Optimal packings of one to seven unit squares in a disk
For n = 1, …, 7: the least radius of a closed disk holding n unit squares with disjoint interiors, each placed and rotated freely, and every packing that attains it. The radii are √2/2, √5/2, 5√17/16, √2, √(5/2), √q* ≈ 1.68854 and √13/2, where q* is a quartic irrational. Up to rotation about the disk centre and relabelling, the optimal packing is unique for n ≤ 6: the square, the 2 × 1 rectangle, the T, the 2 × 2 block, the plus, and for six squares a central square with four neighbours, two of them pushed along its sides, and a sixth square turned by π/4 between those two. For n = 7 the…
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