Table of Contents

2D shapes and tessellation Level 4

Introduction

Have you ever noticed how tiles fit perfectly together on a floor or how honeycombs are structured? This is all about 2D shapes and tessellation! In this article, we’ll dive into the exciting world of 2D shapes, learn about their properties, and discover how they can fit together in beautiful patterns without any gaps. Let’s get started!

Definition and Concept

2D shapes are flat figures that have length and width but no height. Examples include squares, rectangles, triangles, and circles. Tessellation refers to a pattern made of one or more shapes that fit together perfectly without any gaps or overlaps.

Relevance:

  • Mathematics: Understanding shapes helps in geometry and spatial reasoning.
  • Real-world applications: Found in art, architecture, nature, and design.

Historical Context or Origin​

Tessellation has a rich history, dating back to ancient civilizations. The Islamic art of tessellation is famous for its intricate geometric patterns. Artists like M.C. Escher in the 20th century also explored tessellation in their works, creating mesmerizing images that challenge our perception of space.

Understanding the Problem

To understand tessellation, we first need to explore the properties of different 2D shapes. Let’s break down the essential characteristics of common shapes:

  • Triangles: Three sides, can tessellate by repeating.
  • Squares: Four equal sides, fit together perfectly.
  • Hexagons: Six sides, also fit together without gaps.
  • Circles: Do not tessellate as they leave gaps.

Methods to Solve the Problem with different types of problems​

Method 1: Using Regular Shapes
Regular shapes like squares and hexagons can easily tessellate.
Example:
Using squares, you can create a checkerboard pattern by placing squares next to each other.
Method 2: Combining Shapes
Different shapes can work together to tessellate.
Example:
Triangles and squares can create a tessellation by alternating their placement.
Method 3: Exploring Irregular Shapes
Some irregular shapes can tessellate too.
Example:
Using a shape like a L can fit together in a way that covers a surface without gaps.

Exceptions and Special Cases​

  • Non-tessellating Shapes: Circles and some irregular shapes do not tessellate because they leave gaps when repeated.
  • Complex Tessellations: Some patterns involve intricate designs that require careful planning and arrangement.

Step-by-Step Practice​

Problem 1: Create a tessellation using squares.

Solution:

  • Draw a square grid on paper.
  • Color alternate squares to create a checkerboard effect.

    • Problem 2: Use triangles and squares to create a tessellation.

    Solution:

    1. Draw a square.
    2. Divide the square into two triangles.
    3. Arrange the triangles and squares to fill the space.

    Examples and Variations

    Example 1: Using Hexagons

    • Draw a hexagon and replicate it to cover a surface.

    Example 2: Using a Combination of Shapes

    • Combine triangles and squares to form a unique pattern.

    Interactive Quiz with Feedback System​

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    Common Mistakes and Pitfalls

    • Forgetting to leave no gaps or overlaps when creating tessellations.
    • Using shapes that do not tessellate.
    • Not checking the arrangement of shapes visually.

    Tips and Tricks for Efficiency

    • Start with simple shapes like squares or triangles.
    • Experiment with different arrangements to find unique patterns.
    • Use graph paper to help keep shapes aligned.

    Real life application

    • Architecture: Used in building designs and floor patterns.
    • Art: Found in many artworks and designs, especially in Islamic art.
    • Nature: Patterns in honeycombs and other natural formations.

    FAQ's

    Shapes like squares, triangles, and hexagons can tessellate. Some irregular shapes can also tessellate.
    No, circles cannot tessellate because they leave gaps when arranged together.
    You can create your own tessellation by drawing a shape and repeating it in a pattern without gaps.
    Regular tessellation uses one type of regular polygon, while irregular tessellation uses a mix of shapes.
    Tessellation is important in art, architecture, and understanding geometry, showing how shapes can interact in our world.

    Conclusion

    Understanding 2D shapes and tessellation opens up a world of creativity and problem-solving. By exploring how shapes fit together, you can appreciate the beauty of mathematics in art, architecture, and nature. Keep practicing, and soon you’ll be creating your own stunning tessellations!

    References and Further Exploration

    • Khan Academy: Interactive lessons on shapes and tessellation.
    • Book: “Tessellations: A New Approach” by David A. Smith.

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