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Certain polygons, yes. Squares, Triangles and Hexagons are all shapes which, in their regular form, can tessellate. Other polygons cannot.
Look up tesselation on google images, there are also different types of tesselation. Shapes connected together from left to right, or in rotations
No.
No.
An irregular tessellation is a tesselation made of irregular shapes. Escher is famous for drawing these with animals in them.