Class 8 Ganita Prakash Part 2 Chapter 4 Exploring Some Geometric Themes: Complete NCERT Solutions, Fractals, Nets & Isometric Projections

Introduction

“Exploring Some Geometric Themes” is Chapter 4 of Ganita Prakash Part-II (Class 8). It introduces two beautiful topics: fractals (self-similar patterns like Sierpinski Carpet, Triangle and Koch Snowflake) and visualising solids (nets, projections, isometric drawings, shortest paths on cube surface and shadows). You will learn to construct fractals step-by-step, draw nets of cubes/prisms/pyramids, find shortest paths using nets, and represent solids in front/top/side views and isometric grids. The chapter uses hands-on cutouts, visualisation exercises and real-life art examples to build deep spatial reasoning.

Basic Knowledge Required

• Properties of square, equilateral triangle and regular polygons
• Area and perimeter formulas
• Nets and folding of 3D shapes
• Parallel lines and perpendiculars
• Baudhāyana-Pythagoras Theorem (for shortest paths)
• 360° in a circle and angle measurement

Key Definitions

• Fractal: Self-similar shape that repeats the same pattern at smaller and smaller scales (e.g., Sierpinski Carpet, Koch Snowflake).
• Net: 2D pattern that folds into a 3D solid without overlaps or gaps.
• Isometric Projection: Projection where all three principal axes appear equal in length (60° angles on paper).
• Front/Top/Side View: Orthographic projections on vertical, horizontal and side planes.

Important Formulas Used

1. Sierpinski Carpet: Remaining squares $R_n = 8^n$, Holes $H_n = \dfrac{8^n - 1}{7}$, Area remaining $\left( \dfrac{8}{9} \right)^n$
2. Sierpinski Triangle: Remaining triangles $R_n = 3^n$, Holes $H_n = \dfrac{3^n - 1}{2}$, Area remaining $\left( \dfrac{3}{4} \right)^n$
3. Koch Snowflake: Sides at step $n$: $S_n = 3 \times 4^n$, Perimeter $P_n = 3 \times \left( \dfrac{4}{3} \right)^n$
4. Shortest path on cube surface: Straight line on unfolded net (use Pythagoras on different nets)

4.1 Fractals

Sierpinski Carpet Start with square → divide into 9 equal squares → remove centre → repeat on 8 squares.

Figure it Out (Page 71)

1. Draw steps:
   • Step 0: Full square
   • Step 1: 8 small squares + 1 hole
   • Step 2: 64 tiny squares + 9 holes
2. $R_n = 8^n$ (remaining squares) $H_n = 1 + 8 + 8^2 + \dots + 8^{n-1} = \dfrac{8^n - 1}{7}$
3. Area remaining (original = 1): $\left( \dfrac{8}{9} \right)^n$

Sierpinski Gasket (Triangle) Start with equilateral triangle → join midpoints → remove centre triangle → repeat on 3 triangles.

Figure it Out (Page 72)

1. Draw steps (similar to carpet but triangles).
2. Remaining triangles $R_n = 3^n$, Holes $H_n = \dfrac{3^n - 1}{2}$.
3. Area remaining $\left( \dfrac{3}{4} \right)^n$.

Koch Snowflake Start with equilateral triangle (side 1) → divide each side into 3 → add equilateral bump on middle third → repeat.

Figure it Out (Page 73)

1. Draw steps:
   • Step 0: Triangle (3 sides)
   • Step 1: 12 sides (star shape)
   • Step 2: 48 sides
2. Sides at step $n$: $3 \times 4^n$.
3. Perimeter at step $n$: $3 \times \left( \dfrac{4}{3} \right)^n$.

Fractals in Art Kandariya Mahadev Temple and Nigerian Fulani blanket show self-similarity.

4.2 Visualising Solids

Build it in Your Imagination

1. Read name backwards visually.
2. Cut 4 corners of square (midpoints) → octagon; 4 corners make another square.
3. Triangle thirds → hexagon.
4. Square thirds → octagon.

Profiles 5–7: Cube (square), sphere (circle), pyramid (triangle). 8–12: Cylinder (rectangle + circle), cone (triangle + circle), cuboid (rectangle + triangle), etc.

Making Solids (Nets)

Cube Nets Figure it Out (Page 80)

1. (i) Yes, (ii) No, (iii) Yes, (iv) No, (v) Yes, (vi) No.
2. 11 distinct nets (rotations/flips considered same).
3. Cuboid nets: Cross or zigzag arrangements with lengths 5-3-1 cm and 6-3-2 cm.

Regular Tetrahedron Figure it Out (Page 81) Only 2 nets. Draw equilateral triangles net.

Square Pyramid Draw base square + 4 triangles.

Cylinder Net Two circles + rectangle (height = cylinder height, width = circumference).

Cone Net Sector of circle + base circle.

Triangular Prism Two triangles + three rectangles.

Octahedron & Dodecahedron Octahedron: 8 equilateral triangles. Dodecahedron: 12 pentagons (43,380 nets).

Shortest Paths on Cube Unfold net → straight line is shortest (use Pythagoras). Different unfoldings give different lengths; choose minimum.

Figure it Out (Page 84–86) Use nets to find straight-line distance (red path shortest).

Representation on Plane (Projections)

Figure it Out (Page 92)

1. Lengths vary by angle.
2. Cube: square all views; Cylinder: rectangle front/top circle; etc.

Figure it Out (Page 96) Match objects to views (mug, funnel, hammer, car, etc.).

Shadows Same as projections under perpendicular light (sun). Parallel lines remain parallel.

Isometric Projections Three axes at 120° (paper: |, /, ).

Drawing on Isometric Grid Figure it Out (Page 100–101)

1. Additional tetrominoes: skew, T, etc. 2–4. Draw given shapes; impossible triangle has conflicting profiles.

15 Most Asked FAQs – Exploring Some Geometric Themes (Click to Expand)

1. Draw Step 2 of Sierpinski Carpet.

Answer: 64 small squares + 9 holes (remove centre of each of 8 squares from Step 1).

2. Formula for remaining squares in Sierpinski Carpet at step n?

Answer: \( R_n = 8^n \).

3. How many nets does a cube have?

Answer: 11 distinct nets (rotations/flips same).

4. Draw net of regular tetrahedron.

Answer: 4 equilateral triangles in chain or central triangle with 3 attached.

5. Shortest path on cube surface – method?

Answer: Unfold to net → straight line (Pythagoras) between points; try all unfoldings, pick shortest.

6. Perimeter of Koch Snowflake at step n (side 1)?

Answer: \( 3 \times (4/3)^n \).

7. Front/Top/Side view of cylinder?

Answer: Front: rectangle, Top: circle, Side: rectangle.

8. Isometric projection of cube – shape?

Answer: Regular hexagon.

9. Area remaining in Sierpinski Triangle at step n?

Answer: \( (3/4)^n \) (original area 1).

10. Can projection of parallelogram be non-parallelogram?

Answer: No – parallel lines project parallel.

11. Draw 2×2×2 cube on isometric grid.

Answer: Start with base layer 2×2 squares, stack another layer offset.

12. Number of faces/edges/vertices in pentagonal prism?

Answer: 7 faces, 15 edges, 10 vertices (n=5 → F= n+2, E=3n, V=2n).

13. Impossible triangle on isometric grid – why impossible?

Answer: Conflicting front/top/side profiles (cannot exist in 3D).

14. How many nets for dodecahedron?

Answer: 43,380.

15. Shadow of cube under perpendicular light = projection?

Answer: Yes (sunlight example).