A293260 Number of adventitious quadrangles (convex, noncyclic, not kite) such that Pi/n is the largest number that divides all the angles.

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 12, 0, 9, 0, 14, 0, 75, 0, 26, 0, 35, 0, 110, 0, 54, 0, 57, 0, 436

Offset: 1

Author

Albert Lau, Oct 04 2017

Keywords

• nonn
• more

Comments

"All the angles" in the title means any angle formed by 3 vertices. There are 8 nonoverlapping angles in total.

Consider convex quadrilateral ABCD. Let a,b,c,d,e,f,g,h be the angles ABD,DBC,BCA,ACD,CDB,BDA,DAC,CAB, respectively. A quadrangle is adventitious if all these angles are rational multiples of Pi.

Cyclic quadrilaterals have properties a=d, b=g, c=f, e=h, thus making the adventitious case trivial.

Kites have properties a=b, c=h, d=g, e=f, thus making the adventitious case trivial.

Some properties:

1. b+c = f+g := x, d+e = h+a := y, x+y = Pi.

2. sin(a)sin(c)sin(e)sin(g) = sin(b)sin(d)sin(f)sin(h).

3. In an adventitious quadrangle, swapping angles (b,c) with (f,g) or (a,h) with (d,e) gives another adventitious quadrangle.

From empirical observation, it seems that no adventitious quadrangles exist for odd numbers n. For example, take n=9: 180 degrees/9 = 20 degrees, and forming a quadrangle in which all angles are multiples of 20 degrees is impossible (proven by brute force). It seems to hold for all odd numbers n.

Perhaps the most famous case is Langley's problem (where n=18).

Examples

			a(8) = 1 because there is one quadrangle where all angles are divisible by 180/8 = 22.5 degrees.
  a=90, b=45, c=22.5, d=45, e=67.5, f=45, g=22.5, h=22.5.
a(10) = 2 (180/10 = 18):
   72  54  18  36  72  36  36  36
  108  36  18  54  72  36  18  18
a(12) = 12 (180/12 = 18):
   75  30  45  45  60  60  15  30
   75  60  15  45  60  30  45  30
   90  30  30  45  75  45  15  30
   90  45  15  45  75  30  30  30
   90  45  30  45  60  60  15  15
   90  45  30  75  30  45  30  15
   90  60  15  45  60  45  30  15
  105  30  15  30 105  30  15  30
  105  30  30  75  45  45  15  15
  105  45  15  30  90  45  15  15
  105  45  15  75  45  30  30  15
  120  30  15  60  75  30  15  15
		

Links

• Kevin S. Brown's Mathpages, Adventitious Solutions
• Wikipedia, Langley’s Adventitious Angles

Programs

Mathematica

Remove[f];
f[n_Integer] := Do[
      If[A == B < n/2 - C, Continue[]];(* if A == B then C >= H *)
      If[A == B == n/2 - C || C == D == n/2 - B, Continue[]];(* remove kite *)
      F = n/\[Pi] ArcTan[(Sin[d] Sin[a + b])/(Sin[a] Sin[c] Sin[e]) -
           Cot[e], 1] /. Thread[{a, b, c, d, e} -> \[Pi]/n {A, B, C, D, E}];
      F = Round[F, 10^-6];
      If[A < F, Continue[]];
      If[GCD[A, B, C, D, E, F] != 1, Continue[]];
      If[A == E && B < F, Continue[]];(* if A == E then B >= F *)
      If[A == F && B < E, Continue[]];(* if A == F then B >= E *)
      {A, B, C, D, E, F, B + C - F, D + E - A} // Sow;
      , {A, n/4 // Ceiling, n - 3}
      , {B, Max[1, n - 3 A + 2], Min[A, n - A - 2]}(* B <= A and C < A and H < A *)
      , {C, Max[1, n - 2 A - B + 1], Min[A - 1, n - A - B - 1]}(* C < A and H < A *)
      , {D, n - A - B - C, A - 1}(* D < A and E <= A *)
      , {E, {n - B - C - D}}
      ] // Reap // Last // If[# == {}, {}, # // Last] &;
Table[f[n] // Length, {n, 30}]
(* 180/n f[n] /. n -> 18 // TableForm *)