Dilations and Composition

🔮 Transformations: Dilations & Composition

Stretch, Shrink, and Tile Your World!

🎯 The Magic of Shape-Shifting

Imagine you have a magic magnifying glass. When you look through it, things get BIGGER or SMALLER—but they keep the same shape! That’s exactly what dilation does in geometry.

And guess what? You can also combine different transformations (like flipping, sliding, and stretching) to create amazing patterns. Some patterns are so perfect they can tile forever without any gaps—like the tiles on your bathroom floor!

Let’s discover this magic together! 🪄

🔍 What is Dilation?

Dilation is like using a copy machine with a zoom button!

• Zoom IN → Makes things BIGGER (enlargement)
• Zoom OUT → Makes things SMALLER (reduction)

The special part? The shape stays the same—only the size changes!

Simple Example:

Think of a photo on your phone:

• Pinch OUT with your fingers → Photo gets BIGGER
• Pinch IN → Photo gets SMALLER
• But it’s still the SAME photo!

graph TD
    A["Original Shape 📦"] --> B{Dilation}
    B --> C["BIGGER 📦📦"]
    B --> D["SMALLER 📦"]

📍 Dilation Center and Scale Factor

Every dilation needs TWO things:

1. The CENTER (Fixed Point)

The center is the special point that stays in place. Everything else moves toward it or away from it.

Think of it like a thumbtack holding a balloon:

• The thumbtack doesn’t move
• The balloon expands outward from it!

2. The SCALE FACTOR (How Much?)

The scale factor tells us HOW MUCH bigger or smaller.

🎈 Balloon Example:

Imagine a dot (center) with a small square around it:

• Scale factor = 2: Square grows to TWICE the size, but the center dot stays put!
• Scale factor = 0.5: Square shrinks to HALF, center still stays!

graph TD
    A["Center Point •"] --> B["Scale = 2"]
    A --> C["Scale = 0.5"]
    B --> D["2× Bigger! ⬜"]
    C --> E["Half Size ▪"]

How to Calculate New Position:

New distance = Old distance × Scale factor

If a point is 4 units from the center:

• Scale factor 2 → New distance = 4 × 2 = 8 units
• Scale factor 0.5 → New distance = 4 × 0.5 = 2 units

🔄 Negative Scale Factor

Here’s where things get WILD! 🎢

A negative scale factor does TWO things at once:

1. Changes the SIZE (just like positive)
2. FLIPS the shape to the opposite side of the center!

Think of it Like a Mirror-Stretch:

Imagine you’re standing in front of a funhouse mirror:

• Regular mirror → Shows you on the other side (flipped)
• Negative dilation → Shows your reflection AND stretches/shrinks it!

Example:

graph LR
    A["Original 🏠"] --> B{Scale = -2}
    B --> C["Flipped + 2× Bigger 🏠🔄"]

Real World Example:

• Original triangle is 3 cm tall, sitting to the RIGHT of center
• Scale factor = -2
• New triangle is 6 cm tall (2× bigger)
• AND it’s now on the LEFT of center (flipped!)

🎨 Composition of Transformations

Composition means doing multiple transformations in a row—like a combo move in a video game! 🎮

The Order Matters!

Watch this:

1. Rotate a square, THEN translate (slide) it
2. Translate first, THEN rotate

You get different results! Try it yourself:

• Spin, then slide → Ends up in one spot
• Slide, then spin → Ends up somewhere else!

Common Compositions:

🍕 Pizza Example:

Making a pizza pattern:

1. Rotate one slice 60°
2. Repeat 5 more times
3. You get a full pizza with 6 identical slices!

graph TD
    A["Start: One Slice 🍕"] --> B["Rotate 60°"]
    B --> C["Rotate 60° again"]
    C --> D["Keep going..."]
    D --> E["Full Pizza! 🍕🍕🍕🍕🍕🍕"]

Glide Reflection:

A special combo = Translate + Reflect

Think of footprints in the sand:

• Left foot, right foot, left foot…
• Each step is a slide AND a flip!

🧱 Tessellation

A tessellation is a pattern of shapes that covers a surface completely with:

• ✅ NO gaps
• ✅ NO overlaps

You See Them Everywhere!

• 🧱 Brick walls
• 🐝 Honeycomb
• 🛁 Bathroom tiles
• ⚽ Soccer ball patterns
• 🎮 Pixel art!

How Do They Work?

Shapes fit together perfectly like puzzle pieces. The angles around each meeting point must add up to exactly 360° (a full circle).

graph TD
    A["Shapes Meet at a Point"] --> B{Angles Add Up?}
    B -->|= 360°| C["✅ Tessellation!"]
    B -->|≠ 360°| D["❌ Gaps or Overlaps"]

Example:

Squares tessellate because:

• Each corner = 90°
• 4 squares meet at one point
• 90° + 90° + 90° + 90° = 360° ✅

🔷 Regular Tessellation

A regular tessellation is extra special—it uses only ONE type of regular polygon, and they’re all the same size!

What’s a Regular Polygon?

A shape where:

• ALL sides are equal
• ALL angles are equal

Examples: Square, equilateral triangle, regular hexagon

The Big Secret: Only 3 Work!

Out of ALL regular polygons, only THREE can tessellate by themselves:

graph TD
    A["Regular Tessellations"] --> B["Triangles △△△"]
    A --> C["Squares □□□"]
    A --> D["Hexagons ⬡⬡⬡"]

🐝 Nature Knows Best!

Bees make hexagonal honeycomb because:

• Hexagons fit perfectly (no gaps!)
• Uses the LEAST amount of wax
• Holds the MOST honey

That’s math in nature! 🍯

🎯 Quick Summary

🚀 You Did It!

Now you know how to:

• 🔍 Stretch and shrink shapes with dilation
• 📍 Use center points and scale factors
• 🔄 Handle negative scales (flip + resize)
• 🎨 Combine transformations like a pro
• 🧱 Understand why some shapes tile perfectly
• 🔷 Know the only 3 regular tessellations

You’re ready to see geometry EVERYWHERE—in buildings, art, nature, and games!

Keep exploring, keep discovering! 🌟