  We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group
$\mathbf H_4$. They are identical with the Markoff and Lagrange spectra of the
unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff
spectrum of index 2 sublattices by Vulakh and the Markoff spectrum of 2-minimal
forms or $C$-minimal forms by Schmidt, who characterized the spectrum up to the
first accumulation point. After the first accumulation point, we show that both
spectra have positive Hausdorff dimension. Then we find gaps and show the
existence of Hall's ray.
