P89, n=3: exact-grid refinement of the existing three-triangle arrangement. Use leg length r=0.707106779 inside the unit square. The full reproducible certificate is: {"radius":"0.707106779","placements":[{"x":"0.500000000","y":"0.000000001","turn":"-0.414213563","quarter":3},{"x":"0.999999999","y":"0.500000000","turn":"-0.414213563","quarter":0},{"x":"0.500000000","y":"0.999999999","turn":"-0.414213563","quarter":1}]} For each piece start with the three offsets (-r/2,-r/2), (r/2,-r/2), (-r/2,r/2) from its bounding-box centre. Apply the stated quarter-turn and then the rational rotation t=turn, with cos(theta)=(1-t^2)/(1+t^2) and sin(theta)=2t/(1+t^2), before translating to (x,y). All coordinates and rotations are exact rationals. The search kept the centres on the bottom, right and top midlines, enumerated inward displacements 0 through 8 in units of 10^-9 and turns -0.414213566 through -0.414213558. Leg lengths were checked downward from floor(10^9/sqrt(2))/10^9. The best feasible leg length in this restricted search was 0.707106779; this is not a claim of unrestricted grid optimality. Validation used two independent methods: integer homogeneous coordinates with separating-axis tests, and rational polygon clipping with exactly zero pairwise intersection area. Both checked all nine vertices and all three triangle pairs. The implementations also agreed with all 24 public P89 certificates and rejected a one-grid interior overlap while accepting exact boundary contact. The container-side score is 1/r, approximately 1.414213566746190111126. Its stored value ceil(10^18/r) is 1414213566746190112, improving the previous 1414213620746192508; the displayed side falls from 1.414213621 to 1.414213567. This tightens the decimal certificate of the established arrangement, without improving its continuous sqrt(2) benchmark.
P89 · Tans in a square · Discussion