4 ms·

Mandatory reminder that hexagons are the bestagons. https://m.youtube.com/watch?v=thOifuHs6eY https://m.youtube.com/watch?v=thOifuHs6eY

• amelius 5y ago
  How does that generalize to three dimensions?
  • LeifCarrotson 5y ago
    As a Hexagonal Close Packing lattice, which is the correct way to represent the maximum density arrangement of equal spheres. If you've seen a stack of spheres, with the bottom layer in rows offset by one radius such that each sphere touches six others at the same height, and the next layer nestled in the intersections of this bottom layer, that's a hexagonal close packing. Also, the "hexagons are the bestagons" attitude generalizes to three dimensions by dismissing face-centered cubic packing as an inferior way to look at the arrangement. To generalize the Youtube video aspect, see also this Matt Parker/Steve Mould video on spherical packing: https://www.youtube.com/watch?v=3inLMXcetUA https://www.youtube.com/watch?v=3inLMXcetUA
    • Yajirobe 5y ago
      How does that generalize to four dimensions?
    • amelius 5y ago
      Given that the stacking of spheres is built from a stack of planar layers, wouldn't that mean that the stack is prone to shearing forces, sliding the layers over each other? Wouldn't a packing that is irregular in all directions be more robust? In the "bestagons" video, it is explained that the hexagons win because there are no straight lines in the packing, but I'd expect these to correspond to planes in the 3d case.
      • LeifCarrotson 5y ago
        There are no continuous planes in the spherical packing layers, either - they all nestle down into gaps in the lower layers.