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    Rectangular Equation

    Result

    golden rectangle | -a/2<=x<=a/2 and -(a ϕ)/2<=y<=(a ϕ)/2 half rectangle | -a/2<=x<=a/2 and 0<=y<=b/2 half square | -a/2<=x<=a/2 and 0<=y<=a/2 quarter rectangle | 0<=x<=a/2 and 0<=y<=b/2 quarter square | 0<=x<=a/2 and 0<=y<=a/2 rectangle | abs(x)<=a/2 and abs(y)<=b/2 square | -a/2<=x<=a/2 and -a/2<=y<=a/2

    Definitions

    Definitions Golden rectangle

    Definitions Half rectangle

    Definitions Half square

    Lamina properties

    golden rectangle | (a/2, -(ϕ a)/2) | (a/2, (ϕ a)/2) | (-a/2, (ϕ a)/2) | (-a/2, -(ϕ a)/2) half rectangle | (-a/2, 0) | (a/2, 0) | (a/2, b/2) | (-a/2, b/2) half square | (-a/2, 0) | (a/2, 0) | (a/2, a/2) | (-a/2, a/2) quarter rectangle | (0, 0) | (a/2, 0) | (a/2, b/2) | (0, b/2) quarter square | (0, 0) | (a/2, 0) | (a/2, a/2) | (0, a/2) rectangle | (-a/2, -b/2) | (a/2, -b/2) | (a/2, b/2) | (-a/2, b/2) square | (-a/2, -a/2) | (a/2, -a/2) | (a/2, a/2) | (-a/2, a/2)

    golden rectangle | 4 half rectangle | 4 half square | 4 quarter rectangle | 4 quarter square | 4 rectangle | 4 square | 4

    golden rectangle | a>0 half rectangle | a>0 and b>0 half square | a>0 quarter rectangle | a>0 and b>0 quarter square | a>0 rectangle | a>0 and b>0 square | a>0

    golden rectangle | a sqrt(ϕ^2 + 1) | a sqrt(ϕ^2 + 1) rectangle | sqrt(a^2 + b^2) | sqrt(a^2 + b^2) square | sqrt(2) a | sqrt(2) a

    square | r = a/2

    square | h = 1/2 (sqrt(2) - 1) a

    golden rectangle | a ϕ half rectangle | b/2 half square | a/2 quarter rectangle | b/2 quarter square | a/2 rectangle | b square | a

    golden rectangle | A = a^2 ϕ half rectangle | A = (a b)/2 half square | A = a^2/2 quarter rectangle | A = (a b)/4 quarter square | A = a^2/4 rectangle | A = a b square | A = a^2

    golden rectangle | x^_ = (0, 0) half rectangle | x^_ = (0, b/4) half square | x^_ = (0, a/4) quarter rectangle | x^_ = (a/4, b/4) quarter square | x^_ = (a/4, a/4) rectangle | x^_ = (0, 0) square | x^_ = (0, 0)

    Mechanical properties

    golden rectangle | J_x invisible comma x = (a^4 ϕ^3)/12 half rectangle | J_x invisible comma x = (a b^3)/24 half square | J_x invisible comma x = a^4/24 quarter rectangle | J_x invisible comma x = (a b^3)/48 quarter square | J_x invisible comma x = a^4/48 rectangle | J_x invisible comma x = (a b^3)/12 square | J_x invisible comma x = a^4/12

    golden rectangle | J_y invisible comma y = (a^4 ϕ)/12 half rectangle | J_y invisible comma y = (a^3 b)/24 half square | J_y invisible comma y = a^4/24 quarter rectangle | J_y invisible comma y = (a^3 b)/48 quarter square | J_y invisible comma y = a^4/48 rectangle | J_y invisible comma y = (a^3 b)/12 square | J_y invisible comma y = a^4/12

    golden rectangle | J_zz = 1/12 a^4 ϕ (ϕ^2 + 1) half rectangle | J_zz = 1/24 a b (a^2 + b^2) half square | J_zz = a^4/12 quarter rectangle | J_zz = 1/48 a b (a^2 + b^2) quarter square | J_zz = a^4/24 rectangle | J_zz = 1/12 a b (a^2 + b^2) square | J_zz = a^4/6

    golden rectangle | J_x invisible comma y = 0 half rectangle | J_x invisible comma y = 0 half square | J_x invisible comma y = 0 quarter rectangle | J_x invisible comma y = -1/64 a^2 b^2 quarter square | J_x invisible comma y = -a^4/64 rectangle | J_x invisible comma y = 0 square | J_x invisible comma y = 0

    golden rectangle | r_x = 1/2 sqrt(1/6 (2 + sqrt(5))) a r_y = a/(2 sqrt(3)) half rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) half square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3)) quarter rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) quarter square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3)) rectangle | r_x = b/(2 sqrt(3)) r_y = a/(2 sqrt(3)) square | r_x = a/(2 sqrt(3)) r_y = a/(2 sqrt(3))

    golden rectangle | K = 1/3 a^4 ϕ (1 - (192 sum_(n=1)^∞ tanh(1/2 π (2 n - 1) ϕ)/(2 n - 1)^5)/(π^5 ϕ)) rectangle | K = 1/3 a^3 b (1 - (192 a sum_(n=1)^∞ tanh((π b (2 n - 1))/(2 a))/(2 n - 1)^5)/(π^5 b)) square | K = 1/3 a^4 (1 - (192 sum_(n=1)^∞ tanh(1/2 π (2 n - 1))/(2 n - 1)^5)/π^5)

    Distance properties

    golden rectangle | a ϕ | a | a ϕ | a half rectangle | a | b/2 | a | b/2 half square | a | a/2 | a | a/2 quarter rectangle | a/2 | b/2 | a/2 | b/2 quarter square | a/2 | a/2 | a/2 | a/2 rectangle | a | b | a | b square | a | a | a | a

    golden rectangle | p = 2 a (ϕ + 1) half rectangle | p = 2 a + b half square | p = 3 a quarter rectangle | p = a + b quarter square | p = 2 a rectangle | p = 2 (a + b) square | p = 4 a

    quarter square | r = a/4 square | r = a/2

    golden rectangle | R = 1/2 a sqrt(ϕ^2 + 1) half rectangle | R = 1/4 sqrt(4 a^2 + b^2) half square | R = (sqrt(5) a)/4 quarter rectangle | R = 1/4 sqrt(a^2 + b^2) quarter square | R = a/(2 sqrt(2)) rectangle | R = 1/2 sqrt(a^2 + b^2) square | R = a/sqrt(2)

    golden rectangle | a sqrt(ϕ^2 + 1) half rectangle | (sqrt(5) a)/2 half square | (sqrt(5) a)/2 quarter rectangle | 1/2 sqrt(a^2 + b^2) quarter square | a/sqrt(2) rectangle | sqrt(a^2 + b^2) square | sqrt(2) a

    golden rectangle | χ = 1 half rectangle | χ = 1 half square | χ = 1 quarter rectangle | χ = 1 quarter square | χ = 1 rectangle | χ = 1 square | χ = 1

    square | s^_ = 1/15 a (2 + sqrt(2) + 5 sinh^(-1)(1))

    square | A^_ = (11 a^2)/144

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