Tiling a flat plane using one or more geometric shapes with no overlaps or gaps is known as Tessellation.

Only two shapes tessellate a plane perfectly: the Parallelogram and Hexagon. Parallelograms include the Square, Rectangle, Diamond, Rhombus, and Rhomboid.

Other tessellations are possible if one is allowed to flip or rotate tiles, as is the case with the Triangle, Irregular Pentagon, Kite, Half Regular Hexagon, and Quadrilaterals.

see Irregular Tessellation.

References

• [wikipedia: Tessellation]
• [wikipedia: Regular Tiling]
• [wikipedia: Square Tiling]
• [wikipedia: Hexagonal Tiling]
• [Wolfram MathWorld: Tiling]
• [Wolfram MathWorld: Tessellation]