Nex7 (2,1) - Parallel Calc
Calculation parallel struts
Direction vectors of Red- and Green-strut
v1 = (cos(α),sin(α),0)
v2 = M5 v1
Find angle α such that z(v2)=0
z(v2) = (ϕ-1)/2*cos(α) - 1/2*sin(α) = 0
(ϕ-1)*cos(α) = sin(α)
tan(α) = ϕ-1
α = arctan(ϕ-1)
cos(α) = cos(arctan(1/ϕ))
= ϕ/sqrt(2+ϕ)
sin(α) = sin(arctan(1/ϕ))
= 1/sqrt(2+ϕ)
Calculate intersection of strut-planes
Normal strut1 = (1,-ϕ,0)
Rotate 72° around P5Axis:
M5 = {{1,ϕ,ϕ-1}, {-ϕ,ϕ-1,1}, {ϕ-1,-1,ϕ}} / 2
Normal strut2 = M5(-1,ϕ,0)
= (-1+ϕ^2,ϕ+ϕ^2-ϕ,1-ϕ-ϕ)/2
= (ϕ,1+ϕ,1-2ϕ)/2
plane1= ( 1*x - ϕ *y = 0)
plane2= ( ϕ*x +(1+ϕ)*y + (1-2ϕ)*z = 0)
v12 = ( 1,-ϕ,0) ⊗ (ϕ,1+ϕ,1-2ϕ)
= ( ϕ+2 , 2ϕ-1, 2ϕ+2)
|v12| = sqrt((ϕ+2)^2 + (2ϕ-1)^2 + (2ϕ+2)^2 )
= sqrt( 17ϕ+18 )
Normalized vectors of red-strut endpoints
S1 = ( ϕ+2, 2ϕ-1, 2ϕ+2) / sqrt(17ϕ+18)
S2 = (-ϕ-2, 1-2ϕ, 2ϕ+2) / sqrt(17ϕ+18)
Setup Script
ϕ = (sqrt(5)+1)/2
O = (0, 0, 0)
S1 = ( ϕ+2, 2ϕ-1, 2ϕ+2)/sqrt(17ϕ+18)
S2 = (-ϕ-2, 1-2ϕ, 2ϕ+2)/sqrt(17ϕ+18)
cyl = Cylinder(S1, S2, 0.075)
L1cyl = Zip(Rotate(cyl, k*2pi/5, Line(O,(ϕ,1,0))),k,0..5)
L2cyl = Rotate(L1cyl, 2pi/5, Line(O, (1,0,ϕ)) )
L3cyl = Rotate(L2cyl, 2pi/5, Line(O, (1,0,ϕ)) )
L4cyl = Rotate(L3cyl, 2pi/5, Line(O, (1,0,ϕ)) )
L5cyl = Rotate(L4cyl, 2pi/5, Line(O, (1,0,ϕ)) )
L6cyl = Rotate(L5cyl, 2pi/3, Line(O,(ϕ-1,ϕ,0)))