A186282 Decimal expansion of non-right-angle, in degrees, of unique 14th class of convex pentagonal tiling.

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, 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, 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

Offset

3,4

Comments

From Jean-François Alcover, Mar 11 2013: (Start)

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

A = Pi/2 = 90 deg,

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

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

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

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

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)

References

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

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

Links

Table of n, a(n) for n=3..102.

Jean-François Alcover, Pentagonal tile

P. L. Bowers and K. Stephenson, A 'Regular' Pentagonal Tiling of the Plane, Conformal Geom. Dyn. 1, 58-68, 1997.

Ed Pegg, Jr., The 14 Different Types of Pentagons that Tile the Plane"

M. ten Have, Pentagons and the Golden Section

Eric W. Weisstein, Pentagon Tiling

Formula

Solution to 3*cos(x/2) + 2*cos(x) = 1. [Jean-François Alcover, Mar 09 2013]

Example

110.676672523... degrees.

Mathematica

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

Prog

(PARI) solve(x=Pi, Pi/2, 3*cos(x/2) + 2*cos(x) - 1)*180/Pi \\ Michel Marcus, Aug 19 2019

Crossrefs

Sequence in context: A355922 A261024 A343436 * A171909 A199079 A010723

Adjacent sequences: A186279 A186280 A186281 * A186283 A186284 A186285

Keyword

nonn,cons

Author

Jonathan Vos Post, Feb 16 2011

Extensions

More terms from Jean-François Alcover, Mar 09 2013

Status

approved