The pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice (formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons) has now been proved. http://arxiv.org/abs/1602.07220