  It has been argued that the existence of a zero-point length is the hallmark
of quantum gravity. In this letter we suggest a thermal mechanism whereby this
quantum of length arises in flat, Euclidean spacetime $\mathbb{R}^d$. For this
we consider the infinite sequence of all flat, Euclidean spacetimes
$\mathbb{R}^{d'}$ with $d'\geq d$, and postulate a probability distribution for
each $d'$ to occur. The distribution considered is that of a canonical ensemble
at temperature $T$, the energy levels those of a 1-dimensional harmonic
oscillator. Since both the harmonic energy levels and the spacetime dimensions
are evenly spaced, one can identify the canonical distribution of
harmonic-oscillator eigenvalues with that of dimensions $d'$. The state
describing this statistical ensemble has a mean square deviation in the
position operator, that can be interpreted as a quantum of length. Thus placing
an oscillator in thermal equilibrium with a bath provides a thermal mechanism
whereby a zero-point length is generated. The quantum-gravitational
implications of this construction are then discussed. In particular, a model is
presented that realises a conjectured duality between a weakly gravitational,
strongly quantum system and a weakly quantum, strongly gravitational system.
