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Geometry and Trigonometry - Tessellations

Grade 10IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A tessellation (or tiling) is a pattern of shapes that covers a plane without any gaps or overlaps. In a regular tessellation, the sum of the interior angles of the polygons meeting at any vertex must exactly equal 360∘360^{\circ}. This explains why only regular triangles, squares, and hexagons can form regular tessellations.

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Semi-regular (or Archimedean) tessellations are formed by two or more types of regular polygons. Every vertex in the tiling must be identical, meaning the same sequence of polygons surrounds each point. For example, a common semi-regular tessellation uses squares and octagons.

A vertex where a square meets two regular octagons.
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Every quadrilateral, whether regular or irregular, can tessellate. This is because the sum of the interior angles of any quadrilateral is 360∘360^{\circ}. By rotating and translating the quadrilateral, the four different angles (a,b,c,da, b, c, d) can be brought together at each vertex.

An irregular quadrilateral with labeled interior angles.
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Non-regular tessellations can be created using transformations such as translation, reflection, and rotation on a basic tiling unit (like a square or rectangle) to create interlocking 'Escher-style' shapes.

📐Formulae

Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n - 2) \times 180^{\circ}

Interior angle of a regular n-gon=(n−2)×180∘n\text{Interior angle of a regular } n\text{-gon} = \frac{(n - 2) \times 180^{\circ}}{n}

Exterior angle of a regular n-gon=360∘n\text{Exterior angle of a regular } n\text{-gon} = \frac{360^{\circ}}{n}

Condition for regular tessellation: 360∘Interior Angle=k, where k∈Z+\text{Condition for regular tessellation: } \frac{360^{\circ}}{\text{Interior Angle}} = k, \text{ where } k \in \mathbb{Z}^{+}

💡Examples

Problem 1:

Explain why a regular pentagon cannot form a regular tessellation.

Solution:

  1. Calculate the interior angle of a regular pentagon (n=5n = 5): Interior Angle=(5−2)×180∘5=540∘5=108∘\text{Interior Angle} = \frac{(5 - 2) \times 180^{\circ}}{5} = \frac{540^{\circ}}{5} = 108^{\circ}
  2. Check if 108∘108^{\circ} is a divisor of 360∘360^{\circ}: 360∘108∘=3.33…\frac{360^{\circ}}{108^{\circ}} = 3.33\dots
  3. Since 3.33…3.33\dots is not an integer, the pentagons will either overlap or leave a gap.

Explanation:

For a regular polygon to tessellate, the interior angle must divide 360∘360^{\circ} perfectly so that a whole number of polygons meet at a vertex.

Problem 2:

A semi-regular tessellation is formed by two regular octagons and one other regular polygon meeting at each vertex. Identify the third polygon.

Solution:

  1. Find the interior angle of a regular octagon (n=8n = 8): Angleoct=(8−2)×180∘8=1080∘8=135∘\text{Angle}_{oct} = \frac{(8 - 2) \times 180^{\circ}}{8} = \frac{1080^{\circ}}{8} = 135^{\circ}
  2. Let the interior angle of the unknown polygon be xx. The sum of angles at the vertex must be 360∘360^{\circ}: 135∘+135∘+x=360∘135^{\circ} + 135^{\circ} + x = 360^{\circ} 270∘+x=360∘270^{\circ} + x = 360^{\circ} x=360∘−270∘=90∘x = 360^{\circ} - 270^{\circ} = 90^{\circ}
  3. A regular polygon with an interior angle of 90∘90^{\circ} is a square.

Explanation:

In semi-regular tessellations, the sum of all interior angles of the polygons meeting at a single vertex must equal 360∘360^{\circ}.

Problem 3:

Calculate the interior angle of a regular hexagon and show that it can form a regular tessellation.

Solution:

  1. Calculate the interior angle for n=6n = 6: Interior Angle=(6−2)×180∘6=720∘6=120∘\text{Interior Angle} = \frac{(6 - 2) \times 180^{\circ}}{6} = \frac{720^{\circ}}{6} = 120^{\circ}
  2. Divide 360∘360^{\circ} by the interior angle: 360∘120∘=3\frac{360^{\circ}}{120^{\circ}} = 3
  3. Since 33 is an integer, exactly 33 hexagons meet at each vertex without gaps.

Explanation:

Because 120∘120^{\circ} is a factor of 360∘360^{\circ}, regular hexagons can tile the plane perfectly.

Problem 4:

Determine if a regular dodecagon (12-sided polygon) can form a regular tessellation. Support your answer by calculating the interior angle.

Interior angle of a dodecagon shown as 150 degrees.

Solution:

n=12n = 12 Interior angle=(12−2)×180∘12=1800∘12=150∘\text{Interior angle} = \frac{(12 - 2) \times 180^{\circ}}{12} = \frac{1800^{\circ}}{12} = 150^{\circ} Check for tessellation: 360∘150∘=2.4\text{Check for tessellation: } \frac{360^{\circ}}{150^{\circ}} = 2.4 Since 2.42.4 is not an integer, a regular dodecagon cannot form a regular tessellation as there would be a gap of 360∘−(2×150∘)=60∘360^{\circ} - (2 \times 150^{\circ}) = 60^{\circ} if two dodecagons meet, or an overlap if three meet.

Explanation:

For a regular polygon to tessellate, its interior angle must be a factor of 360∘360^{\circ}. Since 150150 does not divide 360360 evenly, a gap is left.

Problem 5:

A semi-regular tessellation is made of regular triangles and regular hexagons. If two triangles and two hexagons meet at each vertex in the order (3, 6, 3, 6), verify that they form a valid tessellation at the vertex.

Vertex showing two 60-degree angles and two 120-degree angles adding to 360.

Solution:

Interior angle of triangle=60∘\text{Interior angle of triangle} = 60^{\circ} Interior angle of hexagon=120∘\text{Interior angle of hexagon} = 120^{\circ} Sum at vertex=(2×60∘)+(2×120∘)\text{Sum at vertex} = (2 \times 60^{\circ}) + (2 \times 120^{\circ}) Sum at vertex=120∘+240∘=360∘\text{Sum at vertex} = 120^{\circ} + 240^{\circ} = 360^{\circ} Since the sum is exactly 360∘360^{\circ}, the combination forms a valid tessellation.

Explanation:

A tiling is valid if the sum of interior angles at any vertex is exactly 360∘360^{\circ}. Here, the combination of two triangles and two hexagons satisfies this condition.