The packing problem for twisted pairs

When wires are packed together in a bundle, as in a cable or on a shelf of a main distribution frame, the packing fraction f is the fraction of cross-sectional area of the bundle occupied by wire. With wires all the same radius, packing fractions as high as 0.90690 can be achieved. However, when the wires are pairs that have been twisted to avoid crosstalk, the packing fraction is much smaller. The largest obtainable packing fraction depends on other properties of the packing. For example, with pairs twisted by machine, all pairs twist at the same rate, and that influences the packing fraction. Several packing problems are considered, but most attention is given to a particularly regular kind of packing in which pairs twist about straight parallel axes located in a lattice arrangement. The densest lattice packing has packing fraction 0.56767. The densest lattice is a complicated one in which each wire touches 10 wires belonging to 6 other pairs. The numbers 10 and 6 cannot be increased even with nonlattice packings of pairs with straight parallel axes. These other packings are also conjectured to have packing fractions less than 0.56767, although only f < 0.62240 is proved.