  The program of understanding Shape Theory layer by layer topologically and
geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$.
Topological shape space graphs are far more complex here, whereas metric shape
spaces are (pieces of) spheres which admit an intricate shape-theoretically
significant tessellation. Metric shapes covers a far wider range of notions of
inhomogeneity: collisions, symmetric states, mergers and uniform states are all
distinctly realized in this model. We furthermore provide quantifiers for the
extent to which various ways which configurations maximally and minimally
realize these. Some of the uniform states additionally form cusps and higher
catastrophes in the indistinguishable-particle and Leibniz shape spaces. We
also provide shape-theoretically significant notions of centre for the
indistinguishable-particle and Leibniz shape spaces. 4 points in 1-$d$
constitutes a useful and already highly nontrivial model of inhomogeneity and
of uniformity -- both topics of cosmological interest -- and also of background
independence: of interest in the foundations of physics and in quantum gravity.
We finally give the automorphism groups of the topological shape space graphs
and the metric shape space (pieces of) manifolds, which is a crucial
preliminary toward quantizing the indistinguishable-particle and Leibniz
versions of the model.
