[1] The surface's Fuchsian group can be constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups.
in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7.
It is possible to realize the resulting triangulated surface as a non-convex polyhedron without self-intersections.
[2] This surface was originally discovered by Robert Fricke (1899), but named after Alexander Murray Macbeath due to his later independent rediscovery of the same curve.
[3] Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by Serre in a 24.vii.1990 letter to Abhyankar".