How many lines are needed on the plane to get all angles from $1$ to $359$ degrees?

Question

I am trying to solve following problem:

How many lines are needed on the plane to get all angles from $1$ to $359$ degrees? We can move lines in parallel.

I am thinking that since moving in parallel doesn't change angles we can put them intersecting in one point. It is obvious that $180$ angles are enough ($0,...,179$ degrees). Need to find minimal number.

For every number from $1$ to $359$ there should be two lines that intersect in that angle.

Answer 1

The complete sparse Wichmann ruler $W(2,5)$ $$0, 1, 2, 5, 10, 15, 26, 37, 48, 59, 70, 76, 82, 88, 89, 90 $$ can generate all integer differences up to $90$. As $\alpha$ immediately also gives us $180^\circ-\alpha$, $180^\circ +\alpha$ and $360^°-\alpha$, this is a solution, though not necessarily minimal (it is minimal for a set giving all numbers up to $90$ as immediate differences).

There might exist a solution with less than $16$ lines and some of the numbers larger than $90$. But we cannot do much better than these $16$ lines: In order to generate $90$ different angles between pairs, we need at least $90$ pairs, hence at least $14$ lines because ${13\choose 2}<90$.

Answer 2

Trying to find $15$ such lines, I found only $2$ dozens of (non-equivalent) such sets but with defect in $1$ angle.

And it is interesting, that all found sets have these defects only:

• $9^{\circ} \;(171^{\circ})$;
• $27^{\circ} \;(153^{\circ})$;
• $63^{\circ} \;(117^{\circ})$;
• $81^{\circ} \;(99^{\circ})$.

Examples:

defect $9^{\circ}$:
$0,4,23,29,39,41,56,61,87,101,126,129,136,137,150$;
$0,8,36,40,42,43,55,60,65,86,98,108,119,135,149$;
$0,5,8,22,28,29,40,55,59,75,98,100,123,136,146$;
$0,7,11,13,25,40,48,50,53,74,85,104,105,121,143$;
$0,8,26,32,39,68,69,73,85,88,90,102,113,123,140$;
$0,5,26,36,38,42,59,60,89,100,103,108,115,128,135$;

defect $27^{\circ}$:
$0,1,3,9,22,38,67,93,103,107,122,127,132,139,150$;
$0,2,6,7,9,22,39,47,57,68,91,96,110,122,146$;
$0,5,23,31,36,53,56,60,62,72,100,114,115,135,146$;
$0,1,5,24,49,62,64,71,74,92,95,103,109,129,145$;
$0,1,8,10,13,31,34,48,59,63,83,102,108,124,144$;
$0,15,16,20,54,60,71,73,82,85,95,96,103,106,132$;

defect $63^{\circ}$:
$0,3,25,32,33,46,51,74,84,86,90,101,110,121,146$;
$0,4,7,15,20,25,37,39,68,94,95,104,118,140,146$;
$0,2,8,31,41,59,66,89,92,102,106,111,126,127,138$;
$0,11,12,17,30,33,53,58,62,82,89,97,99,123,137$;
$0,3,4,21,31,33,40,56,71,76,82,90,95,114,136$;
$0,16,20,21,22,55,56,65,73,80,92,103,105,106,134$;

defect $81^{\circ}$:
$0,1,15,18,20,24,31,52,60,85,95,107,132,142,154$;
$0,5,20,36,45,53,55,58,79,102,106,108,109,120,148$;
$0,1,7,10,25,33,38,50,54,73,84,111,123,125,145$;
$0,5,13,33,54,84,85,98,110,111,121,123,127,130,145$;
$0,3,5,24,41,50,51,54,70,85,93,107,113,118,125$;
$0,6,13,16,18,23,37,38,64,67,71,78,106,114,123$.

So, I am convinced that Hagen von Eitzen's answer "$16$ lines" is the final answer.