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Shape and Space - Tessellations

Grade 5IB

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

🔑Concepts

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A tessellation (or tiling) is a repeating pattern of polygons that covers a flat surface completely without any gaps or overlaps. Imagine a bathroom floor or a honeycomb where every shape fits perfectly against its neighbors.

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Regular tessellations are made using only one type of regular polygon (where all sides and angles are equal). Only three regular polygons can create a regular tessellation: the equilateral triangle, the square, and the regular hexagon. You can visualize this by seeing how 66 triangles, 44 squares, or 33 hexagons meet perfectly at a single point.

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The Vertex Point Rule states that for shapes to tessellate, the sum of the interior angles of the polygons meeting at any vertex (corner) must be exactly 360∘360^\circ. If the sum is less than 360∘360^\circ, there will be a gap; if it is more, the shapes will overlap.

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Semi-regular tessellations are patterns created using two or more different types of regular polygons. To be semi-regular, the arrangement of shapes must be identical at every vertex point throughout the entire pattern, such as a pattern of hexagons surrounded by triangles.

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Non-regular tessellations use irregular shapes, such as rectangles, rhombuses, or even complex 'Escher-style' drawings. As long as the shapes repeat and leave no gaps or overlaps, they form a tessellation. You can see this in brick walls where rectangles are offset but still cover the whole surface.

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Tessellations are created through transformations, specifically translation (sliding a shape), rotation (turning a shape around a point), and reflection (flipping a shape over a line). For example, a square tessellation is often just a simple translation of the original square horizontally and vertically.

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A polygon's interior angle determines if it can tessellate alone. For a regular polygon with nn sides to form a regular tessellation, its interior angle must be a factor of 360∘360^\circ.

📐Formulae

Sum of interior angles in a polygon with nn sides: S=(n−2)×180∘S = (n - 2) \times 180^\circ

Individual interior angle of a regular polygon: A=(n−2)×180∘nA = \frac{(n - 2) \times 180^\circ}{n}

Tessellation condition at a vertex: ∑angles=360∘\sum \text{angles} = 360^\circ

Number of shapes meeting at a vertex in a regular tessellation: m=360∘Interior Anglem = \frac{360^\circ}{\text{Interior Angle}}

💡Examples

Problem 1:

Explain why a regular pentagon (5 sides) cannot form a regular tessellation.

Solution:

  1. Find the interior angle of a regular pentagon: A=(5−2)×180∘5=3×180∘5=540∘5=108∘A = \frac{(5 - 2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ.
  2. Check if 108∘108^\circ is a factor of 360∘360^\circ: 360∘÷108∘≈3.33360^\circ \div 108^\circ \approx 3.33.
  3. Since 3.333.33 is not a whole number, the angles will not sum to exactly 360∘360^\circ. Three pentagons would sum to 324∘324^\circ (leaving a 36∘36^\circ gap), and four would sum to 432∘432^\circ (causing an overlap).

Explanation:

To tessellate, the interior angle of the regular polygon must divide into 360∘360^\circ without a remainder. Because 108∘108^\circ does not, pentagons cannot cover a surface without gaps.

Problem 2:

A semi-regular tessellation is made of regular hexagons and equilateral triangles. If two hexagons and two triangles meet at a vertex, prove that they form a tessellation.

Solution:

  1. Interior angle of a regular hexagon (n=6n=6): (6−2)×180∘6=120∘\frac{(6-2) \times 180^\circ}{6} = 120^\circ.
  2. Interior angle of an equilateral triangle (n=3n=3): (3−2)×180∘3=60∘\frac{(3-2) \times 180^\circ}{3} = 60^\circ.
  3. Sum the angles at the vertex: (2×120∘)+(2×60∘)(2 \times 120^\circ) + (2 \times 60^\circ).
  4. 240∘+120∘=360∘240^\circ + 120^\circ = 360^\circ.

Explanation:

Because the sum of the angles where the shapes meet is exactly 360∘360^\circ, the shapes fit together perfectly around the vertex point.