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How is symmetry related to tessellation?
Tesselation (or tiling) generally implies translational symmetry, because you can generally move one part of a tiling over another a specific distance away and get an exact match (ie the tesselation is periodic). A counterexample (possibly the only one) is Penrose tiling, which is non-periodic. There is certainly no need for a tesselating shape to have either bilateral or rotational symmetry: all triangles and all parallelograms (including squares and rectangles) will tessellate. I'm afraid this is a rather superficial answer to this very interesting question; a deeper one will have to come from someone with a knowledge of group theory.
Another name for tessellation?
Tiling.
What is the first np-hard problem?
You can use the bounded tiling problem. Given a problem in NP A, it has a turing machine M that recognizes its language. We construct tiles for the bounded tiling problem that will simulate a run of M. Given input x, we ask if there is a tiling of the plane and answer that will simulate a run of M on x. We answer true iff there is such a tiling.
Can a hexagon form a tiling pattern?
yes
Can you make a tiling pattern with circles?
no you can not because it is round
