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Ever wondered how bathroom tiles fit together so perfectly, or been mesmerized by intricate patterns on fabrics? The secret often lies in tessellations! But which shapes can be used for tessellation? It’s a fascinating question that delves into the world of geometry and pattern. We will explore the criteria for creating tessellations and discover the surprising variety of shapes that can seamlessly cover a surface without any gaps or overlaps.
The Geometry of Perfect Fit Which Shapes Can Be Used For Tessellation
Which shapes can be used for tessellation, requires a look at the fundamental principles of geometry. A tessellation, also known as a tiling, is essentially a repeating pattern of shapes that covers a plane without any gaps or overlaps. Think of it as a jigsaw puzzle where the pieces fit together perfectly to form a continuous surface. But not just any shape can be used! The key lies in understanding the angles and how they meet at each vertex (corner point). The angles around each vertex must add up to 360 degrees for a shape to tessellate.
Let’s consider some examples. Regular polygons, which have equal sides and equal angles, are often the first shapes that come to mind when thinking about tessellations. Three regular polygons can tessellate on their own, meaning you can repeat the same shape to cover a plane: equilateral triangles, squares, and hexagons. Why these and not others? Let’s look at a table!
| Regular Polygon | Interior Angle | Divides 360? | Tessellates? |
|---|---|---|---|
| Equilateral Triangle | 60° | Yes (6 times) | Yes |
| Square | 90° | Yes (4 times) | Yes |
| Regular Pentagon | 108° | No | No |
| Regular Hexagon | 120° | Yes (3 times) | Yes |
Beyond regular polygons, many other shapes can tessellate. Irregular polygons, which don’t have equal sides or angles, can also form tessellations. For instance, any triangle or quadrilateral (four-sided shape) will tessellate, regardless of the lengths of their sides or the measures of their angles. This is because the angles in a triangle always add up to 180 degrees, and the angles in a quadrilateral always add up to 360 degrees. You can manipulate the shapes to fit together seamlessly, rotating and translating them to cover the plane. Certain irregular pentagons, hexagons, and other polygons can also tessellate, but the rules are more complex and often require specific angle and side length relationships. The possibilities become even greater when you start combining different shapes to create more complex tessellations.
Want to learn more about the specific rules and examples of shapes that can be used for tessellation? Refer to the provided material in the next section to see visual examples, and detailed explanation!