Summary

I did a three strand braid earlier, but I wanted to experiment with higher orders. It was quite a bit harder to work out. I have included the .scad code so that anyone interested can generate similar braids.

faces=20;
step=5;

for(th=[0:step:360]) {
translate([th/PI,20sin(th ),wave((th+168 )%180)])sphere(2,$fn=faces);
translate([th/PI,20sin(th+72 ),wave((th+168+72 )%180)])sphere(2,$fn=faces);
translate([th/PI,20sin(th+144),wave((th+168+144)%180)])sphere(2,$fn=faces);
translate([th/PI,20sin(th+216),wave((th+168+216)%180)])sphere(2,$fn=faces);
translate([th/PI,20*sin(th+288),wave((th+168+288)%180)])sphere(2,$fn=faces);
}

function wave(th)=(((((((1.53850006563052E-16th-1.27387805434208E-13)th+4.34669346542468E-11)th-7.88147244501425E-9)th+ 8.16830636022657E-7)th-4.84392712381305E-5)th+1.54722865226176E-3)th-2.17013888888711E-2)th*th+1;