Shapes mein patterns kya hain?
Short mein jawab
Shape sequences shapes ke patterns hain, jinhe geometry mein padhte hain. Regular polygons mein bhujaayein 3, 4, 5, … badhti hain; complete graphs mein 1, 3, 6, 10, … lines hain; stacked squares aur stacked triangles dono mein 1, 4, 9, 16, … tukde hain; aur Koch snowflake mein 3, 12, 48, … segments. Shape sequences aksar number sequences se jude hote hain.
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Shape sequences
Numbers ke alaawa maths ke doosre buniyaadi patterns shapes ke patterns hain. Inhe padhne waali shaakha geometry hai. Shapes ek, do ya teen dimensions (1D, 2D ya 3D), ya isse bhi zyaada mein ho sakti hain. Shape sequence shapes ki aisi list hai jo kisi niyam pe chale. Chapter inme se paanch dikhaata hai, aur har ek kisi number sequence se juda hai.
Regular polygons
Regular polygons hain regular triangle, quadrilateral (square), pentagon, hexagon, heptagon, octagon, nonagon, decagon, aur aage. Inki bhujaayein 3 se shuru hone waale counting numbers hain: 3, 4, 5, 6, 7, 8, 9, 10, …. Regular ka matlab hai bhujaayein barabar lambi aur angles barabar, isliye saari bhujaayein aur saare kone ek jaise dikhte hain. Polygon mein jitni bhujaayein utne kone hote hain, kyunki har bhuja do konon pe khatam hoti hai aur har kona do bhujaaon ko jodta hai.
Complete graphs
Complete graph mein har bindu baaki har bindu se ek line se juda hota hai. K2 mein 2 bindu hain, K3 mein 3, aur aise hi K6 tak. Inki lines 1, 3, 6, 10, 15 hain: triangular numbers. Kyun? Har naye bindu ko pichhle saare binduon se jodna padta hai, isliye K2 se K3 mein 2 lines judti hain, phir 3, phir 4, phir 5.
Stacked squares aur stacked triangles
Stacked squares mein 1, 4, 9, 16, 25, … chhote squares hain, kyunki n bhuja waali shape mein n squares ki n rows hoti hain. Stacked triangles mein bhi yahi numbers hain, 1, 4, 9, 16, 25, …! Har shape ki rows mein 1, 3, 5, 7, … chhote triangles hain, baari-baari upar aur neeche ki taraf, aur odd numbers jodne se square numbers milte hain.
Koch snowflake
Koch snowflake ek triangle se shuru hota hai. Agli shape ke liye har line segment ki jagah 4 chhote segments ka “speed bump” lagaaya jaata hai. Ise baar-baar karne se kinaara aur aur ubad-khaabad hota jaata hai, aur segments chhote se chhote. Segments ki ginti 3, 12, 48, 192, 768, … hai, yaani 3 guna 4 ki powers. Ye sequence chapter ki number sequences waali Table 1 mein nahi hai.
Agli shape banaana
Niyam pata ho to har sequence ki agli shape banaayi ja sakti hai. Decagon ke baad 11 bhujaaon waala regular polygon aata hai. K6 ke baad K7 aata hai: saat bindu, har ek baaki chhe se juda, yaani 15 + 6 = 21 lines. Agla stacked square 36 chhote squares ka 6 by 6 square hai, aur agle stacked triangle mein 6 rows aur 36 chhote triangles hain. Agla Koch snowflake mushkil hai: uske segments itne chhote ho jaate hain ki haath se banaana lagbhag namumkin hai.
Shapes aur numbers saath
| Shape sequence | Ginti | Number sequence |
|---|---|---|
| Regular polygons | bhujaayein ya kone | 3, 4, 5, 6, … |
| Complete graphs | lines | 1, 3, 6, 10, 15, … |
| Stacked squares | chhote squares | 1, 4, 9, 16, 25, … |
| Stacked triangles | chhote triangles | 1, 4, 9, 16, 25, … |
| Koch snowflake | line segments | 3, 12, 48, 192, … |
Number sequence pata ho to agli shape ka andaaza lagta hai, aur shape samjhaati hai ki number pattern kyun chalta hai. Number patterns ke liye padho Chapter 1 ke notes.
Practice karo
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Regular octagon mein kitni bhujaayein hain?
Jawab dekho
Jawab: (c) 8
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Complete graph K6 mein kitni lines hain?
Jawab dekho
Jawab: (c) 15
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Stacked triangles ki chauthi shape mein kitne chhote triangles hain?
Jawab dekho
Jawab: (c) 16 Iski rows mein 1, 3, 5 aur 7 chhote triangles hain: 1 + 3 + 5 + 7 = 16.
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Koch snowflake ki chauthi shape mein kitne line segments hain?
Jawab dekho
Jawab: 3 × 4 × 4 × 4 = 192.
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Regular polygon mein jitni bhujaayein utne kone kyun hote hain?
Jawab dekho
Jawab: Har bhuja do konon ko jodti hai aur har kona do bhujaaon ko; shape ke chaaron taraf ghoomne pe bhuja, kona, bhuja, kona, … baari-baari milte hain jab tak shuruaat pe na laut aao, isliye dono ginti barabar hain.
Students ke sawal
Geometry kya hai?
Maths ki shaakha jo shapes ke patterns padhti hai. Shapes ek, do ya teen dimensions mein, ya isse bhi zyaada mein ho sakti hain.
Regular polygon mein regular ka kya matlab hai?
Iski saari bhujaayein barabar lambi aur saare angles barabar hote hain, isliye bhujaayein ek jaisi aur kone ek jaise dikhte hain.
Complete graphs mein 1, 3, 6, 10, … lines kyun hoti hain?
Har naya bindu pichhle har bindu se joda jaata hai. K2 se K3 mein 2 lines judti hain, K4 mein 3, K5 mein 4, isliye kul 1, 1 + 2, 1 + 2 + 3, …: triangular numbers.
Koch snowflake kaise banta hai?
Ek triangle se shuru karo. Har line segment ki jagah 4 chhote segments ka 'speed bump' lagao, aur ye dohraate raho. Segments ki ginti 3, 12, 48, 192, … hoti hai, yaani 3 guna 4 ki powers.
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