Truncated Geometry on the Circle

In this letter we prove that the pure state space on the $n\times n$ complex Toeplitz matrices converges in Gromov-Hausdorff sense to the state space on $C(S^1)$ as $n$ grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on $S^1$ with density functions $c{\prod }_{j=1}^n(1-\cos (t-{\theta }_j))$ is dense in the set of all positive Borel measures on $S^1$ in the weak${}^{\ast }$ topology.