  Relational particle mechanics models bolster the relational side of the
absolute versus relational motion debate, and are additionally toy models for
the dynamical formulation of General Relativity and its Problem of Time. They
cover two aspects that the more commonly studied minisuperspace General
Relativity 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 Problem of Time facets.
  I previously solved relational triangleland classically, quantum mechanically
and as regards a local resolution of the Problem of Time. 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 Problem of Time
subtleties. The shape space now being CP^2 with isometry group SU(3)/Z_3, 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.
