  Except for crystalline or random structures, an agreed definition of
complexity for intermediate and hence interesting cases does not exist. We fill
this gap with a notion of complexity that characterises shapes formed by any
finite number of particles greater than or equal to the three needed to define
triangle shapes. The resulting shape complexity is a simple scale-invariant
quantity that measures the extent to which a collection of particles has a
uniform or clustered distribution. As a positive-definite number with an
absolute minimum realised on the most uniform distribution the particles can
have, it not only characterises all physical structures from crystals to the
most complex that can exist but also determines for them a measure that makes
richly structured shapes more probable than bland ones. Strikingly, the
criterion employed to define the shape complexity forces it to be the product
of the two functions that define Newtonian universal gravitation. This suggests
both the form and solutions the law of a universe of such particles should have
and leads to a theory that not only determines the complexity and probability
of any individual shape but also its creation from the maximally uniform shape.
It does this moreover in a manner which makes it probable that the cosmological
principle, according to which on a sufficiently large scale the universe should
have the same appearance everywhere, holds. Our theory relies on universal
group-theoretical principles that may allow generalisation to include all
forces and general relativity.
