the north polar circle? Normal growth (steady state) clings to the sphere. But in the last six months of its life, the agave first switches to the cone tangent to the sphere on the north polar circle, through a complete grain boundary $[5,5,5,5,5,6,6,6,7,7,7,7,7]$ (a slight increase of the rate of growth replaces the five innermost hexagons by heptagons and the gaussian curvature falls from $+1/R^2$ to zero) (Fig.\[f6\]b). Then, immediately, the agave switches to a less open cone, the mast in phyllotaxis $(5,3,2)$, through another smaller grain boundary. The mast terminates as seeds-loaded branches arranged on an open, uncapped cylinder in the $(3,2,1)$ phyllotaxis, through the grain boundary \[5,5,6,7,7\] (Fig.\[f7\]). The cylinder in the $(3,2,1)$ phyllotaxis is indeed the topological endgame, and, unlike a cone, it can be extended without end. If a “Big Bang” had occurred on the south polar cap, growth of a conical mast and a cylinder are the simplest means of avoiding a “Big Crunch” on the north polar cap.

Projection on the tangent cone (as in the mast of the agave) is different from the projection on the sphere because the generative spiral on the cone cannot have constant pitch $h(s)$ and keep a constant 