  In this PhD thesis, we develop a new approach to classical gravity starting
from Mach's principles and the idea that the local shape of spatial
configurations is fundamental. This new theory, "shape dynamics", is equivalent
to general relativity but differs in an important respect: shape dynamics is a
theory of dynamic conformal 3-geometry, not a theory of spacetime. Equivalence
is achieved by trading foliation invariance for local conformal invariance (up
to a global scale). After the trading, what is left is a gauge theory invariant
under 3d diffeomorphisms and conformal transformations that preserve the volume
of space. The local canonical constraints are linear and the constraint algebra
closes with structure constants. Shape dynamics, thus, provides a novel new
starting point for quantum gravity.
  The procedure for the trading of symmetries was inspired by a technique
called "best matching". We explain best matching and its relation to Mach's
principles. The key features of best matching are illustrated through finite
dimensional toy models. A general picture is then established where relational
theories are treated as gauge theories on configuration space. Shape dynamics
is then constructed by applying best matching to conformal geometry. We then
study shape dynamics in more detail by computing its Hamiltonian and
Hamilton-Jacobi functional perturbatively.
  This thesis is intended as a pedagogical but complete introduction to shape
dynamics and the Machian ideas that led to its discovery. The reader is
encouraged to start with the introduction, which gives a conceptual outline and
links to the relevant sections in the text for a more rigorous exposition. When
full rigor is lacking, references to the literature are given. It is hoped that
this thesis may provide a starting point for anyone interested in learning
about shape dynamics.
