File:GeometricAnimation.gif

English: Distributivity-based asymptotic solutions (using Σ = 1+mΣ) for the geometric series Σ_imⁱ, summed over i=0,1,2,...,∞, create cyclic triplets along the real manifold as illustrated here in this inverse stereographic projection plot. Such cycles can be used to illustrate the branch-cut dependence, and cyclic nature, of ordering on the real number line.

If the opaque black arrow represents the transition from m to Σ_imⁱ, the intermediate-darkness arrow represents the transition from Σ_imⁱ to Σ_j(Σ_imⁱ)^j, and the most translucent arrow represents the transition from Σ_j(Σ_imⁱ)^j to Σ_k(Σ_j(Σ_imⁱ)^j)^k ~ m. Note that only m (the black dot) moves around the circle at a constant speed in the animation.

Solid arrows along triangle sides represent convergent series solutions, while dashed arrows represent the non-convergent series solutions. To provide scale, integer points in green around the circle go from -10 to +10.