A uniformly distributed discrete set of points in the plane called lattices are considered. The most well-known and studied are square lattices, a special representative of such lattices is the lattice of all points with integer coefficients in the plane R×R. We are dealing with rectangular lattices, parallelogram lattices and triangle lattices. By exploring the circles, positioned on different lattices, we establish a link between the number of lattice points inside and on the edge of a circle and the number π. Determining the upper and lower bound of the error occurring, is called Gauss circle problem, since it was him who first explored lattices and circles on it. We also show the upper bound of the shortest distance between two lattice points. With the help of a theorem of number theory, we connect the number of lattice points inside and on the edge of a circle with Leibniz series. Generalizations of Pick's theorem on general lattices in the plane are considered and with a counterexample it is shown that the theorem does not apply to the hexagonal lattices. Separately we introduce two programs: a program for calculating the number of lattice points inside and on the edge of a circle, which also calculates an approximation for the number π and the error occurring, and a program for calculating the area of a polygon with Pick's theorem.
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