Tessera can be any shape. Often they are assumed to be polygonal but that is not a requirement. To see tessellation based on other shapes see the set of "Symmetry" artwork by MC Escher.
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Look up tesselation on google images, there are also different types of tesselation. Shapes connected together from left to right, or in rotations
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An irregular tessellation is a tesselation made of irregular shapes. Escher is famous for drawing these with animals in them.
Tessellation is using multiple copies of one - or a few - basic shapes to cover a plane space without gaps or overlaps.
Certain polygons, yes. Squares, Triangles and Hexagons are all shapes which, in their regular form, can tessellate. Other polygons cannot.