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Tesselation

5.4 Tessellation - Pandai Notes
MATHEMATICS • Form 5 • Chapter 5: Congruency, Enlargement and Combined Transformations

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.

Valid Tessellation
  • No gaps
  • No overlaps
  • Shapes recur
Not a Tessellation
  • Has empty gaps
  • Has non-recurring shape
Translation
Reflection
Rotation

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
1
Translation
Shifting a shape along a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) without rotating or flipping it.
2
Reflection
Flipping a shape across a line of reflection (mirror line).
3
Rotation
Turning a shape through a specified angle, such as \(60^\circ\), \(120^\circ\), or \(150^\circ\), clockwise or anticlockwise about a centre point.

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.

1
Start with a regular polygon
Begin with a regular polygon, such as a square.
2
Modify two adjacent sides
Cut out specific shapes or curves from two adjacent sides.
3
Attach cut-out pieces
Attach the cut-out pieces onto the opposite sides via translation or rotation.
4
Repeat the tile
The resulting modified tile can be repeated continuously across a plane without gaps or overlapping.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

What are the two essential conditions for a pattern of recurring shapes to be classified as a tessellation?

Answer: The pattern must fill a plane without leaving empty spaces (gaps) and without overlapping.
2

Why is a pattern containing an isolated square surrounded by recurring triangles and pentagons not considered a tessellation?

Answer: Because the square shape does not recur throughout the pattern.
3

What is an Escher tessellation?

Answer: An Escher tessellation is a tessellation in which the repeating shapes are easily identified as real-life objects or figures.

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

Misconception

Tessellations can only be made from a single type of regular polygon, like regular hexagons or equilateral triangles.

Correct Fact

Tessellations can be formed from combinations of different regular or irregular polygons, such as squares combined with regular octagons, or irregular interlocking shapes.

Misconception

In an Escher tessellation, the area of the modified tile changes from the initial square.

Correct Fact

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 3

4 A pattern is not a tessellation if it contains a shape which does not throughout the pattern.

Answer for question 4

C. 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.

Concept / Transformation
Translation
Reflection
Rotation
Tessellation
Description
Shifting a tile along a vector without turning or flipping itCorrectTry againEmpty
Producing an image by flipping a tile across a straight lineCorrectTry againEmpty
Turning a tile through a specific angle about a fixed centre pointCorrectTry againEmpty
A pattern of recurring shapes that fills a plane without gaps or overlapsCorrectTry againEmpty
 

Key Terms

Tessellation Recurring shapes Isometric transformation Translation Reflection Rotation Escher tessellation
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.

Tesselation

5.4 Tessellation - Pandai Notes
MATHEMATICS • Form 5 • Chapter 5: Congruency, Enlargement and Combined Transformations

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.

Valid Tessellation
  • No gaps
  • No overlaps
  • Shapes recur
Not a Tessellation
  • Has empty gaps
  • Has non-recurring shape
Translation
Reflection
Rotation

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
1
Translation
Shifting a shape along a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) without rotating or flipping it.
2
Reflection
Flipping a shape across a line of reflection (mirror line).
3
Rotation
Turning a shape through a specified angle, such as \(60^\circ\), \(120^\circ\), or \(150^\circ\), clockwise or anticlockwise about a centre point.

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.

1
Start with a regular polygon
Begin with a regular polygon, such as a square.
2
Modify two adjacent sides
Cut out specific shapes or curves from two adjacent sides.
3
Attach cut-out pieces
Attach the cut-out pieces onto the opposite sides via translation or rotation.
4
Repeat the tile
The resulting modified tile can be repeated continuously across a plane without gaps or overlapping.

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

1

What are the two essential conditions for a pattern of recurring shapes to be classified as a tessellation?

Answer: The pattern must fill a plane without leaving empty spaces (gaps) and without overlapping.
2

Why is a pattern containing an isolated square surrounded by recurring triangles and pentagons not considered a tessellation?

Answer: Because the square shape does not recur throughout the pattern.
3

What is an Escher tessellation?

Answer: An Escher tessellation is a tessellation in which the repeating shapes are easily identified as real-life objects or figures.

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

Misconception

Tessellations can only be made from a single type of regular polygon, like regular hexagons or equilateral triangles.

Correct Fact

Tessellations can be formed from combinations of different regular or irregular polygons, such as squares combined with regular octagons, or irregular interlocking shapes.

Misconception

In an Escher tessellation, the area of the modified tile changes from the initial square.

Correct Fact

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 3

4 A pattern is not a tessellation if it contains a shape which does not throughout the pattern.

Answer for question 4

C. 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.

Concept / Transformation
Translation
Reflection
Rotation
Tessellation
Description
Shifting a tile along a vector without turning or flipping itCorrectTry againEmpty
Producing an image by flipping a tile across a straight lineCorrectTry againEmpty
Turning a tile through a specific angle about a fixed centre pointCorrectTry againEmpty
A pattern of recurring shapes that fills a plane without gaps or overlapsCorrectTry againEmpty
 

Key Terms

Tessellation Recurring shapes Isometric transformation Translation Reflection Rotation Escher tessellation
The views expressed in the notes here are those of the contributor and don’t necessarily represent the views of Pandai.
Chapter : Congruency, Enlargement and Combined Transformations
Topic : Tessellation
Form 5 Mathematics
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