Different-shape polygons can cover planes {tiling, geometry} with no gaps and no overlaps.
shapes
One triangle, square, or hexagon shape can tile. One pentagon shape cannot tile.
Pairs of shapes can tile, such as two different irregular pentagons. Such tilings have repeated parallelograms {periodic tiling}.
Spirals can tile without repeated parallelogram {aperiodic tiling}. For example, nine-sided triangle-shaped tile {versatile} can tile periodically and non-periodically. Square-shaped tiles with protruding points and corresponding concave depressions can tile aperiodically. Other four-sided shapes {Penrose tiles} have protrusions and depressions, have five-fold symmetry, make repeated patterns, and can tile aperiodically {quasi-periodic tiling}.
periodicity
Algorithms can decide if tiles can tile the plane periodically. No algorithm can decide generally if tiles can tile the plane aperiodically.
Identical shapes, such as triangles, rectangles, hexagons, or special five-sided polygons, or shape sets can fill planes, polyhedrons, or curved surfaces without gaps or overlaps {tessellation}|.
types
Tessellation {regular tessellation} can use equilateral triangles, squares, or regular hexagons. Tessellation {homogeneous tessellation} {semiregular tessellation} can have congruent common vertices {nodal point, homogeneous tessellation} and regular polygons. Tessellation {non-homogeneous tessellation} can use irregular shapes, different sizes of one shape, or both.
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Date Modified: 2022.0225