new cube packing record for n=12 at 2.931514578 found with this method. beats Haowei Lin's 2026 record (2.93277+). s=2.931514577965 # i x y z w qx qy qz # cube=c+R(w,qx,qy,qz)[-1/2,1/2]^3, Hamilton 1 1.5243879843 2.4315145780 1.3320162706 0.5838886869 -0.5838886869 -0.3988408220 -0.3988408220 2 2.4315145780 2.4315145780 0.5000000000 0.7071067812 0.7071067812 0 0 3 0.5000000000 1.4369539605 1.4377198700 0.5395289954 0.4570650535 0.4570650535 -0.5395289954 4 0.5098735814 0.5769311229 2.3545747284 0.4560511351 -0.5403863020 -0.4560472394 0.5403896022 5 0.6086093613 2.2929283270 2.3887741023 0.0258332833 0.7609821302 -0.6482073268 -0.0081302199 6 2.4315145780 1.4315145780 0.5000000000 0 0 0 -1 7 2.4298401750 2.4215249296 2.2620514795 0.0000269579 0.0016621483 -0.9999712786 0.0073944948 8 1.5800388387 1.4806909324 2.4315145780 0 -0.9765112947 -0.2154662186 0 9 1.6870669871 0.7046048868 1.3931621797 0.1630483719 -0.8266039691 -0.3023204668 0.4458065074 10 0.6714192000 0.5125505069 0.5042784926 0.0019279356 0.0017767559 -0.7100527881 -0.7041435680 11 2.4315145780 0.5648562958 2.4315145780 0.5 -0.5 -0.5 0.5 12 0.5 2.3777488832 0.5 0.7071067812 -0.7071067812 0 0