Exploring Some
Geometric Themes
Fractals · Self-Similar Shapes · Visualising Solids · Nets · Projections · Isometric Drawing
4.1 What are Fractals?
🔍 Definition
A fractal is a shape that looks the same at smaller and smaller scales. If you zoom into any part of it, you see the same pattern again — endlessly!
- 🌿 Fern — sub-leaves look like the whole frond
- 🌳 Trees — trunk → limbs → branches → branchlets
- ☁️ Clouds and ⛰️ Mountains
- 🌊 Coastlines and ⚡ Lightning
Sierpinski Carpet
📐 How to Build It
- Start with a solid square (Step 0)
- Divide it into a 3×3 grid of 9 equal smaller squares
- Remove the central square — creating 1 hole
- Repeat the same process on each of the 8 remaining squares
- Continue forever → the Sierpinski Carpet fractal appears!
Rₙ = remaining squares at step n
Hₙ = number of holes at step n
Coloured squares after n steps
Each square splits into 9 smaller ones
Sierpinski Gasket (Triangle)
📐 How to Build It
- Start with a solid equilateral triangle (Step 0)
- Join the midpoints of all 3 sides → creates 4 smaller equilateral triangles
- Remove the central inverted triangle — 3 corner triangles remain
- Repeat on each of the 3 remaining triangles
- Continue forever → the Sierpinski Triangle / Gasket emerges!
Koch Snowflake
❄️ How to Build It
- Start with an equilateral triangle
- Divide each side into 3 equal parts
- Build a smaller equilateral triangle on the middle part, pointing outward
- Remove the middle segment — each side becomes a bump ∧
- Repeat on every new side created — snowflake keeps growing!
Fractals in Art & Culture
🛕 Indian Temple Architecture
The Kandariya Mahadev Temple in Khajuraho, Madhya Pradesh (~1025 CE) is one of the oldest fractal artworks. The tall structure is made of smaller copies of itself, which have even smaller copies — a true architectural fractal!
Fractal-like patterns also appear at Madurai, Hampi, Rameswaram, Varanasi and many others.
🪢 Nigerian Fulani Wedding Blanket
Traditional African art uses fractals beautifully. Nigerian Fulani wedding blankets show diamond patterns inside diamond patterns inside still smaller diamonds — fractal structure woven into fabric!
🦎 M.C. Escher — Modern Fractal Master
The Dutch artist M.C. Escher is the modern maestro of fractal art. His famous work "Smaller and Smaller" shows the same pattern of lizards at smaller and smaller scales — a mathematical fractal in fine art form.
4.2 Visualising Solids
📦 Key Terms
- Face — a flat/plane surface forming the boundary of the solid
- Edge — a line segment where two faces meet
- Vertex — a corner point where 3 or more edges meet
Faces of a Cube / Cuboid
Edges of a Cube / Cuboid
Vertices of a Cube / Cuboid
📊 Prism vs Pyramid — Quick Comparison
| Feature | Prism | Pyramid |
|---|---|---|
| Bases | 2 congruent polygons (top & bottom) | 1 polygonal base |
| Other Faces | Parallelograms | Triangles meeting at apex |
| Top | A flat polygon | A single point (apex) |
| Named by | Shape of cross-section | Shape of base |
👁️ Profiles from Different Viewpoints
- The same solid looks very different from different directions
- Cylinder from side → Rectangle | from top → Circle
- Cone from side → Triangle | from top → Circle
- Cube from any face → Square
- A circular + rectangular profile pair → think Cylinder!
Nets of Solids
📋 What is a Net?
A net is a flat 2D shape that can be folded up to form a 3D solid. It is what you get by "unfolding" a solid onto a plane. Different unfoldings give different nets of the same solid!
🔵 Nets of Curved Solids
- Cylinder net → 2 circles + 1 rectangle (rectangle width = circumference = 2πr)
- Cone net → 1 circle (base) + 1 sector of a larger circle (lateral surface)
- Sphere → ❌ No flat net! You cannot unfold a sphere without gaps or wrinkles
Shortest Paths on a Cuboid
🤔 The Problem
An ant is on the surface of a cuboid. A laddu is also on the surface. The ant can only travel along the surface. What is the shortest path the ant can take?
💡 The Net Trick — The Key Insight!
- Unfold the cuboid into its flat net
- On a flat surface, the shortest path = a straight line
- Every path on the cuboid surface = path of same length on the net
- So: shortest path on cuboid = straight line on the unfolded net
- ⚠️ The way you unfold matters — different unfoldings give different line lengths
- Try all possible unfoldings and pick the one with the shortest line!
After unfolding, use the Baudhayana (Pythagoras) Theorem:
Projections & Three Views
📐 What is a Projection?
A projection is the "shadow" of an object on a flat plane when light falls perpendicularly on it. The projection of point P on plane M is the foot of the perpendicular from P to M.
When sunlight is perpendicular to a surface, the shadows cast are identical to mathematical projections!
Front View
Projection on the Vertical Plane
(looking from front)
Top View
Projection on the Horizontal Plane
(looking from above)
Side View
Projection on the Side Plane
(looking from the side)
✨ Key Properties of Projections
- Projected length ≤ actual length (equal only when line is parallel to the plane)
- Projection of parallel lines = parallel lines (always!)
- Projection of a parallelogram = always a parallelogram
- A given projection can come from many different objects — it is not unique!
- So we use all 3 views together to fully describe a solid
Isometric Projections & Drawing
🔷 What is Isometric Projection?
When a cube is balanced perfectly on one corner vertex and projected downward, all 12 edges appear as equal length. This is called the isometric projection.
The isometric view of a glass cube looks like a regular hexagon. Tiling the plane with hexagons gives the isometric grid — widely used in engineering drawings!
📐 The 3 Axes on Isometric Grid
| Grid Direction | Axis | What It Represents |
|---|---|---|
| | | Height | Up / Down |
| / | Depth | Front / Back |
| \ | Length | Left / Right |
✅ Why Isometric Drawing Works
- Parallel lines on the solid → parallel lines on the isometric grid
- Equal distances along all 3 axes appear as equal lengths on the grid
- You can measure height, length, and depth directly from the drawing
- Widely used in engineering, architecture, and game design
⭐ Chapter Summary
- Fractals are self-similar objects — same pattern at smaller and smaller scales — found in nature and art
- The Sierpinski Carpet removes the central square from a 3×3 grid repeatedly → Rₙ = 8ⁿ squares remain at step n
- The Sierpinski Gasket removes the central triangle repeatedly → Tₙ = 3ⁿ triangles remain at step n
- The Koch Snowflake has finite area but infinite perimeter → sides = 3×4ⁿ at step n
- Fractals appear in Indian temple architecture (~1025 CE), African textiles, and Escher's artwork
- Solids have Faces (flat surfaces), Edges (line segments), Vertices (corner points) → Cube has 6F · 12E · 8V
- A net is a flat shape that folds into a solid → Cube: 11 nets · Tetrahedron: 2 nets · Dodecahedron: 43,380 nets!
- Shortest surface path on a cuboid = straight line on the correctly chosen net, found using Baudhayana theorem
- Three views — Front (vertical plane) · Top (horizontal plane) · Side (side plane) — together fully describe a solid
- Isometric projection: all cube edges project to equal lengths → drawn on isometric grid using | / \ for height, depth, length