In 2005, so-called grid cells have been discovered in the rodent brain. With their activity, grid cells express a remarkably regular metric tessellation of a flat surface - a triangular grid. Their spiking activity is concentrated when the animal is at the nodes of an imaginary grid, different from neuron to neuron. Later found also in crawling bats, it is not clear yet whether in flying bats grid units will provide a regular 3D tiling - such as a face-cubic-centered or hexagonal-close-packed arrangement. Most computational models of this phenomenon are based on various forms of wiring instructions, whereas we have been analyzing a model that demonstrates how grid units can spontaneously self-organize during animal development. Our model predicts that if rats are raised in a cage with non-flat geometry, such as a ball or a hyperbolic surface of negative curvature, grid units will express a non-Euclidean regular tiling, with firing rate peaks that have e.g. 5 or 7 nearest neighbors instead of the Euclidean 6 so far observed with flat cages. Experiments are underway to test this prediction.