Tiling Plane Surfaces with Regular Polygons

To determine which regular polygon is the odd one out in tiling a plane surface, we need to check if the polygon can tessellate. A tessellation (or tiling) covers a plane with shapes having no overlaps or gaps. For a regular polygon to tessellate, the sum of the interior angles meeting at any vertex must equal exactly 360 degrees.

Calculating Interior Angles

The formula for the interior angle of a regular n-sided polygon is:

$ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} $

Analyzing Each Polygon

• Equilateral Triangle (n=3):
  • Interior Angle = $ \frac{(3-2) \times 180^\circ}{3} = \frac{180^\circ}{3} = 60^\circ $
  • Number of triangles meeting at a vertex = $ \frac{360^\circ}{60^\circ} = 6 $.
  • Since 6 triangles fit perfectly ($6 \times 60^\circ = 360^\circ$), equilateral triangles can tile a plane.
• Square (n=4):
  • Interior Angle = $ \frac{(4-2) \times 180^\circ}{4} = \frac{360^\circ}{4} = 90^\circ $
  • Number of squares meeting at a vertex = $ \frac{360^\circ}{90^\circ} = 4 $.
  • Since 4 squares fit perfectly ($4 \times 90^\circ = 360^\circ$), squares can tile a plane.
• Regular Pentagon (n=5):
  • Interior Angle = $ \frac{(5-2) \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ $
  • Number of pentagons meeting at a vertex = $ \frac{360^\circ}{108^\circ} \approx 3.33 $.
  • Since $360^\circ$ is not perfectly divisible by $108^\circ$, a whole number of pentagons cannot meet at a vertex without gaps or overlaps. Regular pentagons cannot tile a plane by themselves.
• Regular Hexagon (n=6):
  • Interior Angle = $ \frac{(6-2) \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ $
  • Number of hexagons meeting at a vertex = $ \frac{360^\circ}{120^\circ} = 3 $.
  • Since 3 hexagons fit perfectly ($3 \times 120^\circ = 360^\circ$), regular hexagons can tile a plane.

Conclusion on Tiling

The regular polygons that can tile a plane are the equilateral triangle, the square, and the regular hexagon. The regular pentagon is the odd one out because its interior angles do not allow for a combination that sums to exactly 360 degrees at any vertex, preventing it from forming a tessellation.