Welcome to the World of Tessellations!
Have you ever looked closely at the tiles on a bathroom floor, the pattern on a soccer ball, or the surface of a honeycomb? If you have, you have already seen tessellations in action! In this chapter, we are going to explore how shapes can fit together perfectly to cover a surface. It is like a mathematical jigsaw puzzle that never ends.
Don't worry if this seems a bit "shapy" at first—once you see the patterns, you will start spotting them everywhere you go!
What is a Tessellation?
A tessellation (sometimes called a tiling) is a pattern made of one or more shapes that fits together perfectly on a flat surface. To be a true tessellation, it must follow two very important rules:
1. No Gaps: There should be no empty spaces between the shapes.
2. No Overlaps: The shapes should not lay on top of each other; they must touch edge-to-edge.
Think of it like this: If you were tiling a floor, you wouldn't want gaps where dust could hide, and you wouldn't want shapes overlapping, or people would trip!
Quick Review: The "Perfect Fit" Check
Imagine you have a handful of \( 5 \)-cent coins. If you lay them flat on a table and try to push them together, you will notice little diamond-shaped gaps between them. Because of these gaps, circles cannot tessellate.
Regular Polygons and Tessellations
In our other chapter on Regular and irregular polygons, we learned that regular polygons have sides that are all the same length and angles that are all the same size. Only three regular shapes can tessellate all by themselves:
1. Equilateral Triangles: These fit together to make a pointy, star-like grid.
2. Squares: This is the most common tessellation, like a chessboard or a grid on graph paper.
3. Regular Hexagons: These look like the cells in a beehive. They are very strong and fit together beautifully.
Why do only these three work?
It all comes down to the points where the shapes meet (called a vertex). For shapes to tessellate perfectly, the angles meeting at that point must add up to exactly \( 360 \) degrees. If they add up to less, you get a gap. If they add up to more, the shapes will overlap!
Congruency: The Secret Ingredient
In a simple tessellation, we use congruent shapes. This is a fancy math word that means the shapes are exactly the same size and the same shape.
When you use congruent squares to cover a wall, each square is a perfect copy of the one next to it. This ensures that every side matches up perfectly with the neighbor's side. You can learn more about this in our chapter on Congruency and similarity in 2D shapes.
Finding Tessellations in the Real World
Nature and humans both love using tessellations because they are organized and efficient. Here are some places you might see them:
In Nature:
- Honeycombs: Bees use hexagons because they use the least amount of wax to hold the most amount of honey.
- Snake Skin: The scales often fit together in a repeating pattern to allow the snake to move easily.
- Dried Mud: When ground dries up, it often cracks into patterns that look like irregular tessellations.
In Human Design:
- Brick Walls: Rectangles (which are a type of polygon) are stacked to build strong houses.
- Quilts: Many traditional blankets are made by sewing together triangles or squares of fabric.
- Art: Artists use tessellations to create optical illusions and beautiful decorations.
Did you know? The word "tessellation" comes from the Latin word tessella, which means a small square stone or tile used to make mosaics!
How to Create Your Own Tessellation
You don't need to be a professional builder to make a tessellation. You can try it with paper!
1. Start with a simple shape: A square or a rectangle is easiest.
2. The "Cut and Slide" method: Cut a wiggly shape out of one side of your square and slide it directly across to the opposite side. Tape it there!
3. Trace and Repeat: Now, trace your new "funky" shape on a piece of paper. You will find that the "wiggle" you cut out of one side fits perfectly into the "wiggle" on the next shape.
4. Check: Does it have gaps? No. Does it overlap? No. You've made a tessellation!
Common Mistakes to Avoid
- The "Gap" Trap: Using shapes like pentagons or octagons by themselves. Regular pentagons will always leave a small gap!
- Mixing Sizes: If your triangles aren't congruent (the same size), they won't line up in a repeating pattern.
- Forgetting the Surface: Remember, tessellations usually happen on a flat 2D surface (like a piece of paper or a wall).
Key Takeaways Summary
- Tessellation: A repeating pattern of shapes with no gaps and no overlaps.
- Vertex: The corner point where the shapes meet.
- Regular Tessellations: Patterns made of only one regular polygon (Triangles, Squares, or Hexagons).
- Congruency: Using shapes that are identical in size and shape helps them fit together perfectly.
Next time you are walking to school, look down at the sidewalk or up at a brick building. Can you spot the tessellating patterns? Once you start looking, you won't be able to stop!