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The program of understanding Shape Theory layer by layer topologically andgeometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$.Topological shape space graphs are far more complex here, whereas metric shapespaces are (pieces of) spheres which admit an intricate shape-theoreticallysignificant tessellation. Metric shapes covers a far wider range of notions ofinhomogeneity: collisions, symmetric states, mergers and uniform states are alldistinctly realized in this model. We furthermore provide quantifiers for theextent to which various ways which configurations maximally and minimallyrealize these. Some of the uniform states additionally form cusps and highercatastrophes in the indistinguishable-particle and Leibniz shape spaces. Wealso provide shape-theoretically significant notions of centre for theindistinguishable-particle and Leibniz shape spaces. 4 points in 1-$d$constitutes a useful and already highly nontrivial model of inhomogeneity andof uniformity -- both topics of cosmological interest -- and also of backgroundindependence: of interest in the foundations of physics and in quantum gravity.We finally give the automorphism groups of the topological shape space graphsand the metric shape space (pieces of) manifolds, which is a crucialpreliminary toward quantizing the indistinguishable-particle and Leibnizversions of the model.

The Smallest Shape Spaces. II. 4 Points in 1-d Suffices to have a Complex Background-Independent Theory of Inhomogeneity