Symmetry, Patterns and Tessellations
Symmetry is a distance-preserving transformation that maps a figure or an entire pattern onto itself.
Summary
Symmetry is a distance-preserving transformation that maps a figure or an entire pattern onto itself. Geometric patterns are organized through repetition and transformations of a basic motif.
| Transformation | Defining action |
|---|---|
| Translation | Moves every point the same distance in the same direction |
| Reflection | Maps points to mirror images across a fixed line |
| Rotation | Turns points through the same angle about a fixed center |
| Glide reflection | Combines reflection with a nonzero translation parallel to the reflection line |
A tessellation covers the entire plane without gaps or overlapping tile interiors. In a polygonal tessellation, the tile angles meeting at any vertex must sum to .
For a regular polygon with sides, each interior angle is . When identical regular polygons meet at a vertex, the angle condition is , where counts the polygons meeting there. The only regular tessellations use equilateral triangles, squares, or regular hexagons.
🧠 Key Concepts
- Distance preservation
- Translation
- Reflection
- Rotation
- Glide reflection
- Plane coverage
- Vertex angle condition
- Regular tessellations
🧠 Quick Check
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Statement 1: A symmetry preserves distances and maps a figure or pattern onto itself. Statement 2: Every symmetry of an individual motif must also preserve the entire pattern. Which is correct?
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Symmetry, Patterns and Tessellations
📘 Overview Symmetry describes transformations that leave a figure or pattern unchanged, while tessellations cover the plane without gaps or overlaps. Repeated shapes form geometric patterns whose structure can be analyzed through transformations and angle relationships.
🧠 Key Idea A geometric pattern is organized by rules of repetition and transformation. A tessellation must also satisfy geometric constraints that allow its tiles to fit together throughout the plane.
⚔️ Core Details: - A symmetry is a distance-preserving transformation that maps a figure or an entire pattern onto itself. - Translations, rotations, reflections, and glide reflections generate repeated arrangements of a basic shape or motif. - A tessellation covers the entire plane with tiles whose interiors do not overlap and which leave no gaps. - The tile angles meeting at any vertex of a polygonal tessellation must sum to . - A regular polygon has interior angle
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