  In scaled relational particle mechanics, only relative times, relative angles
and relative separations are meaningful. It arose in the study of the absolute
versus relative motion debate. It has then turned out to be a useful toy model
of classical and quantum general relativity, such as for investigating
conceptual strategies for the problem of time. This paper studies the
3-particle 2-d scaled relational particle model, for which the configurations
are scaled triangles. The configuration space for these is R^3 with a
conformally flat metric thereupon (it is the cone over the corresponding shape
space S^2. I use multiple harmonic oscillator type potentials and other
potentials suggested by analogy with cosmology, and solve for some of these by
using a partial analogy with the treatment of the atom in spherical and
parabolic coordinates. Spherical coordinates are here the total moment of
inertia $I$ for radius and two pure-shape coordinates. These are \Theta, a
function of the ratio of the two relative separations of subsystems, and \Phi,
the relative angle between the two subsystems. Parabolic coordinates are \Phi
again and twice the partial moments of inertia of each subsystem. I interpret
these solutions using 1) a `Bohr moment of inertia' for the model universe
(playing the role of the scalefactor). 2) Expectations and spreads of sizes and
shapes. 3) Superimposing the probability density function on the labelled
tessellation of the configuration space that encodes meaningful subregions such
as collinear configurations, equilateral triangles and isosceles triangles.
Applications include hidden time, emergent semiclassical time, timeless and
histories theory problem of time strategies, and comparing reduced and Dirac
methods of quantization.
