Tilings

The plane can be tiled with one regular polygon only if it is a triangle, a square or an hexagon.
It is impossible to tile the plane with regular pentagons.
But the tiling of the plane by irregular pentagons might be possible, depending on the shape of these pentagons.
It is not hard to tile the plane with any irregular triangle or quadrilateral.
In case of the pentagons, we must satisfy conditions on the angles and the sides.

Irregular triangles

   Solution

Irregular quadrilaterals

   Solution

Irregular pentagons

There are 14 known shapes of irregular convex pentagons which can tile the plane :
  1. D + E = 180°
  2. C + E = 180°, a = d
  3. A = C = D = 120°, a = b, d = c + e
  4. A = C = 90°, a = b, c = d
  5. C = 2A = 120°, a = b, c = d
  6. C + E = 180°, A = 2C, a = b = e, c = d
  7. 2B + C = 360°, 2D + A = 360°, a = b = c = d
  8. 2A + B = 360°, 2D + C = 360°, a = b = c = d
  9. 2E + B = 360°, 2D + C = 360°, a = b = c = d
  10. E = 90°, A + D = 180°, 2B - D = 180°, 2C + D = 360°, a = e = b + d
  11. A = 90°, C + E = 180°, 2B + C = 360°, d = e = 2a + c
  12. A = 90°, C + E = 180°, 2B + C = 360°, 2a = c + e = d
  13. A = C = 90°, 2B = 2E = 360° - D, c = d, 2c = e
  14. D = π/2, 2E + A = 2π, C + A = π, 2a = 2d = b = c
To construct such pentagons and tile the plane. (recall : the sum of the angles in a pentagon is 540°)
Details 1-6     Details 7-9      Details 10-14
Tiling with not convex pentagons ? Details

Irregular hexagons

Same question (easier) with the three shapes of irregular hexagons which tile the plane.
  1. A + B + C = 360°, a = d
  2. A + B + D = 360°, a = d and c = e
  3. A = C = E = 120°, a = b, c = d and e = f

Details

 

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