Lie Subgroups of the 2D Torus Group

This Demonstration draws the one-dimensional Lie group

(α)={(exp(iθ),exp(iαθ))|θ∈}

plotted as a subgroup embedded (or immersed) in the two-dimensional torus group

2



={(exp(iθ),exp(iθ))|θ∈}

. The slope

α

is set by the direction of the velocity, and the subgroup is plotted from

θ=0

up to a maximum value set by the "length" value; this latter is the product of the "multiplier" value and the length of the velocity, so you can watch the thread grow by putting the velocity near

(0,0)

(left-bottom corner) and dragging it to longer and longer lengths.