The Cannon-Thurston map \(\phi:\overline{\mathbb{R}}\longrightarrow \overline{\mathbb{C}}\) associated to the Gieseking manifold (a complete, finite-volume, non-orientable, hyperbolic three-manifold) determines a set \(E_{\overline{\mathbb{C}}}\) of those points of \(\overline{\mathbb{C}}\) which are images of at least two points of \(\overline{\mathbb{R}}\). \noindent The authors evaluate the Hausdorff dimension \(d\) of \(E_{\overline{\mathbb{C}}}\) in a metric \(dist_{chord}\) in \(\overline{\mathbb{C}}\) given by \[ dist_{chord}(w,z):=\frac{2|w-z|}{\sqrt{(1+|w|^2)(1+|z|^2)}} \] for complex numbers \(w\), \(z\), with the natural extension for \(\infty\). Namely, they show that \(E_{\overline{\mathbb{C}}}\) is a countable union of Jordan curves, each of Hausdorff dimension \(d\). It is proved that \(1.15