L3 · Tessellations and symmetry in design

Learning goal

“I can say whether a shape tessellates by checking the angles around a point, and I can describe the reflection and rotation symmetry in a tessellation - including the eight-pointed star design used in First Nations and Métis star blankets.”

Tessellation — a pattern of shapes covering a flat surface with no gaps and no overlaps.

Tile — one shape repeated to make a tessellation.

Vertex point (of a tessellation) — a point where the corners of tiles meet.

Interior angle — the angle inside a shape at one of its corners.

Worked example

Do regular hexagons tessellate? Step 1: The interior angle of a regular hexagon is 120°. Step 2: Around one vertex point, 3 hexagons can meet: 120° + 120° + 120° = 360°. Step 3: The angles fill the full turn exactly - no gap, no overlap. So regular hexagons DO tessellate (this is the honeycomb pattern). The test is always the same: the interior angles meeting at a point must add to exactly 360°. Squares pass (4 x 90° = 360°) and equilateral triangles pass (6 x 60° = 360°).

Check yourself — explain it

A shape tessellates on its own when...

Paired problem — example still visible

Do regular pentagons tessellate? Interior angle of a regular pentagon = 108°. -> No. 3 pentagons give 3 x 108° = 324°, leaving a 36° gap, and 4 pentagons give 432°, which overlaps. No whole number of pentagons makes exactly 360°, so regular pentagons do NOT tessellate.

Faded completion — last step is yours

"Square tiles meet 4 at a point: 4 x 90° = ___°, so squares tessellate." -> 360°

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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