  For the heat equation on a bounded subdomain $\Omega$ of $\mathbb{R}^d$, we
investigate the optimal shape and location of the observation domain in
observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat
packets allows us to remove the randomisation procedure and assumptions on the
geometry of $\Omega$ in previous works. The explicit nature of the heat packets
gives new information about the observability constant in the inverse problem.
