Tessellations: Creating Patterns with Geometric Shapes
How can simple shapes cover a floor or wall without leaving holes or overlapping? The key is a quick angle check: around every point where the shapes meet, their angles must add to exactly 360°. Learn the rule, test regular polygons, and try a mixed-shape pattern of your own.
The main idea: make every vertex fit
A tessellation covers a flat surface with shapes that repeat without gaps or overlaps. A vertex is a corner point where shapes meet. If the shapes leave an opening at a vertex, or crowd past a full turn, the pattern will not fit together there.
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The regular-polygon angle formula
For a regular polygon with n sides, every interior angle has the same measure:
If you are testing one kind of regular polygon by itself, divide 360° by its interior angle. The result must be a whole number of polygons at each vertex.
Which regular polygons tile by themselves?
Use the angle formula and test how many identical polygons can meet at one point.
| Regular shape | Interior angle | Count at a vertex | Tiles by itself? |
|---|---|---|---|
| Equilateral triangle | 60° | 6 | Yes |
| Square | 90° | 4 | Yes |
| Regular pentagon | 108° | 360° ÷ 108° is not whole | No |
| Regular hexagon | 120° | 3 | Yes |
| Regular heptagon | About 128.57° | 360° ÷ angle is not whole | No |
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So the regular polygons that tile the plane by themselves, edge-to-edge, are equilateral triangles, squares, and regular hexagons. This statement is about regular polygons; some nonregular polygons, including some pentagons, can also tessellate.
Worked example 1: a regular hexagon
A regular hexagon has 6 sides. Find each interior angle, then determine how many hexagons meet around one vertex.
Three equal angles make exactly one full turn, so three regular hexagons meet snugly at each vertex. With equal side lengths, their edges line up and the pattern continues without gaps or overlaps.
Worked example 2: why regular pentagons do not tile alone
A regular pentagon’s interior angle is 108°. Check the two nearest whole-number possibilities:
Three leave a gap, while four overlap. A regular pentagon cannot tile the plane by itself in an edge-to-edge pattern, although other shapes can be combined with it in different tessellations.
A mixed-shape pattern can work too
The 360° rule also works when more than one regular shape meets at a vertex. A square angle is 90°, and a regular octagon angle is 135°.
Thus one square and two regular octagons can meet around each vertex in the familiar square-and-octagon tiling. For a clean edge-to-edge pattern, all the shapes must also use matching side lengths where their edges meet.
Common mistakes to avoid
- Using 180° instead of 360°. A flat polygon’s interior angles add to a different total; the tessellation check is around one point, a full turn of 360°.
- Mixing up side count and shapes-at-a-vertex count. A hexagon has 6 sides, but 3 regular hexagons meet at each vertex in its regular tiling.
- Rounding an almost-whole result. The count must be an exact whole number; 3.33 shapes cannot meet at a vertex.
- Assuming every polygon behaves like a regular polygon. The result that regular pentagons do not tile alone does not mean every pentagon is unable to tile.
- Checking angles but forgetting the edges. Mixed shapes need compatible side lengths for an edge-to-edge tessellation.
Practice questions
- How many regular hexagons meet at one point in a regular hexagon tessellation? Show the angle check.
- A vertex is formed by two regular octagons and one square. Find the total angle and decide whether they fit exactly around the point.
- Three regular pentagons meet at a point. How large is the gap? What happens if a fourth pentagon is added?
- A regular heptagon has interior angle 900° ÷ 7, about 128.57°. Can three meet at a vertex without overlap? Explain.
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Answers
- Three regular hexagons. Each angle is 120°.3 × 120° = 360°
- They fit exactly.135° + 135° + 90° = 360°
- Three leave a 36° gap.360° − (3 × 108°) = 36°; 4 × 108° = 432°; 432° − 360° = 72°
- No. Three angles total more than one full turn.3 × (900° ÷ 7) = 2700° ÷ 7 ≈ 385.71° > 360°
The takeaway
To test a tessellation, check the angles where shapes meet: they must add to 360°. For a single regular polygon, its interior angle must divide 360° into a whole number of shapes. That quick check explains why regular triangles, squares, and hexagons tile by themselves—and why mixed shapes can make many more patterns.
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