5.4 Tessellation
A tessellation is an arrangement of shapes fitted together in a repeated pattern without leaving any gaps or overlapping. This geometric concept is seen naturally in beehives as well as in art, floor tiling, and fabric designs. By applying isometric transformations—such as translation, reflection, and rotation—various geometric shapes can be transformed to create repetitive patterns across a plane.
Learning Objectives
- Explain the meaning of tessellation.
- Determine whether a given pattern is a tessellation.
- Design tessellations involving isometric transformations (translation, reflection, and rotation).
- Identify isometric transformations used to produce shapes within a tessellation.
- Understand the concept and construction of an Escher tessellation.
Characteristics and Types of Tessellations
Compare valid tessellations with non-tessellations, and see how isometric operations transform a base tile.
- No gaps
- No overlaps
- Shapes recur
- Has empty gaps
- Has non-recurring shape
Brief Explanation
What is a Tessellation?
A tessellation is a pattern of recurring shapes that fills a plane without leaving empty spaces or overlapping.
Natural Example
The structure of a beehive is a tessellation of regular hexagons providing maximum space to store honey.
| Type of pattern | Examples | Condition |
|---|---|---|
| Single shape | Equilateral triangles, rhombuses, regular hexagons | One recurring shape |
| Combination of shapes | Squares and regular octagons, equilateral triangles and squares | Two or more recurring shapes |
| Not a tessellation | Isolated square surrounded by triangles and pentagons | Has gaps, overlaps, or a shape that does not recur |
Escher Tessellation
Named after Maurits Cornelis Escher, an Escher tessellation is a tessellation in which its shapes are easily identified as real-life figures or objects, such as animals, birds, or reptiles.
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
What are the two essential conditions for a pattern of recurring shapes to be classified as a tessellation?
Why is a pattern containing an isolated square surrounded by recurring triangles and pentagons not considered a tessellation?
What is an Escher tessellation?
Common Mistakes
- Assuming any geometric pattern is a tessellation even when small empty gaps exist between the shapes.
- Thinking shapes in a tessellation can overlap as long as the entire plane is covered.
- Forgetting that every shape in a tessellation must recur; a single unique shape surrounded by repeating shapes is not a tessellation.
- Confusing non-isometric transformations, such as enlargement, with isometric transformations; enlargement changes size, so it cannot form a standard tessellation of congruent shapes.
Misconceptions
Tessellations can only be made from a single type of regular polygon, like regular hexagons or equilateral triangles.
Tessellations can be formed from combinations of different regular or irregular polygons, such as squares combined with regular octagons, or irregular interlocking shapes.
In an Escher tessellation, the area of the modified tile changes from the initial square.
The area remains identical because the pieces cut out from one side are translated and reattached onto the opposite side.
Summary
- Tessellation: A pattern of recurring shapes that completely fills a flat surface without gaps or overlaps.
- Isometric Transformations: Transformations (translation, reflection, rotation) that preserve congruency, used to generate tessellations.
- Escher Tessellation: A tessellation where the repeating tiles are altered into recognisable real-life figures while still tiling the plane seamlessly.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following conditions must be fulfilled for a recurring pattern to be a tessellation?
2 What distinguishes an Escher tessellation from a standard geometric tessellation?
B. Fill in the Blanks
3 An isometric transformation produces an image that is with the object.
Answer for question 34 A pattern is not a tessellation if it contains a shape which does not throughout the pattern.
Answer for question 4C. Drag and Drop: Match Each Term with Its Description
Drag each term to the correct description. On a phone, tap the answer choice first, then tap the description box.
