Lattice Points and Pi Relationship
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This example demonstrates the phenomenon that the ratio of lattice points with coprime coordinates to the total number of lattice points approaches 6/π² as the grid size increases, as per Euler's theorem. It involves core math concepts like coordinate geometry, probability & statistics, limits, and inequalities. Key GeoGebra commands and objects used include Intersect, RemoveUndefined, Length, Vector, and elements like a, b, A, n. Suitable for high school and above students exploring the intersection of geometry, probability, and number theory.
GeoGebra Commands & Parameters
Intersect(xAxis, yAxis) → ARemoveUndefined(Join[Sequence[Sequence[If[GCD[j, k] ≟ 1, (j, k)], j, 1, n], k, 1, n]]) → blauPLength(blauP) → blauZRemoveUndefined(Join[Sequence[Sequence[If[GCD[j, k] > 1, (j, k)], j, 1, n], k, 1, n]]) → rotPLength(rotP) → rotZVector(Element[blauP, einzeln]) → vP