The Deligne Complex for the $B_3$ Artin Group

We show that the piecewise Euclidean Moussong metric on the Deligne complex of the Artin group of type $B_3$ is $\mathrm{C}\mathrm{A}\mathrm{T}(0)$. We do this by establishing a criteria for a complex made of $B_3$ simplices to be $\mathrm{C}\mathrm{A}\mathrm{T}(1)$ in terms of embedded edge paths, which in particular applies to the spherical Deligne complex of type $B_3$. This provides one more step to showing that the Moussong metric is $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ for any 3-dimensional Artin group.