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%I #27 Jun 25 2024 07:24:50
%S 4,5,9,1,3,3,7,2,3,3,1,0,2,0,7,5,3,9,4,6,7,5,1,1,4,6,3,0,0,1,6,5,3,9,
%T 8,6,5,1,3,3,9,0,8,8,2,1,9,9,5,3,4,4,6,5,4,5,4,6,4,2,8,8,5,6,8,7,0,9,
%U 4,4,9,4,5,5,7,4,3,2,4,5,8,0,0,7,1,7,1,7,7,3,6,4,4,4,9,1,7,9,6,5,1,7,6,3,1,3,3,0
%N Decimal expansion of the real root of the quintic equation x^5 + 3*x^4 + 4*x^3 + x -1 = 0.
%C The root appears in the problem of minimizing the area of self-intersection of a folded rectangle. A rectangle with sides a, b (a<b) is folded along the line that passes through the center of the rectangle in order to get the minimum area of crossing intersections: a unique rectangle exists for two solutions with equal area but different shapes - triangle and pentagon.
%C The unique ratio of sides a/b=T=0.81502370129163... is derived based on the real root of the quintic. If a/b<T ('long' rectangle) the angle to fold is Pi/4. If a/b=1 (square) the angle is Pi/8.
%H Mikhail Gaichenkov, <a href="https://math.stackexchange.com/questions/4734823/quintic-equation-with-integer-coefficients">Quintic equation with integer coefficients</a>, Math Stackexchange, 2023.
%H Mikhail Gaichenkov, <a href="/A366185/a366185.pdf">Folded rectangle</a>
%H <a href="/index/Al#algebraic_05">Index entries for algebraic numbers, degree 5</a>
%e 0.45913372331020753...
%t First[RealDigits[Root[#^5 + 3*#^4 + 4*#^3 + # - 1 &, 1], 10, 100]] (* _Paolo Xausa_, Jun 25 2024 *)
%o (PARI) polrootsreal(x^5 + 3*x^4 + 4*x^3 + x-1)[1]
%K nonn,cons
%O 0,1
%A _Mikhail Gaichenkov_, Oct 03 2023