  We formulate the ferromagnetic Ising model on a two-dimensional sphere using
the Delaunay triangulation of the Fibonacci covering. The Fibonacci approach
generates a uniform isotropic covering of the sphere with approximately
equal-area triangles, thus potentially supporting a smooth thermodynamic limit.
In the absence of a magnetic field, the model exhibits a spontaneous
magnetization phase transition at a critical temperature that depends on the
connectivity properties of the underlying lattice. While in the standard
triangular lattice, every site is connected to 6 neighboring sites, the
triangulated Fibonacci lattice of the curved surface contains a substantial
density of the 5- and 7-vertices. As the number of sites in the Fibonacci
sphere increases, the triangular cover of the sphere experiences a series of
singular transitions that reflect a sudden change in its connectivity
properties. These changes substantially influence the statistical features of
the system leading to a series of first-order-like discontinuities as the
radius of the sphere increases. We found that the Ising model on a uniform,
Fibonacci-triangulated sphere in a large-radius limit possesses the phase
transition at the critical temperature $T_c \simeq 3.33(3) J$, which is
slightly lower than the thermodynamic result for an equilaterally triangulated
planar lattice. This mismatch is a memory effect: the planar Fibonacci lattice
remembers its origin from the curved space.
