Equilateral Pentagon That Tiles The Plane
With Two Consecutive Complementary to 180° Angles
With Two Non-consecutive Complementary to 360° Angles
A+B = 180° & A+C = 360° & (B = 360°/12 or A = 5B)
This pentagon has the property of the 1st family, two consecutive complementary to 180° angles, and the property of the 4th family, two non-consecutive complementary to 360° angles. The angle of 30° and the angle of 60° allow to tile the plane with order 6 and order 12 rotational symmetry. The A=5B property allows other interesting tilings.
You can see below the order 12 rotational symmetry tiling.
The tiling can continue to infinity.
And below there is the order 6 rotational symmetry tiling.
Next tiling is the most famous: 12 pentagons forms a regular dodecagon that tiles the plane with the help of other 4 pieces:
Now we use the A=5B property. The elegant symmetrical configuration of 10 pieces tiles the plane with the help of other two pentagons:
Below we demonstrate that this pentagon can tile 1/4 of the infinite plane.
At last, a labouriosus demonstration that can tile 1/3 of the infinite plane: