  In the tetrad formulation of gravity, the so-called simplicity constraints
play a central role. They appear in the Hamiltonian analysis of the theory, and
in the Lagrangian path integral when constructing the gravity partition
function from topological BF theory. We develop here a systematic analysis of
the corner symplectic structure encoding the symmetry algebra of gravity, and
perform a thorough analysis of the simplicity constraints. Starting from a
precursor phase space with Poincar\'e and Heisenberg symmetry, we obtain the
corner phase space of BF theory by imposing kinematical constraints. This
amounts to fixing the Heisenberg frame with a choice of position and spin
operators. The simplicity constraints then further reduce the Poincar\'e
symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a
particle-like description of (quantum) geometry: The internal normal plays the
role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the
flux that of a relativistic position, and the frame that of a spin harmonic
oscillator. Moreover, we show that the corner area element corresponds to the
Poincar\'e spin Casimir. We achieve this central result by properly splitting,
in the continuum, the corner simplicity constraints into first and second class
parts. We construct the complete set of Dirac observables, which includes the
generators of the local $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincar\'e,
and the components of the tangential corner metric satisfying an
$\mathfrak{sl}(2,\mathbb{R})$ algebra. We then present a preliminary analysis
of the covariant and continuous irreducible representations of the
infinite-dimensional corner algebra. Moreover, as an alternative path to
quantization, we also introduce a regularization of the corner algebra and
interpret this discrete setting in terms of an extended notion of twisted
geometries.
