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)

Non-consecutive angles sum = 360

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.

nc360g tiling

The tiling can continue to infinity.


And below there is the order 6 rotational symmetry tiling.

nc360g tiling


Next tiling is the most famous: 12 pentagons forms a regular dodecagon that tiles the plane with the help of other 4 pieces:

nc360g tiling


Now we use the A=5B property. The elegant symmetrical configuration of 10 pieces tiles the plane with the help of other two pentagons:

nc360g tiling


Below we demonstrate that this pentagon can tile 1/4 of the infinite plane.

nc360g tiling


At last, a labouriosus demonstration that can tile 1/3 of the infinite plane:

nc360g tiling

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