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Shape and Space - Symmetry and Tessellations

Grade 3IB

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

🔑Concepts

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Line Symmetry (Reflective Symmetry): A shape has line symmetry if it can be divided into two identical halves by a line. Imagine a butterfly with a vertical line drawn down its body; if you fold it along that line, the left and right wings align perfectly.

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Lines of Symmetry: This is the imaginary line where you fold a shape. A regular polygon (where all sides and angles are equal) has the same number of lines of symmetry as it has sides. For example, a regular triangle has 33 lines of symmetry, while a square has 44 (vertical, horizontal, and two diagonals).

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Rotational Symmetry: A shape has rotational symmetry if it looks exactly the same after being turned around its center point by less than a full circle. Imagine a star shape or a propeller; as you rotate it 120∘120^{\circ} or 180∘180^{\circ}, it sits in the same position it started in.

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Tessellations (Tiling): A tessellation is a pattern of repeating shapes that covers a flat surface with no gaps and no overlaps. Think of a brick wall where rectangles fit together perfectly, or a checkerboard made of squares.

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Regular Tessellations: These are special patterns made using only one type of regular polygon. Only three regular polygons can create a regular tessellation on their own: the equilateral triangle, the square, and the regular hexagon. This is because their interior angles meet at a point to form exactly 360∘360^{\circ}.

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Vertex in Tessellation: A vertex is a point where the corners of the shapes in a tessellation meet. In a square tessellation, four corners meet at every vertex, looking like a plus sign ++. In a hexagonal tessellation (like a honeycomb), three corners meet at each vertex, looking like a YY shape.

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Non-Tessellating Shapes: Shapes that have curved edges or irregular angles often cannot tessellate because they leave gaps. For example, if you place several circles side-by-side, there will always be a small diamond-shaped gap between them.

📐Formulae

L=nL = n (Number of lines of symmetry in a regular polygon with nn sides)

Sum of angles at a vertex in a tessellation=360∘\text{Sum of angles at a vertex in a tessellation} = 360^{\circ}

Order of Rotational Symmetry=360∘Smallest angle of rotation\text{Order of Rotational Symmetry} = \frac{360^{\circ}}{\text{Smallest angle of rotation}}

💡Examples

Problem 1:

Determine how many lines of symmetry a regular hexagon has and describe where they are located.

Solution:

Step 1: Count the number of sides of a regular hexagon, which is n=6n = 6. Step 2: Use the rule that for regular polygons, the number of lines of symmetry equals the number of sides. Step 3: Identify the positions. There are 33 lines that go through opposite corners (vertices) and 33 lines that go through the midpoints of opposite sides. Total = 66.

Explanation:

In a regular polygon, symmetry lines always pass through the center, either connecting vertices or bisecting sides.

Problem 2:

Can you create a tessellation using only regular octagons (8-sided shapes)? Why or why not?

Solution:

Step 1: Examine the interior angle of a regular octagon, which is 135∘135^{\circ}. Step 2: Try to fit octagons together at a single point (vertex). 135∘+135∘=270∘135^{\circ} + 135^{\circ} = 270^{\circ}. If we add a third octagon, 270∘+135∘=405∘270^{\circ} + 135^{\circ} = 405^{\circ}, which is more than 360∘360^{\circ}. Step 3: Since the angles do not add up exactly to 360∘360^{\circ}, there will be overlaps or gaps.

Explanation:

For shapes to tessellate, the angles meeting at every vertex must sum exactly to 360∘360^{\circ}. Since octagons leave gaps, they cannot tessellate alone.