Relational particle mechanics (RPM) models bolster the relational side of the absolute versus relational motion debate. They are additionally toy models for the dynamical formulation of General Relativity (GR) and its Problem of Time (PoT). They cover two aspects that the more commonly studied minisuperspace GR models do not: 1) by having a nontrivial notion of structure and thus of cosmological structure formation and of localized records. 2) They have linear as well as quadratic constraints, which is crucial as regards modelling many PoT facets.I previously solved relational triangleland classically, quantum mechanically and as regards a local resolution of the PoT. This rested on triangleland's shape space being S 2 with isometry group SO(3), allowing for use of widely-known Geometry, Methods and Atomic/Molecular Physics analogies. I now extend this work to the relational quadrilateral, which is far more typical of the general N -a-gon, represents a 'diagonal to nondiagonal Bianchi IX minisuperspace' step-up in complexity, and encodes further PoT subtleties. The shape space now being CP 2 with isometry group SU (3)/Z3, I now need to draw on Geometry, Shape Statistics and Particle Physics to solve this model; this is therefore an interdisciplinary paper. This Paper treats quadrilateralland at the classical level, and then Paper II provides a quantum treatment.

This paper provides the quantum treatment of the relational quadrilateral. The underlying reduced configuration spaces are CP 2 and the cone over this, C(CP 2 ). We consider exact free and isotropic HO potential cases and perturbations about these. Moreover, our purely relational kinematical quantization is distinct from the usual one for CP 2 , which turns out to carry absolutist connotations instead. Thus this paper is the first to note absolute-versus-relational motion distinctions at the kinematical rather than dynamical level. It is also an example of value to the discussion of kinematical quantization along the lines of Isham 1984. This treatment of the relational quadrilateral is the first relational QM with very new mathematics for a finite QM model. It is far more typical of the general quantum relational N -a-gon than the previously-studied case of the relational triangle. We consider useful integrals as regards perturbation theory and the peaking interpretation of quantum cosmology. We subsequently consider problem of time applications of this: quantum Kuchař beables, the Machian version of the semiclassical approach and the timeless naïve Schrödinger interpretation. These go toward extending the combined Machian semiclassical-Histories-Timeless Approach of [1] to the case of the quadrilateral, which will be treated in subsequent papers.