Two-dimensional examples are helpful in order to get some understanding about the origin of the competition between local rules and geometry in the large. Consider first an arrangement of identical discs (a model for a hypothetical two-dimensional metal) on a plane; we suppose that the interaction between discs is isotropic and locally tends to arrange the disks in the densest way as possible. The best arrangement for three disks is trivially an equilateral triangle with the disk centers located at the triangle vertices. The study of the long range structure can therefore be reduced to that of plane tilings with equilateral triangles. A well known solution is provided by the triangular tiling with a total compatibility between the local and global rules: the system is said to be "unfrustrated".
But now, the interaction energy is supposed to be at a minimum when atoms sit on the vertices of a regular pentagon. Trying to propagate in the long range a packing of these pentagons sharing edges (atomic bonds) and vertices (atoms) is impossible. This is due to the impossibility of tiling a plane with regular pentagons, simply because the pentagon vertex angle does not divide 2. Three such pentagons can easily fit at a common vertex, but a gap remains between two edges. It is this kind of discrepancy which is called "geometric frustration". There is one way to overcome this difficulty. Let the surface to be tiled be free of any presupposed topology, and let us build the tiling with a strict application of the local interaction rule. In this simple example, we observe that the surface inherits the topology of a sphere and so receives a curvature. The final structure, here a pentagonal dodecahedron, allows for a perfect propagation of the pentagonal order. It is called an "ideal" (defect-free) model for the considered structure.