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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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