Equilateral Pentagon That Tiles The Plane
- With Two Non-consecutive Complementary to 360 Angles
- With Two Non-consecutive Complementary to 180 Angles
A+C = 360 & A+D = 180

Livio's pentagon


The story began the day that I checked the tiling of the convex (non-equilateral) pentagon that you see below. This tiling is not present on the list of the 14 convex pentagons that tile the plane because this pentagon is considered a subset of the Type 2, despite the topology is very different.

Livio's pentagon


This convex pentagon has 4 equal edges. If you want to impose equal the 5th, you'll obtain the false-pentagon (blue), a trapezium. But if you continue the quest, you'll find a concave equilateral pentagon (red).

Obviously this new pentagon tiles the plane with the same topology of the non-equilateral first:

Livio's pentagon tiling


It allows also the Cairo tiling below and all the general tilings for the property: A+C = 360.

Livio's pentagon


For the property B+C+E = 360, it's possible this hexagon with parallel edges:

Livio's pentagon


For the property 2B+2D+2E = 360, this interesting decagon:

Livio's pentagon


For the property 2B+6D = 360, this decagon that we can multiply to infinity:

Livio's pentagon


For all its properties, this chaotic tiling:

Livio's pentagon


An example for each type of knot:

Livio's pentagon


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