Minimum planar sets with maximum equidistance counts

AbstractLet g(k) be the smallest integer n for which there are n planar points each of which has k others equidistant from it. Every equilateral triangle realizes g(2) = 3. We prove that g(3) = 6, g(4) = 8 and g(5) ≤ 16. Every realizer of g(3) = 6 consists of the vertices of two similarly-oriented equilateral triangles of side length d with distance d between each vertex of a triangle and its congruent twin in the other triangle. Our constructions for k = 4,5 feature squares and equilateral triangles