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Over cosmic distances, astrophysical neutrino oscillations average out to aclassical flavor propagation matrix $\mathscr{P}$. Thus, flavor ratios injectedat the cosmic source $W_e,W_\mu,W_\tau$ evolve to flavor ratios at Earthlydetectors $w_e,w_\mu,w_\tau$ according to $\vec{w}=\mathscr{P} \vec{W}$. Theunitary constraint reduces the Euclidean octant to a "flavor triangle". Weprove a theorem that the area of the Earthly flavor triangle is proportional toDet$(\mathscr{P})$. One more constraint would further reduce the dimensionalityof the flavor triangle at Earth (two) to a line (one). We discuss fourmotivated such constraints. The first is the possibility of a vanishingdeterminant for $\mathscr{P}$. We give a formula for a unique$\delta(\theta_{ij}$'s) that yields the vanishing determinant. Next we considerthe thinness of the Earthly flavor triangle. We relate this thinness to thesmall deviations of the two angles $\theta_{32}$ and $\theta_{13}$ from maximalmixing and zero, respectively. Then we consider the confusion resulting fromthe tau neutrino decay topologies, which are showers at low energy,"double-bang" showers in the PeV range, and a mixture of showers and tracks ateven higher energies. We examine the simple low-energy regime, where there arejust two topologies, $w_{\rm shower}=w_e+w_\tau$ and $w_{\rm track}=w_\mu$. Weapply the statistical uncertainty to be expected from IceCube to this model.Finally, we consider ramifications of the expected lack of $\nu_\tau$ injectionat cosmic sources. In particular, this constraint reduces the Earthly triangleto a boundary line of the triangle. Some tests of this "no $\nu_\tau$injection" hypothesis are given.

Aspects of the Flavor Triangle for Cosmic Neutrino Propagation