A186282 - OEIS

%I #33 Aug 19 2019 15:37:12

%S 1,1,0,6,7,6,6,7,2,5,2,3,0,0,7,1,1,8,6,2,9,1,9,7,5,2,4,0,3,3,8,6,5,8,

%T 7,0,8,0,5,4,8,8,6,7,1,8,6,5,1,6,2,4,0,5,4,8,9,9,0,7,3,8,9,5,3,5,0,7,

%U 7,2,9,7,6,8,1,5,5,9,6,6,8,5,3,4,8,5,2,2,5,2,5,3,4,9,5,3,1,7,6,2

%N Decimal expansion of non-right-angle, in degrees, of unique 14th class of convex pentagonal tiling.

%C From _Jean-François Alcover_, Mar 11 2013: (Start)

%C These are the 5 angles, in radians and in degrees:

%C A = Pi/2 = 90 deg,

%C B = Pi/2 + arccos((sqrt(57)-3)/8) =  145.338336261... deg,

%C C = Pi - 2*arccos((sqrt(57)-3)/8) = 69.323327476... deg,

%C D = Pi - arccos((sqrt(57)-3)/8) = 124.661663738... deg,

%C E = 2*arccos((sqrt(57)-3)/8) = 110.676672523... deg.

%C Ratios of sides are AB:BC:CD:DE:EA = d:1:2:2:1 with d = sqrt(22*sqrt(57)-50)/4 = 2.693700493... (End)

%D Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

%D Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 177-179, 208, and 211, 1991.

%H Jean-François Alcover, <a href="/A186282/a186282.gif">Pentagonal tile</a>

%H P. L. Bowers and K. Stephenson, <a href="https://doi.org/10.1090/S1088-4173-97-00014-3 ">A 'Regular' Pentagonal Tiling of the Plane</a>, Conformal Geom. Dyn. 1, 58-68, 1997.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/tilepent.html">The 14 Different Types of Pentagons that Tile the Plane"</a>

%H M. ten Have, <a href="http://www.mtenhave.net/5/pentagon.htm">Pentagons and the Golden Section</a>

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PentagonTiling.html">Pentagon Tiling</a>

%F Solution to 3*cos(x/2) + 2*cos(x) = 1. [_Jean-François Alcover_, Mar 09 2013]

%e 110.676672523... degrees.

%t 2*ArcCos[(Sqrt[57]-3)/8]*180/Pi // RealDigits[#, 10, 100]& // First (* _Jean-François Alcover_, Mar 09 2013 *)

%o (PARI) solve(x=Pi, Pi/2, 3*cos(x/2) + 2*cos(x) - 1)*180/Pi \\ _Michel Marcus_, Aug 19 2019

%K nonn,cons

%O 3,4

%A _Jonathan Vos Post_, Feb 16 2011

%E More terms from _Jean-François Alcover_, Mar 09 2013