Particular Examples of Planar Integral Point Sets and Their Classification

A planar integral point set (PIPS) is a finite set of non-collinear points in the Euclidean plane such that the Euclidean distance between any pair of points is an integer. These sets are characterized by their cardinality (the finite number of points), diameter (the maximum pairwise distance), and characteristic (the smallest positive integer \(q\) such that all triangular areas are commensurable with \(\sqrt{q}\)). The characteristic remains invariant under translations, dilations, reflections, and even the addition or removal of points. Existing classifications include sets in semi-general position (no three points collinear) and general position (no three collinear and no four concyclic). Circular sets and facher sets (all but one point on a line) are prominent examples, but finding sets of general position is difficult problem. For instance, the~largest known set has seven points, and no eight-point example is currently known. This work introduces new examples to advance the classification, including rails sets (points on two parallel lines) and sets with multiple symmetries. We also present sets with many shared points that cannot be merged. These constructions highlight the potential of sequential extensions and limitations of merging sets, offering insights into the structure and properties of planar integral point sets.