliye har bhuja 4 bhujaayein banti hai aur har kadam par parimaap 4/3 se guna hota hai. Padho fractal kya hai.
Kala mein fractal
Insaan ke banaaye sabse puraane fractal mein se kuch Bharatiya mandiron mein hain. Khajuraho ke Kandariya Mahadev Mandir ka shikhar, jo lagbhag 1025 CE mein poora hua, apni hi chhoti nakalon se bana hai, jin par aur chhoti nakalein hain; Madurai, Hampi, Rameswaram aur Varanasi ke mandir bhi aise pattern dikhaate hain. Nigeria ke Fulani shaadi ke kambal heere ke andar heere bunte hain, aur Dutch kalaakaar M.C. Escher ne chhipkaliyon ko chhote se chhote size mein banaaya.
Thos ko dekhna
Nikola Tesla jaise aavishkaarak machine ko asal mein banaane se pehle man mein banaate the. Aazmaao: ek kaalpanik varg ke kone uski bhujaaon ke madhya-binduon par kaato, to ek chhota varg bachta hai. Thos ka jo hum dekhte hain wo ek drishti-bindu se uska profile hai, aur jagah badalne par wo poori tarah badal sakta hai: cone upar se vritt aur bagal se tribhuj dikhta hai. Samtal falak wale thos ke falak, kinaare aur sheersh hote hain; n-bhuja prism ke n + 2 falak, 3n kinaare aur 2n sheersh hote hain, aur n-bhuja pyramid ke n + 1 falak, 2n kinaare aur n + 1 sheersh. Padho thos aakritiyon ko samajhna.
Net aur sabse chhota raasta
Net ek samtal aakaar hai jo modkar thos banta hai. Cube ke 11 alag net hain aur niyamit tetrahedron ke sirf 2; cylinder ek aayat aur do vritton mein khulta hai, aur cone ek vritt ke sector aur ek vritt mein. Net ek paheli bhi suljhaate hain: dabbe par baitha ek bhookha keeda laddu tak sabse chhota raasta chahta hai. Dabbe ko kholo, aur sabse chhota raasta sahi net par ek seedhi rekha hai. 30 × 12 × 12 ke dabbe mein sahi tareeke se kholne par 24 cm aur 32 cm bhujaaon wala samkon tribhuj banta hai, isliye raasta 40 cm hai, seedhe 42 cm wale raaste se chhota. Padho thos aakritiyon ke net.
Prakshep aur drishya
Kisi bindu ka samtal par prakshep bindu se giraaye lamb ka paanv hai; vastu ke saare binduon ke prakshep milkar vastu ka prakshep banaate hain. Prakshep rekha se kabhi lamba nahi hota, aur samaantar rekhaayein samaantar rehti hain. Kyunki alag vastuon ka ek hi prakshep ho sakta hai, engineer teen istemaal karte hain: khade tal par saamne ka drishya, letay tal par upar ka drishya aur bagal ke tal par bagal ka drishya. Deewar par seedhi padti dhoop jo parchhaaiyaan banaati hai, ve prakshep jaisi hi hoti hain.
Isometric chitra
Cube ko ek kone par santulit karke seedha neeche dekho: saare 12 kinaare barabar lambe dikhte hain, aur baahari rekha ek niyamit shatbhuj hai. Ye isometric prakshep isometric grid kaagaz ka aadhaar hai, jiski teen dishaon ki rekhaayein lambaai, gehraai aur oonchaai dikhaati hain. Ispar kinaara dar kinaara banaane se thos aakritiyaan dikhaana aasaan hota hai, jaise Tetris ke chaar vargon wale paanch aakaar cube ban jaayein. Yahi grid asambhav chitron ko bhi samjhaata hai, jaise cube ka aisa tribhuj jo band lagta hai par kabhi banaaya nahi ja sakta.
Saar
Fractal har paimaane par khud ko dohraate hain, aur saade niyam (9 mein se 8 rakho, 4 mein se 3, ya 1 bhuja ko 4 banaao) sateek ginti dete hain. Thos ko man mein banaaya, net mein khola aur prakshep se banaaya ja sakta hai; teen drishya thos tay karte hain, aur isometric kaagaz teeno dishaayein ek chitra mein rakhta hai. Har sawaal ka jawaab Part 2 Chapter 4 ke NCERT solutions mein hai.
Key terms (zaroori shabd)
| Term | Matlab |
|---|---|
| Fractal | Swa-samroop aakriti jo apna wahi ya milta-julta pattern baar-baar chhote se chhote paimaane par dikhaati hai. |
| Sierpinski carpet | Varg ko 9 mein kaatkar beech wala hataana, phir har bache varg par yahi dohraana: isse bana fractal. |
| Sierpinski triangle (gasket) | Tribhuj ki bhujaaon ke madhya-binduon ko jodkar beech wala tribhuj hataana aur dohraana: isse bana fractal. |
| Koch snowflake | Har bhuja ke beech ke tihaai hisse ko ek chhote samabahu tribhuj ki do bhujaaon se baar-baar badalne se bana fractal. |
| Profile | Kisi khaas jagah se dekhne par thos aakriti ki baahari rekha. |
| Falak, kinaare aur sheersh | Thos ki samtal sathein (faces), jahan sathein milti hain wo rekhaayein (edges), aur jahan kinaare milte hain wo kone (vertices). |
| Prism (praanjal) | Aisa thos jiske do aamne-saamne falak sarvaangsam bahubhuj hon aur samaantar chaturbhujon se jude hon. |
| Pyramid (piramid) | Bahubhuj aadhaar wala thos jiske sheersh ek baahari bindu, apex, se jude hon. |
| Net | Samtal aakaar jo modkar thos ban jaaye; thos ko samtal par khola hua roop. |
| Prakshep (projection) | Kisi bindu ya vastu ki samtal par lamb giraakar bani chhaaya jaisi chhavi. |
| Saamne, upar aur bagal ke drishya | Vastu ke khade, letay aur bagal wale talon par prakshep: front, top aur side view. |
| Isometric prakshep | Cube ka aisa drishya jismein uske saare kinaare barabar lambe dikhte hain; baahari rekha niyamit shatbhuj hoti hai. |
Is chapter ke topics
Practice karo
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Sierpinski carpet ke kadam 2 ke baad kitne varg bachte hain?
Jawab dekho
Jawab: (c) 64
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Koch snowflake ka har kadam parimaap ko kisse guna karta hai?
Jawab dekho
Jawab: (c) 4/3
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Shatbhujiya (hexagonal) prism ke kitne kinaare hote hain?
Jawab dekho
Jawab: (b) 18
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Cube ke kitne alag net hote hain?
Jawab dekho
Jawab: (c) 11
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Kaun sa thos upar se vritt aur bagal se tribhuj dikhta hai?
Jawab dekho
Jawab: (b) cone
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Cube ka isometric prakshep hai:
Jawab dekho
Jawab: (c) niyamit shatbhuj
Students ke sawal
Class 8 Maths Part 2 Chapter 4 ke mukhya topics kya hain?
Fractal: Sierpinski carpet aur triangle, Koch snowflake, aur prakriti aur kala mein fractal. Thos aakritiyon ko samajhna: man mein aakritiyaan banaana, profile, prism aur pyramid ke falak, kinaare aur sheersh, thos ke net, cuboid par sabse chhota raasta, prakshep, saamne, upar aur bagal ke drishya, parchhaaiyaan, aur isometric chitra.
Kya Exploring Some Geometric Themes naye NCERT Class 8 Maths book mein hai?
Haan. Ye Ganita Prakash, Grade 8 Part 2 ka Chapter 4 hai, pages 70 se 102 tak.
Aasaan shabdon mein fractal kya hai?
Fractal aisi aakriti hai jismein uski chhoti nakalein hoti hain, aur unmein aur chhoti nakalein, aur aage bhi. Fern ki patti apne jaisi chhoti pattiyon se bani hoti hai; Sierpinski triangle teen chhote Sierpinski triangle se bana hai.
Dabbe ki sateh par do binduon ke beech sabse chhota raasta kaise nikaalein?
Dabbe ko aise kholo ki dono bindu net par aa jaayein, aur unhe net par hi rehne wali seedhi rekha se jodo. Alag tareeke se kholne par alag seedhi rekhaayein milti hain, isliye sab aazmaao aur sabse chhoti chuno; book ke 30 × 12 × 12 ke dabbe mein ye Baudhayana theorem se 40 cm aata hai.
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