1,10

"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).

Kevin S. Brown's Mathpages, Adventitious Solutions

Wikipedia, Langley’s Adventitious Angles

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

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 *)

Sequence in context: A075533 A355783 A053814 * A095238 A167345 A292496

Adjacent sequences: A293257 A293258 A293259 * A293261 A293262 A293263

nonn,more

Albert Lau, Oct 04 2017

approved